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69 results on '"Control And GEometry (CaGE )"'

1. Solving chance constrained optimal control problems in aerospace via Kernel Density Estimation

Author: Emmanuel Trélat, Achille Sassi, Max Cerf, Jean-Baptiste Caillau, Hasnaa Zidani, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Mathematics for Control, Transport and Applications ( McTAO ), Inria Sophia Antipolis - Méditerranée ( CRISAM ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Airbus Safran Launchers, Optimisation et commande ( OC ), Unité de Mathématiques Appliquées ( UMA ), École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ) -École Nationale Supérieure de Techniques Avancées ( Univ. Paris-Saclay, ENSTA ParisTech ), Control And GEometry ( CaGE ), Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria de Paris, Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), ANR : IDEX PARIS SACLAY,ANR-11-IDEX-0003-02 IDEX Paris-Saclay, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Mathematics for Control, Transport and Applications (McTAO), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Optimisation et commande (OC), Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU), ANR-11-IDEX-0003,IPS,Idex Paris-Saclay(2011), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS), ANR: IDEX PARIS SACLAY,ANR-11-IDEX-0003-02 IDEX Paris-Saclay, Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB)
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], Mathematical optimization, Control and Optimization, chance constrained optimization, Kernel density estimation, 0211 other engineering and technologies, Probability density function, 02 engineering and technology, 01 natural sciences, Kernel Density Estimation, 010104 statistics & probability, 0101 mathematics, Mathematics, 021103 operations research, Applied Mathematics, Constrained optimization, Trajectory optimization, stochastic optimization, Optimal control, Distribution (mathematics), Aerospace engineering, Control and Systems Engineering, Stochastic optimization, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Random variable, Software
Abstract: International audience; The goal of this paper is to show how non-parametric statistics can be used to solve some chance constrained optimization and optimal control problems. We use the Kernel Density Estimation method to approximate the probability density function of a random variable with unknown distribution , from a relatively small sample. We then show how this technique can be applied and implemented for a class of problems including the God-dard problem and the trajectory optimization of an Ariane 5-like launcher.
Published: 2017

2. Highly corrupted image inpainting through hypoelliptic diffusion

Author: Roman Chertovskih, Jean-Paul Gauthier, Ugo Boscain, A. O. Remizov, Dario Prandi, Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Universidade do Porto, Samara State University (SSAU), Laboratoire d'Informatique et Systèmes (LIS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), This work was supported by the ERC POC project ARTIV1 contract number 727283, by the ANR project 'SRGI' ANR-15-CE40-0018, by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH (in a joint call with Programme Gaspard Monge en Optimisation et Recherche Opérationnelle), by the iCODE institute, research project of the Idex Paris-Saclay, by the POCI-01-0145-FEDER006933/SYSTEC project financed by ERDF and FCT through COMPETE2020., ANR-15-CE40-0018,SRGI,Géométrie sous-Riemannienne et Interactions(2015), Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU), Universidade do Porto = University of Porto, Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), Control And GEometry ( CaGE ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), University of Porto, Samara State University ( SSAU ), Laboratoire des Sciences de l'Information et des Systèmes ( LSIS ), Aix Marseille Université ( AMU ) -Université de Toulon ( UTLN ) -Arts et Métiers Paristech ENSAM Aix-en-Provence-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire des signaux et systèmes ( L2S ), Université Paris-Sud - Paris 11 ( UP11 ) -CentraleSupélec-Centre National de la Recherche Scientifique ( CNRS ), Centre de Mathématiques Appliquées - Ecole Polytechnique ( CMAP ), École polytechnique ( X ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-15-CE40-0018,SRGI,Sub-Riemannian Geometry and Interactions ( 2015 ), Faculdade de Engenharia, Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Sud - Paris 11 (UP11), and ANR-15-CE40-0018,SRGI,Sub-Riemannian Geometry and Interactions(2015)
Subjects: FOS: Computer and information sciences, Statistics and Probability, Diffusion (acoustics), Computer science, Computer Vision and Pattern Recognition (cs.CV), Inpainting, Computer Science - Computer Vision and Pattern Recognition, 02 engineering and technology, Iterative reconstruction, 01 natural sciences, Image (mathematics), Mathematics - Analysis of PDEs, [ INFO.INFO-TI ] Computer Science [cs]/Image Processing, FOS: Mathematics, 0202 electrical engineering, electronic engineering, information engineering, 0101 mathematics, Applied Mathematics, Sub-Riemannian hypoelliptic diffusion, Condensed Matter Physics, 010101 applied mathematics, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], Modeling and Simulation, Hypoelliptic operator, Image reconstruction, 020201 artificial intelligence & image processing, Geometry and Topology, Computer Vision and Pattern Recognition, Focus (optics), Algorithm, Analysis of PDEs (math.AP)
Abstract: We present a new image inpainting algorithm, the Averaging and Hypoelliptic Evolution (AHE) algorithm, inspired by the one presented in [SIAM J. Imaging Sci., vol. 7, no. 2, pp. 669--695, 2014] and based upon a semi-discrete variation of the Citti-Petitot-Sarti model of the primary visual cortex V1. The AHE algorithm is based on a suitable combination of sub-Riemannian hypoelliptic diffusion and ad-hoc local averaging techniques. In particular, we focus on reconstructing highly corrupted images (i.e. where more than the 80% of the image is missing), for which we obtain reconstructions comparable with the state-of-the-art., 15 pages, 10 figures
Published: 2015

3. Regularization of chattering phenomena via bounded variation controls

Author: Emmanuel Trélat, Marco Caponigro, Roberta Ghezzi, Benedetto Piccoli, Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Conservatoire National des Arts et Métiers, Département Ingénierie Mathématique (IMATH), Equipe M2N, Modélisation mathématique et numérique ( M2N ), Conservatoire National des Arts et Métiers [CNAM] : EA7340-Conservatoire National des Arts et Métiers [CNAM] : EA7340, Rutgers University, Department of Mathematics, Department of Mathematics - Rutgers School of Arts and Sciences, Rutgers, The State University of New Jersey [New Brunswick] ( RUTGERS ) -Rutgers, The State University of New Jersey [New Brunswick] ( RUTGERS ), Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), European Project : 264735,EC:FP7:PEOPLE,FP7-PEOPLE-2010-ITN,SADCO ( 2011 ), Department of Mathematical Sciences [Camden], Rutgers University [Camden], Control And GEometry ( CaGE ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria de Paris, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Modélisation mathématique et numérique (M2N), Conservatoire National des Arts et Métiers [CNAM] (CNAM), Rutgers, The State University of New Jersey [New Brunswick] (RU), Rutgers University System (Rutgers)-Rutgers University System (Rutgers)-Rutgers, The State University of New Jersey [New Brunswick] (RU), Rutgers University System (Rutgers)-Rutgers University System (Rutgers), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), European Project: 264735,EC:FP7:PEOPLE,FP7-PEOPLE-2010-ITN,SADCO(2011), HESAM Université - Communauté d'universités et d'établissements Hautes écoles Sorbonne Arts et métiers université (HESAM)-HESAM Université - Communauté d'universités et d'établissements Hautes écoles Sorbonne Arts et métiers université (HESAM), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Center for Computational and Integrative Biology [Camden] (CCIB), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS), Bouzat, Estelle, and Sensitivity Analysis for Deterministic Controller Design - SADCO - - EC:FP7:PEOPLE2011-01-01 - 2014-12-31 - 264735 - VALID
Subjects: [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC], 0209 industrial biotechnology, State constraints, Boundary (topology), 02 engineering and technology, Interval (mathematics), 01 natural sciences, 020901 industrial engineering & automation, Shooting method, Convergence (routing), FOS: Mathematics, Applied mathematics, Hybrid problems, 0101 mathematics, Electrical and Electronic Engineering, Mathematics - Optimization and Control, Mathematics, Total variation, 010102 general mathematics, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Optimal control, Computer Science Applications, Controllability, Control and Systems Engineering, Optimization and Control (math.OC), Chattering control, Bounded variation, Trajectory, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Fuller phenomenon
Abstract: In control theory, the term chattering is used to refer to strong oscillations of controls, such as an infinite number of switchings over a compact interval of times. In this paper we focus on three typical occurences of chattering: the Fuller phenomenon, referring to situations where an optimal control switches an infinite number of times over a compact set; the Robbins phenomenon, concerning optimal control problems with state constraints, meaning that the optimal trajectory touches the boundary of the constraint set an infinite number of times over a compact time interval; the Zeno phenomenon, referring as well to an infinite number of switchings over a compact set, for hybrid optimal control problems. From the practical point of view, when trying to compute an optimal trajectory, for instance by means of a shooting method, chattering may be a serious obstacle to convergence. In this paper we propose a general regularization procedure, by adding an appropriate penalization of the total variation. This produces a quasi-optimal control, and we prove that the family of quasi-optimal solutions converges to the optimal solution of the initial problem as the penalization tends to zero. Under additional assumptions, we also quantify the quasi-optimality property by determining a speed of convergence of the costs.
Published: 2014

4. Model Predictive Control, Cost Controllability, and Homogeneity

Author: Jean-Michel Coron, Karl Worthmann, Lars Grüne, Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Mathematisches Institut [Bayreuth], Universität Bayreuth, Technische Universität Ilmenau (TU ), ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
Subjects: 0209 industrial biotechnology, Control and Optimization, Homogeneous approximation, Applied Mathematics, Homogeneity (statistics), 010102 general mathematics, Stability guarantee, 02 engineering and technology, 01 natural sciences, Controllability, Model predictive control, Cost controllability, 020901 industrial engineering & automation, Exponential stability, Optimization and Control (math.OC), Linearization, Control theory, FOS: Mathematics, Homogeneity, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Mathematics - Optimization and Control, Closed loop, Mathematics
Abstract: International audience; We are concerned with the design of Model Predictive Control (MPC) schemes such that asymptotic stability of the resulting closed loop is guaranteed even if the linearization at the desired set point fails to be stabilizable. Therefore, we propose to construct the stage cost based on the homogeneous approximation and rigorously show that applying MPC yields an asymp-totically stable closed-loop behavior if the homogeneous approximation is asymptotically null controllable. To this end, we verify cost controllability-a condition relating the current state, the stage cost, and the growth of the value function w.r.t. time-for this class of systems in order to provide stability and performance guarantees for the proposed MPC scheme without stabilizing terminal costs or constraints.
Published: 2020

5. Adaptive control of Lipschitz time-delay systems by sigma modification with application to neuronal population dynamics

Author: Alain Destexhe, Jakub Orlowski, Antoine Chaillet, Mario Sigalotti, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), Institut des Neurosciences Paris-Saclay (NeuroPSI), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), This paper has received support from the Institute for Control and Decision of Paris Saclay (iCODE), France and Digiteo, France. AD was supported by the CNRS, France and the European Commission (Human Brain Project H2020-945539)., Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
Subjects: 0209 industrial biotechnology, Time-delay systems, Adaptive control, General Computer Science, Computer science, Mechanical Engineering, Dynamics (mechanics), Sigma, 02 engineering and technology, Dissipation, Sigma modification, Lipschitz continuity, Nonlinear system, Neuronal populations, 020901 industrial engineering & automation, Quadratic equation, Control and Systems Engineering, Control theory, Stability in the mean, 0202 electrical engineering, electronic engineering, information engineering, 020201 artificial intelligence & image processing, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Electrical and Electronic Engineering, Neuronal population
Abstract: International audience; Adaptive control using the -modification provides an easily implementable way to stabilize systems with uncertain or fluctuating parameters. Motivated by a specific application from neuroscience, we extend here this methodology to nonlinear time-delay systems ruled by globally Lipschitz dynamics. In order to make the result more handy in practice, we provide an explicit construction of a Lyapunov–Krasovskii functional (LKF) with linear bounds and strict dissipation rate based on the knowledge of an LKF with quadratic bounds and point-wise dissipation rate. When applied to a model of neuronal populations involved in Parkinson’s disease, the benefits with respect to a pure proportional stabilization scheme are discussed through numerical simulations.
Published: 2022

6. Les systèmes bilinéaires de dimension finie contrôlables de façon approchée sont contrôlables

Author: Daniele Cannarsa, Mario Sigalotti, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), ANR-17-CE40-0007,QUACO,Contrôle quantique : systèmes d'EDPs et applications à l'IRM(2017), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: Bilinear systems, Controllability, 0209 industrial biotechnology, General Computer Science, 02 engineering and technology, 01 natural sciences, Foliations, 020901 industrial engineering & automation, FOS: Mathematics, 0101 mathematics, Electrical and Electronic Engineering, Bilinear control systems, Mathematics - Optimization and Control, Mathematics, Mechanical Engineering, 010102 general mathematics, Mathematical analysis, 93B03, 53C12, 37C86, Codimension, Radial direction, Optimization and Control (math.OC), Control and Systems Engineering, Homogeneous, Bilinear control, Transversal (combinatorics), Foliation (geology), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: We show that a bilinear control system is approximately controllable if and only if it is controllable in $\mathbb{R}^{n}\setminus\{0\}$. We approach this problem by looking at the foliation made by the orbits of the system, and by showing that there does not exist a codimension-one foliation in $\mathbb{R}^{n}\setminus\{0\}$ with dense leaves that are everywhere transversal to the radial direction. The proposed geometric approach allows to extend the results to homogeneous systems that are angularly controllable., Comment: 9 pages, 2 figures
Published: 2021

7. An auditory cortex model for sound processing

Author: Ludovic Sacchelli, Rand Asswad, Dario Prandi, Giuseppina Turco, Ugo Boscain, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Laboratoire de Linguistique Formelle (LLF UMR7110), Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Laboratoire d'automatique, de génie des procédés et de génie pharmaceutique (LAGEPP), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-École Supérieure Chimie Physique Électronique de Lyon-Centre National de la Recherche Scientifique (CNRS), Nielsen F., Barbaresco F., Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire de Linguistique Formelle (LLF - UMR7110), Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Université de Lyon-Université de Lyon-École Supérieure de Chimie Physique Électronique de Lyon (CPE)-Centre National de la Recherche Scientifique (CNRS), ANR-20-CE48-0003,RUBIN-VASE,Modélisation sans redondance de la détection visuelle et auditive inspirée de la neurobiologie(2020), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
Subjects: Signal Processing (eess.SP), Kolmogorov operator, Computer science, Speech recognition, 02 engineering and technology, computer.software_genre, Auditory cortex, Heisenberg group, 03 medical and health sciences, Mathematics - Analysis of PDEs, 0302 clinical medicine, FOS: Mathematics, FOS: Electrical engineering, electronic engineering, information engineering, 0202 electrical engineering, electronic engineering, information engineering, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Electrical Engineering and Systems Science - Signal Processing, Audio signal processing, Mathematics - Optimization and Control, Sound (geography), geography, geography.geographical_feature_category, Quantitative Biology::Neurons and Cognition, Representation (systemics), [SCCO.LING]Cognitive science/Linguistics, Human auditory system, Wilson-Cowan equation, Optimization and Control (math.OC), Computer Science::Sound, 020201 artificial intelligence & image processing, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], computer, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing, 030217 neurology & neurosurgery, Analysis of PDEs (math.AP)
Abstract: The reconstruction mechanisms built by the human auditory system during sound reconstruction are still a matter of debate. The purpose of this study is to refine the auditory cortex model introduced in [9], and inspired by the geometrical modelling of vision. The algorithm transforms the degraded sound in an 'image' in the time-frequency domain via a short-time Fourier transform. Such an image is then lifted in the Heisenberg group and it is reconstructed via a Wilson-Cowan differo-integral equation. Numerical experiments on a library of speech recordings are provided, showing the good reconstruction properties of the algorithm., arXiv admin note: substantial text overlap with arXiv:2004.02450
Published: 2021

8. IMODAL: creating learnable user-defined deformation models

Author: Leander Lacroix, Benjamin Charlier, Alain Trouvé, Barbara Gris, Institut National de la Santé et de la Recherche Médicale (INSERM), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay), Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: Theoretical computer science, business.industry, Computer science, 02 engineering and technology, Modular design, Deformation (meteorology), Object (computer science), Data type, 030218 nuclear medicine & medical imaging, Data modeling, Set (abstract data type), 03 medical and health sciences, 0302 clinical medicine, 0202 electrical engineering, electronic engineering, information engineering, 020201 artificial intelligence & image processing, Polygon mesh, Artificial intelligence, [MATH]Mathematics [math], business, Computer Science::Databases, Interpolation
Abstract: International audience; A natural way to model the evolution of an object (growth of a leaf for instance) is to estimate a plausible deforming path between two observations. This interpolation process can generate deceiving results when the set of considered deformations is not relevant to the observed data. To overcome this issue, the framework of deformation modules allows to incorporate in the model structured deformation patterns coming from prior knowledge on the data. The goal of this article is twofold. First defining new deformation modules incorporating structures coming from the elastic properties of the objects. Second, presenting the IMODAL library allowing to perform registration through structured deformations. This library is modular: adapted priors can be easily defined by the user, several priors can be combined into a global one and various types of data can be considered such as curves, meshes or images. It can be downloaded at https://github.com/imodal.
Published: 2021

9. Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation

Author: Jean-Michel Coron, Amaury Hayat, Georges Bastin, Peipei Shang, UCL - SST/ICTM/INMA - Pôle en ingénierie mathématique, Centre for Systems Engineering and Applied Mechanics (CSAM), Université Catholique de Louvain = Catholic University of Louvain (UCL), Département de Mathématiques-Université de Paris XI, Université Paris-Sud - Paris 11 (UP11), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), School of Mathematical Sciences [Shanghai], Tongji University, Department of Mathematics [ETH Zurich] (D-MATH), Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich] (ETH Zürich), Laboratoire International Associé Sino-Français de Mathématiques Appliquées (LIASFMA), French Corps des IPEF, ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), Université Catholique de Louvain (UCL), and Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology in Zürich [Zürich] (ETH Zürich)
Subjects: 0209 industrial biotechnology, Steady state (electronics), boundary feedback controls, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), 02 engineering and technology, 01 natural sciences, Burgers equation, Shock (mechanics), Exponential function, Burgers' equation, 020901 industrial engineering & automation, Inviscid flow, Modelling and Simulation, Modeling and Simulation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Boundary value problem, 0101 mathematics, shock steady state, Control-Lyapunov function, Mathematics
Abstract: In this paper, we study the exponential stabilization of a shock steady state for the inviscid Burgers equation on a bounded interval. Our analysis relies on the construction of an explicit strict control Lyapunov function. We prove that by appropriately choosing the feedback boundary conditions, we can stabilize the state as well as the shock location to the desired steady state in [Formula: see text]-norm, with an arbitrary decay rate.
Published: 2019

10. Null Controllability of a Linearized Korteweg--de Vries Equation by Backstepping Approach

Author: Shengquan Xiang, Sorbonne Université (SU), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Finite4SoS - ANR 15 CE23 0007, ETH Zürich, Institute for Theoretical Studies, Forschungsinstitut für Mathematik, Laboratoire International Associé Sino-Français de Mathématiques Appliquées, and ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015)
Subjects: 0209 industrial biotechnology, Control and Optimization, Spectral theory, Mathematics::Analysis of PDEs, 02 engineering and technology, AMS Subject Classification. 35Q53, 34H05, 35P10, 01 natural sciences, symbols.namesake, 020901 industrial engineering & automation, Korteweg-de Vries, Neumann boundary condition, [MATH]Mathematics [math], 0101 mathematics, Korteweg–de Vries equation, Mathematics, Null controllability, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Null (mathematics), Stabilization, Controllability, Backstepping, Bounded function, Dirichlet boundary condition, symbols
Abstract: International audience; This paper deals with the controllability problem of a linearized Korteweg-de Vries equation on bounded interval. The system has a homogeneous Dirichlet boundary condition and a homogeneous Neumann boundary condition at the right end-points of the interval, a non homogeneous Dirichlet boundary condition at the left end-point which is the control. We prove the null controllability by using a backstepping approach, a method usually used to handle stabilization problems.
Published: 2019

11. PI Regulation of a Reaction-Diffusion Equation with Delayed Boundary Control

Author: Christophe Prieur, Hugo Lhachemi, Emmanuel Trélat, University College Dublin [Dublin] (UCD), GIPSA - Infinite Dimensional Dynamics (GIPSA-INFINITY), GIPSA Pôle Automatique et Diagnostic (GIPSA-PAD), Grenoble Images Parole Signal Automatique (GIPSA-lab), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Grenoble Alpes (UGA), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), This publication has emanated from research supported in part by a research grant from Science Foundation Ireland (SFI) under grant number 16/RC/3872 and is co-funded under the European Regional Development Fund and byI-Form industry partners., Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Lyapunov function, 0209 industrial biotechnology, 1-D reaction-diffusion equation, Partial Differential Equations (PDEs), 1-D Reaction-diffusion equation, PID controller, Boundary (topology), 02 engineering and technology, Systems and Control (eess.SY), Electrical Engineering and Systems Science - Systems and Control, Setpoint, Tracking error, symbols.namesake, 020901 industrial engineering & automation, Control theory, Neumann trace, Reaction–diffusion system, FOS: Mathematics, FOS: Electrical engineering, electronic engineering, information engineering, [INFO.INFO-SY]Computer Science [cs]/Systems and Control [cs.SY], Electrical and Electronic Engineering, Mathematics - Optimization and Control, Delay boundary control, Mathematics, Computer Science Applications, PI regulation control, Control and Systems Engineering, Optimization and Control (math.OC), Time derivative, symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Numerical stability
Abstract: The general context of this work is the feedback control of an infinite-dimensional system so that the closed-loop system satisfies a fading-memory property and achieves the setpoint tracking of a given reference signal. More specifically, this paper is concerned with the Proportional Integral (PI) regulation control of the left Neumann trace of a one-dimensional reaction-diffusion equation with a delayed right Dirichlet boundary control. In this setting, the studied reaction-diffusion equation might be either open-loop stable or unstable. The proposed control strategy goes as follows. First, a finite-dimensional truncated model that captures the unstable dynamics of the original infinite-dimensional system is obtained via spectral decomposition. The truncated model is then augmented by an integral component on the tracking error of the left Neumann trace. After resorting to the Artstein transformation to handle the control input delay, the PI controller is designed by pole shifting. Stability of the resulting closed-loop infinite-dimensional system, consisting of the original reaction-diffusion equation with the PI controller, is then established thanks to an adequate Lyapunov function. In the case of a time-varying reference input and a time-varying distributed disturbance, our stability result takes the form of an exponential Input-to-State Stability (ISS) estimate with fading memory. Finally, another exponential ISS estimate with fading memory is established for the tracking performance of the reference signal by the system output. In particular, these results assess the setpoint regulation of the left Neumann trace in the presence of distributed perturbations that converge to a steady-state value and with a time-derivative that converges to zero. Numerical simulations are carried out to illustrate the efficiency of our control strategy., Comment: Preprint
Published: 2021

12. Nonnegative control of finite-dimensional linear systems

Author: Emmanuel Trélat, Enrique Zuazua, Jérôme Lohéac, Centre de Recherche en Automatique de Nancy (CRAN), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Universidad de Deusto (DEUSTO), Universidad Autónoma de Madrid (UAM), This project has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agree-ment NO. 694126-DyCon), the Alexander von Humboldt-Professorship program, the Grants Fi-nite4SoS ANR-15-CE23-0007-01 and ICON-ANR-16-ACHN-0014 of the French ANR, the Air ForceOffice of Scientific Research under Award NO: FA9550-18-1-0242, Grant MTM2017-92996-C2-1-RCOSNET of MINECO (Spain) and by the ELKARTEK project KK-2018/00083 ROAD2DC of theBasque Government, the European Union Horizon 2020 research and innovation programme underthe Marie Sklodowska-Curie grant agreement NO. 765579-ConFlex., European Project: 0650424(2007), Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Universidad de Deusto [Bilbao] (DEUSTO), Universidad Autonoma de Madrid (UAM), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), University of Deusto, and UAM. Departamento de Matemáticas
Subjects: 0209 industrial biotechnology, Discretization, Matemáticas, Boundary (topology), 02 engineering and technology, 01 natural sciences, Matrix (mathematics), 020901 industrial engineering & automation, Applied mathematics, Minimal time, 0101 mathematics, Nonnegative control, Mathematical Physics, Eigenvalues and eigenvectors, Mathematics, Contrôles positifs, Applied Mathematics, Linear system, Impulsions de Dirac, Kalman filter, 010101 applied mathematics, Controllability, Heat equation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Dirac impulse, Temps minimal, Analysis
Abstract: We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations, This Project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement NO.694126-DyCon), the Alexander von Humboldt-Professorship program, the Grants Finite4SoS ANR-15-CE23-0007-01 and ICON-ANR-16-ACHN-0014 of the French ANR, the Air Force Oÿce of Scientifc Research under Award NO:FA9550-18-1-0242, Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain) and by the ELKARTEK projectKK-2018/00083ROAD2DC of the Basque Government, the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement NO.765579-ConFlex
Published: 2021

13. Small-time global stabilization of the viscous Burgers equation with three scalar controls

Author: Shengquan Xiang, Jean-Michel Coron, Sorbonne Université (SU), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), LIASFMA and ANR Project Finite4SoS (ANR 15-CE23-0007), ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: Controllability, 0209 industrial biotechnology, Applied Mathematics, General Mathematics, AMS Subject Classiﬁcation. 93D15, 93D20, 35K55, 010102 general mathematics, Mathematical analysis, Scalar (mathematics), Small-time global stabilization, 02 engineering and technology, Time-varying feedback laws, 01 natural sciences, Burgers' equation, 020901 industrial engineering & automation, Quadratic equation, Backstepping, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Burgers equations, 0101 mathematics, [MATH]Mathematics [math], Mathematics
Abstract: International audience; We construct explicit time-varying feedback laws leading to the global (null) stabilization in small time of the viscous Burgers equation with three scalar controls. Our feedback laws use first the quadratic transport term to achieve the small-time global approximate stabilization and then the linear viscous term to get the small-time local stabilization.
Published: 2021

14. Multi-shape Registration with Constrained Deformations

Author: Rosa Kowalewski, Barbara Gris, University of Bath [Bath], Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Rosa Kowalewski is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/S022945/1., Frank Nielsen, Frédéric Barbaresco, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: Large deformation, Computer science, Structure (category theory), 010103 numerical & computational mathematics, 02 engineering and technology, Large deformations, Deformation (meteorology), Object (computer science), 01 natural sciences, Sub-Riemannian geometry, Shape registration, Metric (mathematics), 0202 electrical engineering, electronic engineering, information engineering, 020201 artificial intelligence & image processing, Diffeomorphism, [MATH]Mathematics [math], 0101 mathematics, Boundary constraints, Algorithm
Abstract: International audience; Based on a Sub-Riemannian framework, deformation modules provide a way of building large diffeomorphic deformations satisfying a given geometrical structure. This allows to incorporate prior knowledge about object deformations into the model as a means of regularisation [10]. However, diffeomorphic deformations can lead to deceiving results if the deformed object is composed of several shapes which are close to each other but require drastically different deformations. For the related Large Deformation Diffemorphic Metric Mapping, which yields unstructured deformations, this issue was addressed in [2] introducing object boundary constraints. We develop a new registration problem, marrying the two frameworks to allow for different constrained deformations in different coupled shapes.
Published: 2021

15. Feedforward boundary control of $2 \times 2$ nonlinear hyperbolic systems with application to Saint-Venant equations

Author: Jean-Michel Coron, Amaury Hayat, Georges Bastin, Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM), Université Catholique de Louvain = Catholic University of Louvain (UCL), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), This research was supported by the ANR project Finite4SoS (No.ANR 15-CE23-0007),INRIA team CAGE, the NSF CPS Synergy project 'Smoothing Traffic via Energy-efficient Autonomous Driving' (STEAD) CNS 1837481 and the French Corps des IPEF., Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and UCL - SST/ICTM/INMA - Pôle en ingénierie mathématique
Subjects: 0209 industrial biotechnology, Computer science, Linear system, Physical system, Feed forward, General Engineering, Feedforward control, Boundary (topology), 02 engineering and technology, Dynamical Systems (math.DS), Hyperbolic systems, Nonlinear system, 020901 industrial engineering & automation, Control theory, Optimization and Control (math.OC), Frequency domain, 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, Saint-Venant equations, 020201 artificial intelligence & image processing, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Dynamical Systems, Shallow water equations, Mathematics - Optimization and Control
Abstract: International audience; Because they represent physical systems with propagation delays, hyperbolic systems are well suited for feedforward control. This is especially true when the delay between a disturbance and the output is larger than the control delay. In this paper, we address the design of feedforward controllers for a general class of $2 \times 2$ hyperbolic systems with a single disturbance input located at one boundary and a single control actuation at the other boundary. The goal is to design a feedforward control that makes the system output insensitive to the measured disturbance input. We show that, for this class of systems, there exists an efficient ideal feedforward controller which is causal and stable. The problem is first stated and studied in the frequency domain for a simple linear system. Then, our main contribution is to show how the theory can be extended, in the time domain, to general nonlinear hyperbolic systems. The method is illustrated with an application to the control of an open channel represented by Saint- Venant equations where the objective is to make the output water level insensitive to the variations of the input flow rate. Finally, we address a more complex application to a cascade of pools where a blind application of perfect feedforward control can lead to detrimental oscillations. A pragmatic way of modifying the control law to solve this problem is proposed and validated with a simulation experiment.
Published: 2021

16. Null-controllability of linear hyperbolic systems in one dimensional space

Author: Jean-Michel Coron, Hoai-Minh Nguyen, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Ecole Polytechnique Fédérale de Lausanne (EPFL), The authors are partially supported by ANR Finite 4SoS ANR-15-CE23-0007, and ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015)
Subjects: 0209 industrial biotechnology, Work (thermodynamics), General Computer Science, One-dimensional space, Boundary (topology), 02 engineering and technology, Null-controllability, Hyperbolic systems, Hilbert uniqueness method, 020901 industrial engineering & automation, 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, Applied mathematics, Boundary value problem, Electrical and Electronic Engineering, Mathematics - Optimization and Control, Mathematics, Compactness, Mechanical Engineering, 020208 electrical & electronic engineering, Null (mathematics), Term (time), Controllability, Control and Systems Engineering, Optimization and Control (math.OC), Backstepping, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: International audience; This paper is devoted to the controllability of a general linear hyperbolic system in one space dimension using boundary controls on one side. Under precise and generic assumptions on the boundary conditions on the other side, we previously established the optimal time for the null and the exact controllability for this system for a generic source term. In this work, we prove the null-controllability for any time greater than the optimal time and for any source term. Similar results for the exact controllability are also discussed.
Published: 2021

17. Observability of Baouendi-Grushin-Type Equations Through Resolvent Estimates

Author: Cyril Letrouit, Chenmin Sun, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Paris (ENS-PSL), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and ANR-18-CE40-0020,ODA,Ondes déterministes et aléatoires(2018)
Subjects: 0209 industrial biotechnology, General Mathematics, Astrophysics::High Energy Astrophysical Phenomena, Structure (category theory), Mathematics::Analysis of PDEs, 02 engineering and technology, Type (model theory), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Mathematics - Analysis of PDEs, 020901 industrial engineering & automation, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Observability, 0101 mathematics, Mathematics - Optimization and Control, Mathematical physics, Mathematics, Resolvent, Operator (physics), 010102 general mathematics, Functional Analysis (math.FA), Mathematics - Functional Analysis, Controllability, Type equation, Optimization and Control (math.OC), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis of PDEs (math.AP)
Abstract: In this article, we study the observability (or equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the ‘degenerated directions’ of the subelliptic structure.First, for any $\gamma \geq 1$ , we establish a resolvent estimate for the Baouendi–Grushin-type operator $\Delta _{\gamma }=\partial _x^2+\left \lvert x\right \rvert ^{2\gamma }\partial _y^2$ , which has step $\gamma +1$ . We then derive consequences for the observability of the Schrödinger-type equation $i\partial _tu-\left (-\Delta _{\gamma }\right )^{s}u=0$ , where $s\in \mathbb N$ . We identify three different cases: depending on the value of the ratio $(\gamma +1)/s$ , observability may hold in arbitrarily small time or only for sufficiently large times or may even fail for any time.As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial _tu+\left (-\Delta _{\gamma }\right )^su=0$ and establish a decay rate for the damped wave equation associated with $\Delta _{\gamma }$ .
Published: 2020

18. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback

Author: Christophe Zhang, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Mines Paristech, ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
Subjects: 0209 industrial biotechnology, Control and Optimization, Scalar (mathematics), Fredholm transformations, 02 engineering and technology, Space (mathematics), 01 natural sciences, Transport equation, 020901 industrial engineering & automation, FOS: Mathematics, Rapid stabilization, 0101 mathematics, Mathematics - Optimization and Control, Physics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Function (mathematics), Stabilization, Sobolev space, Amplitude, Optimization and Control (math.OC), Internal control, Backstepping, Piecewise, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Convection–diffusion equation
Abstract: We use a variant the backstepping method to study the stabilization of a 1-D linear transport equation on the interval \begin{document}$ (0,L) $\end{document}, by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than \begin{document}$ 1 $\end{document}, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate. The variant of the backstepping method used here relies mainly on the spectral properties of the linear transport equation, and leads to some original technical developments that differ substantially from previous applications.
Published: 2020

19. An Eco-routing algorithm for HEVs under traffic conditions

Author: Arthur Le Rhun, Giovanni De Nunzio, Frédéric Bonnans, Pierre Martinon, Thomas Leroy, IFP Energies nouvelles (IFPEN), Controle, Optimisation, modèles, Méthodes et Applications pour les Systèmes Dynamiques non linéaires (COMMANDS), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Inria Saclay - Ile de France, Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: 0209 industrial biotechnology, Mathematical optimization, Discretization, Computer science, Energy management, 020208 electrical & electronic engineering, Traffic model, Routing algorithm, 02 engineering and technology, Lipschitz continuity, Upper and lower bounds, 020901 industrial engineering & automation, State of charge, Control and Systems Engineering, Shortest path problem, Traffic conditions, Bi-level optimization, 0202 electrical engineering, electronic engineering, information engineering, Eco-routing, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Bi-level
Abstract: International audience; An extension of the bi-level optimization for the energy management of hybrid electric vehicles (HEVs) proposed in Le Rhun et al. (2019a) to the eco-routing problem is presented. Using the knowledge of traffic conditions over the entire road network, we search both the optimal path and state of charge trajectory. This problem results in finding the shortest path on a weighted graph whose nodes are (position, state of charge) pairs for the vehicle, the edge cost being evaluated thanks to the cost maps from optimization at the 'micro' level of a bi-level decomposition. The error due to the discretization of the state of charge is proven to be linear if the cost maps are Lipschitz. The classical A * algorithm is used to solve the problem, with a heuristic based on a lower bound of the energy needed to complete the travel. The eco-routing method is validated by numerical simulations and compared to the fastest path on a synthetic road network.
Published: 2020

20. Reachable sets for a 3D accidentally symmetric molecule

Author: Mario Sigalotti, Ugo Boscain, Eugenio Pozzoli, Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), ANR-17-CE40-0007,QUACO,Contrôle quantique : systèmes d'EDPs et applications à l'IRM(2017), ANR-15-CE40-0018,SRGI,Géométrie sous-Riemannienne et Interactions(2015), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
Subjects: Controllability, Distributed-parameter systems, 0209 industrial biotechnology, Schrödinger equation, 02 engineering and technology, Reachable states, symbols.namesake, 020901 industrial engineering & automation, Quantum mechanics, Electric field, 0202 electrical engineering, electronic engineering, information engineering, Quantum, Physics, 020208 electrical & electronic engineering, Bilinear control, Partial differential equations, Symmetry (physics), Dipole, Electric dipole moment, Control and Systems Engineering, Moment (physics), symbols, Quantum angular momentum, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: International audience; In this paper we study the controllability properties of the quantum rotational dynamics of a 3D symmetric molecule, with electric dipole moment not collinear to the symmetry axis of the molecule (that is, an accidentally symmetric-top). We control the dynamics with three orthogonally polarized electric fields. When the dipole has a nonzero component along the symmetry axis, it is known that the dynamics is approximately controllable. We focus here our attention to the case where the dipole moment and the symmetry axis are orthogonal (that is, an orthogonal accidentally symmetric-top), providing a description of the reachable sets.
Published: 2020

21. Robustness under control sampling of reachability in fixed time for nonlinear control systems

Author: Emmanuel Trélat, Loïc Bourdin, Mathématiques & Sécurité de l'information (XLIM-MATHIS), XLIM (XLIM), Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS)-Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: 0209 industrial biotechnology, Control and Optimization, 02 engineering and technology, Nonlinear control, 01 natural sciences, 020901 industrial engineering & automation, Fixed time, Reachability, Control theory, Robustness (computer science), Nonlinear control systems, 0101 mathematics, Control (linguistics), Sampled-data controls, Mathematics - Optimization and Control, Mathematics, Piecewise constant controls, Applied Mathematics, 010102 general mathematics, Sampling (statistics), 16. Peace & justice, Regular controls, Control and Systems Engineering, Signal Processing, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: International audience; Under a regularity assumption we prove that reachability in fixed time for nonlinear control systems is robust under control sampling.
Published: 2020

22. Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space

Author: Jean-Michel Coron, Hoai-Minh Nguyen, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Ecole Polytechnique Fédérale de Lausanne (EPFL), ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and The authors were partially supported by ANR Finite4SoS ANR-15-CE23-0007
Subjects: 0209 industrial biotechnology, Control and Optimization, One-dimensional space, 02 engineering and technology, 01 natural sciences, 020901 industrial engineering & automation, Mathematics - Analysis of PDEs, Nonlinear 1-D hyperbolic systems, FOS: Mathematics, Initial value problem, Boundary value problem, 0101 mathematics, [MATH]Mathematics [math], Mathematics - Optimization and Control, Mathematics, 010102 general mathematics, Null (mathematics), Mathematical analysis, 1-D quasilinear hyperbolic systems, Hyperbolic systems, Stabilization, Feedback laws, Controllability, Computational Mathematics, Nonlinear system, Control and Systems Engineering, Homogeneous, Optimization and Control (math.OC), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis of PDEs (math.AP)
Abstract: International audience; We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.
Published: 2020

23. Effective adiabatic control of a decoupled Hamiltonian obtained by rotating wave approximation

Author: Mario Sigalotti, Nicolas Augier, Ugo Boscain, COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA), Biological control of artificial ecosystems (BIOCORE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire d'océanographie de Villefranche (LOV), Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), ANR-15-CE40-0018,SRGI,Géométrie sous-Riemannienne et Interactions(2015), ANR-17-CE40-0007,QUACO,Contrôle quantique : systèmes d'EDPs et applications à l'IRM(2017), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: 0209 industrial biotechnology, Mechanical systems, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Quantum control, FOS: Physical sciences, 02 engineering and technology, Dynamical Systems (math.DS), Averaging control, 01 natural sciences, 020901 industrial engineering & automation, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, FOS: Mathematics, Electrical and Electronic Engineering, Mathematics - Dynamical Systems, 010306 general physics, Control (linguistics), Adiabatic process, Robustness, Mathematics - Optimization and Control, Mathematical Physics, Physics, Quantum Physics, Mathematical Physics (math-ph), Classical mechanics, Flow (mathematics), Control and Systems Engineering, Cascade, Optimization and Control (math.OC), Rotating wave approximation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Quantum Physics (quant-ph), Hamiltonian (control theory)
Abstract: International audience; In this paper we study up to which extent we can apply adiabatic control strategies to a quantum control model obtained by rotating wave approximation. In particular, we show that, under suitable assumptions on the asymptotic regime between the parameters characterizing the rotating wave and the adiabatic approximations, the induced flow converges to the one obtained by considering the two approximations separately and by combining them formally in cascade. As a consequence, we propose explicit control laws which can be used to induce desired populations transfers, robustly with respect to parameter dispersions in the controlled Hamiltonian.
Published: 2020

24. Sub-Riemannian Methods in Shape Analysis

Author: Laurent Younes, Barbara Gris, Alain Trouvé, Center for Imaging Science (CIS), Johns Hopkins University (JHU), Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Centre de Mathématiques et de Leurs Applications (CMLA), École normale supérieure - Cachan (ENS Cachan)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Algebra, Geodesic, Homogeneous, Computer science, 010102 general mathematics, 0202 electrical engineering, electronic engineering, information engineering, 020201 artificial intelligence & image processing, 02 engineering and technology, 0101 mathematics, [MATH]Mathematics [math], 01 natural sciences, ComputingMilieux_MISCELLANEOUS, Shape analysis (digital geometry)
Abstract: Because they reduce the search domains for geodesic paths in manifolds, sub-Riemannian methods can be used both as computational and modeling tools in designing distances between complex objects. This chapter provides a review of the methods that have been recently introduced in this context to study shapes, with a special focus on shape spaces defined as homogeneous spaces under the action of diffeomorphisms. It describes sub-Riemannian methods that are based on control points, possibly enhanced with geometric information, and their generalization to deformation modules. It also discusses the introduction of implicit constraints on geodesic evolution, and the associated computational challenges. Several examples and numerical results are provided as illustrations.
Published: 2020

25. Characterization by observability inequalities of controllability and stabilization properties

Author: Emmanuel Trélat, Yashan Xu, Gengsheng Wang, Sorbonne Université (SU), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Tianjin University (TJU), School of Mathematical Sciences [Fudan], Fudan University [Shanghai], Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: 0209 industrial biotechnology, Pure mathematics, Ocean Engineering, 02 engineering and technology, 01 natural sciences, controllability, symbols.namesake, 020901 industrial engineering & automation, FOS: Mathematics, 93C20, Observability, 0101 mathematics, Mathematics - Optimization and Control, Mathematics, Operator (physics), 010102 general mathematics, Null (mathematics), Hilbert space, stabilizability, 93B05, Dual (category theory), Exponential function, 93B07, Controllability, Optimization and Control (math.OC), Bounded function, symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], observability inequality
Abstract: International audience; Given a linear control system in a Hilbert space with a bounded control operator, we establish a characterization of exponential stabilizability in terms of an observability inequality. Such dual characterizations are well known for exact (null) controllability. Our approach exploits classical Fenchel duality arguments and, in turn, leads to characterizations in terms of observability inequalities of approximately null controllability and of α-null controllability. We comment on the relationships between those various concepts, at the light of the observability inequalities that characterize them.
Published: 2020

26. Counterexample to a Lyapunov Condition for Uniform Asymptotic Partial Stability

Author: Mario Sigalotti, Jakub Orlowski, Antoine Chaillet, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Sud - Paris 11 (UP11), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Lyapunov function, 0209 industrial biotechnology, State variable, Control and Optimization, Dynamical systems theory, 020208 electrical & electronic engineering, 02 engineering and technology, Function (mathematics), symbols.namesake, 020901 industrial engineering & automation, Rate of convergence, Exponential stability, Control and Systems Engineering, Total derivative, 0202 electrical engineering, electronic engineering, information engineering, symbols, Applied mathematics, Stability of nonlinear systems, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics, Counterexample, Lyapunov methods
Abstract: International audience; Partial stability characterizes dynamical systems for which only a part of the state variables exhibits a stable behavior. In his book on partial stability, Vorotnikov proposed a sufficient condition to establish this property through a Lyapunov-like function whose total derivative is upper-bounded by a negative definite function involving only the sub-state of interest. In this note, we show with a simple two-dimensional system that this statement is wrong in general. More precisely, we show that the convergence rate of the relevant state variables may not be uniform in the initial state. We also discuss the impact of this lack of uniformity on the connected issue of robustness with respect to exogenous disturbances.
Published: 2020

27. A Comprehensive Introduction to sub-Riemannian Geometry

Author: Andrei Agrachev, Davide Barilari, Ugo Boscain, Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies (SISSA / ISAS), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: 0209 industrial biotechnology, Riemannian geometry, sub-Riemannian geometry, Hamiltonian geometric control, 010102 general mathematics, 02 engineering and technology, 16. Peace & justice, 01 natural sciences, 020901 industrial engineering & automation, Settore MAT/05 - Analisi Matematica, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Settore MAT/03 - Geometria, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics
Abstract: Introduction de l'ouvrage; International audience
Published: 2020

28. Null-controllability, exact controllability, and stabilization of hyperbolic systems for the optimal time

Author: Hoai-Minh Nguyen, Jean-Michel Coron, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Ecole Polytechnique Fédérale de Lausanne (EPFL), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: 0209 industrial biotechnology, One-dimensional space, Boundary (topology), 02 engineering and technology, 01 natural sciences, Mathematics - Analysis of PDEs, 020901 industrial engineering & automation, Zero state response, equations, FOS: Mathematics, Applied mathematics, dissipative boundary-conditions, Boundary value problem, 0101 mathematics, [MATH]Mathematics [math], Mathematics - Optimization and Control, Mathematics, Null (mathematics), rapid stabilization, space, pdes, Term (time), 010101 applied mathematics, Controllability, Optimization and Control (math.OC), Backstepping, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis of PDEs (math.AP)
Abstract: In this paper, we discuss our recent works on the null-controllability, the exact controllability, and the stabilization of linear hyperbolic systems in one dimensional space using boundary controls on one side for the optimal time. Under precise and generic assumptions on the boundary conditions on the other side, we first obtain the optimal time for the null and the exact controllability for these systems for a generic source term. We then prove the null-controllability and the exact controllability for any time greater than the optimal time and for any source term. Finally, for homogeneous systems, we design feedbacks which stabilize the systems and bring them to the zero state at the optimal time. Extensions for the non-linear homogeneous system are also discussed, Comment: To appear in the proceedings of the 2020 59th IEEE Conference on Decision and Control. arXiv admin note: text overlap with arXiv:1910.12268
Published: 2020
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29. Switching systems with dwell time: computation of the maximal Lyapunov exponent

Author: Yacine Chitour, Vladimir Yu. Protasov, Mario Sigalotti, Nicola Guglielmi, Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Gran Sasso Science Institute (GSSI), Istituto Nazionale di Fisica Nucleare (INFN), Università degli Studi dell'Aquila = University of L'Aquila (UNIVAQ), Faculty of Computer Science [Moscow] (CS-HSE), Vysšaja škola èkonomiki = National Research University Higher School of Economics [Moscow] (HSE), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Università degli Studi dell'Aquila (UNIVAQ), National Research University Higher School of Economics [Moscow] (HSE), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Sud - Paris 11 (UP11), and Università degli Studi dell'Aquila [L'Aquila] (UNIVAQ.IT)
Subjects: Lyapunov function, 0209 industrial biotechnology, Polytopic Lyapunov function, Switching systems, Polytope, 02 engineering and technology, Lyapunov exponent, Dynamical Systems (math.DS), symbols.namesake, 020901 industrial engineering & automation, Exponential stability, 37M25, Invariant polytope algorithm, 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, 15A60, Applied mathematics, AMS classification: 37B25, Mathematics - Numerical Analysis, Invariant (mathematics), Mathematics - Dynamical Systems, 15-04, Mathematics, Joint spectral radius, Numerical Analysis (math.NA), Dwell time, Constrained switching, Computer Science Applications, Control and Systems Engineering, Norm (mathematics), symbols, 020201 artificial intelligence & image processing, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis
Abstract: International audience; We study asymptotic stability of continuous-time systems with mode-dependent guaranteed dwell time. These systems are reformulated as special cases of a general class of mixed (discrete-continuous) linear switching systems on graphs, in which some modes correspond to discrete actions and some others correspond to continuous-time evolutions. Each discrete action has its own positive weight which accounts for its time-duration. We develop a theory of stability for the mixed systems; in particular, we prove the existence of an invariant Lyapunov norm for mixed systems on graphs and study its structure in various cases, including discrete-time systems for which discrete actions have inhomogeneous time durations. This allows us to adapt recent methods for the joint spectral radius computation (Gripenberg's algorithm and the Invariant Polytope Algorithm) to compute the Lyapunov exponent of mixed systems on graphs.
Published: 2019

30. Converse Lyapunov theorems for infinite-dimensional nonlinear switching systems

Author: Mario Sigalotti, Ihab Haidar, Yacine Chitour, Paolo Mason, Laboratoire QUARTZ (QUARTZ ), Université Paris 8 Vincennes-Saint-Denis (UP8)-SUPMECA - Institut supérieur de mécanique de Paris (SUPMECA)-Ecole Nationale Supérieure de l'Electronique et de ses Applications (ENSEA)-Ecole Internationale des Sciences du Traitement de l'Information (EISTI), Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Université Paris 8 Vincennes-Saint-Denis (UP8)-Ecole Nationale Supérieure de l'Electronique et de ses Applications (ENSEA)-SUPMECA - Institut supérieur de mécanique de Paris (SUPMECA)-Ecole Internationale des Sciences du Traitement de l'Information (EISTI), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: Lyapunov function, 0209 industrial biotechnology, Lemma (mathematics), Converse Lyapunov theorems, Generalization, Switching systems, Infinite-dimensional systems, 02 engineering and technology, symbols.namesake, Nonlinear system, 020901 industrial engineering & automation, Lyapunov functional, Exponential stability, Converse, symbols, Nonlinear systems, Applied mathematics, Point (geometry), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics
Abstract: International audience; In this paper, we provide two converse Lyapunov theorems in the framework of nonlinear infinite-dimensional switching systems. Our results characterize uniform exponential stability with respect to the switching law through the existence of both coercive and non-coercive Lyapunov functionals. The starting point for our arguments is a generalization of the well-known Datko lemma to the case of nonlinear infinite-dimensional switching systems.
Published: 2019

31. Shape turnpike for linear parabolic PDE models

Author: Emmanuel Trélat, Enrique Zuazua, Gontran Lance, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Departamento de Matemáticas [Madrid], Universidad Autonoma de Madrid (UAM), Department Mathematik [Erlangen], Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), This project has received funding from the Grants ICON-ANR-16-ACHN-0014 and Finite4SoS ANR-15-CE23-0007-01 of the French ANR, the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement 694126-DyCon), the Alexander von Humboldt-Professorship program, the Air Force Office of Scientific Research under Award NO: FA9550-18-1-0242, Grant MTM2017-92996-C2-1-RCOSNET of MINECO (Spain) and by the ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government, Transregio 154 Project *Mathematical Modelling, Simulation and Optimization usingthe Example of Gas Networks* of the German DFG, the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement 765579-ConFlex., Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Universidad Autónoma de Madrid (UAM)
Subjects: 0209 industrial biotechnology, Property (philosophy), General Computer Science, 02 engineering and technology, 020901 industrial engineering & automation, Mathematics - Analysis of PDEs, Strict dissipativity, 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Mathematics - Numerical Analysis, Electrical and Electronic Engineering, Mathematics - Optimization and Control, Mathematics, Parabolic equation, Turnpike, Mechanical Engineering, 020208 electrical & electronic engineering, State (functional analysis), Numerical Analysis (math.NA), Optimal control, Parabolic partial differential equation, Exponential function, Term (time), Hausdorff distance, Control and Systems Engineering, Optimization and Control (math.OC), Optimal shape design, Direct methods, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Focus (optics), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: International audience; We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike property in term of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations.
Published: 2019

32. Turnpike in optimal shape design

Author: Emmanuel Trélat, Gontran Lance, Enrique Zuazua, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Departamento de Matemáticas [Madrid], Universidad Autonoma de Madrid (UAM), Department Mathematik [Erlangen], Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), and Department Mathematik
Subjects: Parabolic equation, 0209 industrial biotechnology, Turnpike, Property (philosophy), 020208 electrical & electronic engineering, Context (language use), 02 engineering and technology, State (functional analysis), Optimal control, Term (time), 020901 industrial engineering & automation, Control and Systems Engineering, Direct methods, Optimal shape design, Strict dissipativity, 0202 electrical engineering, electronic engineering, information engineering, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Heat equation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Focus (optics), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike property in term of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations.
Published: 2019

33. Addendum to Pontryagin’s maximum principle for dynamic systems on time scales

Author: Loïc Bourdin, Emmanuel Trélat, Oleksandr Stanzhytskyi, Mathématiques & Sécurité de l'information (XLIM-MATHIS), XLIM (XLIM), Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS)-Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS), Taras Shevchenko National University of Kyiv, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: 0209 industrial biotechnology, 02 engineering and technology, 01 natural sciences, Pontryagin's minimum principle, optimal control, 020901 industrial engineering & automation, Maximum principle, Pontryagin maximum principle, FOS: Mathematics, Calculus, 0101 mathematics, Mathematics - Optimization and Control, Mathematics, Algebra and Number Theory, packages of needle-like variations, Applied Mathematics, 010102 general mathematics, time scale, Addendum, Ekeland variational principle, Optimal control, 34K35, 34N99, 39A12, 39A13, 49K15, 93C15, 93C55, Optimization and Control (math.OC), [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis, Hamiltonian (control theory)
Abstract: International audience; This note is an addendum to [1, 2], pointing out the differences between these papers and raising open questions.
Published: 2017

34. On the gap between deterministic and probabilistic joint spectral radii for discrete-time linear systems

Author: Mario Sigalotti, Yacine Chitour, Guilherme Mazanti, Laboratoire des signaux et systèmes (L2S), Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Sud - Paris 11 (UP11), Institut Polytechnique des Sciences Avancées (IPSA), Dynamical Interconnected Systems in COmplex Environments (DISCO), Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Sud - Paris 11 (UP11)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), iCODE Institute, Hadamard Mathematics LabEx (LMH), Fondation Mathématique Jacques Hadamard (FMJH), ANR: 11-LABX-0056,LMH,LabEx Mathématique Hadamard(2011), ANR-10-CAMP-0151-02,FMJH, Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), ANR-10-CAMP-0151-02,Fondation Mathématique jacques Hadamard-DNC,Fondation Mathématique jacques Hadamard(2010), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: 93C30, 93C55, 37H15, 0209 industrial biotechnology, 2010 Mathematics Subject Classification: 93C30, 93C55, 37H15, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], 0211 other engineering and technologies, 02 engineering and technology, Dynamical Systems (math.DS), Discrete time, Joint spectral radius, Combinatorics, 020901 industrial engineering & automation, 2020 Mathematics Subject Classification: 93C30, 93C55, 37H15, Condensed Matter::Superconductivity, FOS: Mathematics, Discrete Mathematics and Combinatorics, Markov process, Invariant (mathematics), Mathematics - Dynamical Systems, Finite set, Mathematics - Optimization and Control, Mathematics, Numerical Analysis, Algebra and Number Theory, Random switching, Probability (math.PR), Stochastic matrix, 021107 urban & regional planning, Discrete time nonlinear systems, Infimum and supremum, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Optimization and Control (math.OC), Linear switched systems, Geometry and Topology, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Probability
Abstract: International audience; Given a discrete-time linear switched system $\Sigma(\mathcal A)$ associated with a finite set $\mathcal A$ of matrices, we consider the measures of its asymptotic behavior given by, on the one hand, its deterministic joint spectral radius $\rho_{\mathrm d}(\mathcal A)$ and, on the other hand, its probabilistic joint spectral radii $\rho_{\mathrm p}(\nu,P,\mathcal A)$ for Markov random switching signals with transition matrix $P$ and a corresponding invariant probability $\nu$. Note that $\rho_{\mathrm d}(\mathcal A)$ is larger than or equal to $\rho_{\mathrm p}(\nu,P,\mathcal A)$ for every pair $(\nu, P)$. In this paper, we investigate the cases of equality of $\rho_{\mathrm d}(\mathcal A)$ with either a single $\rho_{\mathrm p}(\nu,P,\mathcal A)$ or with the supremum of $\rho_{\mathrm p}(\nu,P,\mathcal A)$ over $(\nu,P)$ and we aim at characterizing the sets $\mathcal A$ for which such equalities may occur.
Published: 2019

35. On the compatibility between the adiabatic and the rotating wave approximations in quantum control

Author: Mario Sigalotti, Ugo Boscain, Nicolas Augier, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
Subjects: 0209 industrial biotechnology, Quantum control, FOS: Physical sciences, 02 engineering and technology, symbols.namesake, 020901 industrial engineering & automation, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, Energy level, Statistical physics, Adiabatic process, Quantum, Mathematics - Optimization and Control, Mathematical Physics, Physics, Quantum Physics, 020208 electrical & electronic engineering, Mathematical Physics (math-ph), Controllability, Amplitude, Optimization and Control (math.OC), Compatibility (mechanics), symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Hamiltonian (quantum mechanics), Quantum Physics (quant-ph)
Abstract: In this paper, we discuss the compatibility between the rotating-wave and the adiabatic approximations for controlled quantum systems. Although the paper focuses on applications to two-level quantum systems, the main results apply in higher dimension. Under some suitable hypotheses on the time scales, the two approximations can be combined. As a natural consequence of this, it is possible to design control laws achieving transitions of states between two energy levels of the Hamiltonian that are robust with respect to inhomogeneities of the amplitude of the control input., 58th Conference on Decision and Control - Nice, France, Dec 2019, Nice, France
Published: 2019

36. Gaussian Mixture Penalty for Trajectory Optimization Problems

Author: Pierre Martinon, Baptiste Gregorutti, Frédéric Bonnans, Cédric Rommel, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Safety Line [Paris], Controle, Optimisation, modèles, Méthodes et Applications pour les Systèmes Dynamiques non linéaires (COMMANDS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: 020301 aerospace & aeronautics, 0209 industrial biotechnology, Mathematical optimization, [STAT.AP]Statistics [stat]/Applications [stat.AP], Computer science, Applied Mathematics, Gaussian, Aerospace Engineering, Estimator, 02 engineering and technology, Trajectory optimization, Mixture model, Optimal control, Term (time), symbols.namesake, 020901 industrial engineering & automation, 0203 mechanical engineering, Space and Planetary Science, Control and Systems Engineering, symbols, Trajectory, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Electrical and Electronic Engineering, Parametric statistics
Abstract: International audience; We consider the task of solving an aircraft trajectory optimization problem where the system dynamics have been estimated from recorded data. Additionally, we want to avoid optimized trajectories that go too far away from the domain occupied by the data, since the model validity is not guaranteed outside this region. This motivates the need for a proximity indicator between a given trajectory and a set of reference trajectories. In this presentation, we propose such an indicator based on a parametric estimator of the training set density. We then introduce it as a penalty term in the optimal control problem. Our approach is illustrated with an aircraft minimal consumption problem and recorded data from real flights. We observe in our numerical results the expected trade-off between the consumption and the penalty term.
Published: 2019

37. Consensus and Flocking under Communication Failures for a Class of Cucker-Smale Systems

Author: Emilien Flayac, Benoît Bonnet, Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Modelling and Analysis for Medical and Biological Applications (MAMBA), University of Melbourne, The authors were partially supported by the Archimède Labex (ANR-11-LABX-0033) and the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the ``Investissements d'Avenir' French Government program. The authors wish to dearly thank Francesco Rossi for suggesting a preliminary version of the problem that we treated here, as well as Ioannis Sarras for his numerous insights on the topic of strict Lyapunov design for persistent systems., Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Pronostic/Diagnostic Et CommAnde : Santé et Energie (PECASE), Laboratoire d'Informatique et Systèmes (LIS), Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU), Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU), DTIS, ONERA, Université Paris Saclay [Palaiseau], and ONERA-Université Paris-Saclay
Subjects: Lyapunov function, 0209 industrial biotechnology, General Computer Science, Computer science, 02 engineering and technology, Upper and lower bounds, 34D05, 34D23, 91C20, symbols.namesake, 020901 industrial engineering & automation, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, Applied mathematics, MSC2020 Classification :34D05, 34D23, 91C20, Electrical and Electronic Engineering, Mathematics - Optimization and Control, Asymptotic Flocking, Lyapunov methods, Algebraic connectivity, flocking, Flocking (behavior), Mechanical Engineering, Multi-agent system, 020208 electrical & electronic engineering, Multiplicative function, Multi-agent systems, Computer Science::Multiagent Systems, Control and Systems Engineering, Optimization and Control (math.OC), symbols, Graph (abstract data type), Persistence of excitation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Strict Lyapunov Design
Abstract: International audience; In this paper, we study sufficient conditions for the emergence of asymptotic consensus and flocking in a certain class of non-linear generalised Cucker-Smale systems subject to multiplicative communication failures. Our approach is based on the combination of strict Lyapunov design together with the formulation of a suitable persistence condition for multi-agent systems. The latter can be interpreted as a lower bound on the algebraic connectivity of the time-average of the interaction graph generated by the communication weights, and provides quantitative decay estimates for the variance functional along the solutions of the system.
Published: 2019

38. Feedback stabilization of a 1D linear reaction-diffusion equation with delay boundary control

Author: Christophe Prieur, Emmanuel Trélat, GIPSA - Systèmes non linéaires et complexité (GIPSA-SYSCO), Département Automatique (GIPSA-DA), Grenoble Images Parole Signal Automatique (GIPSA-lab ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Grenoble Images Parole Signal Automatique (GIPSA-lab ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Grenoble Images Parole Signal Automatique (GIPSA-lab), and Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)
Subjects: Lyapunov function, 0209 industrial biotechnology, Boundary (topology), 02 engineering and technology, Delay control, symbols.namesake, Mathematics - Analysis of PDEs, 020901 industrial engineering & automation, Control theory, Reaction–diffusion system, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Electrical and Electronic Engineering, Mathematics - Optimization and Control, Mathematics, Partial differential equation, Computer Science Applications, Transformation (function), Control and Systems Engineering, Optimization and Control (math.OC), partial differential equation, symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Reaction-diffusion equation, Constant (mathematics), Convection–diffusion equation, Analysis of PDEs (math.AP)
Abstract: The goal of this work is to compute a boundary control of reaction-diffusion partial differential equation. The boundary control is subject to a constant delay, whereas the equation may be unstable without any control. For this system equivalent to a parabolic equation coupled with a transport equation, a prediction-based control is explicitly computed. To do that we decompose the infinite-dimensional system into two parts: one finite-dimensional unstable part, and one stable infinite-dimensional part. An finite-dimensional delay controller is computed for the unstable part, and it is shown that this controller succeeds in stabilizing the whole partial differential equation. The proof is based on a an explicit form of the classical Artstein transformation, and an appropriate Lyapunov function. A numerical simulation illustrate the constructive design method., Comment: arXiv admin note: substantial text overlap with arXiv:1511.03030
Published: 2019

39. Optimal Control of Endo-Atmospheric Launch Vehicle Systems: Geometric and Computational Issues

Author: Bruno Hérissé, Emmanuel Trélat, Riccardo Bonalli, Sorbonne Université (SU), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), DTIS, ONERA, Université Paris Saclay [Palaiseau], ONERA-Université Paris-Saclay, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
Subjects: 0209 industrial biotechnology, Mathematical optimization, Geometric analysis, Computer science, Initialization, 02 engineering and technology, Geometric optimal control, Numerical homotopy methods, [SPI.AUTO]Engineering Sciences [physics]/Automatic, Vehicle dynamics, symbols.namesake, 020901 industrial engineering & automation, Shooting method, Missile, Local coordinates, FOS: Mathematics, Electrical and Electronic Engineering, Mathematics - Optimization and Control, Indirect methods, Optimal control, Manifold, Computer Science Applications, Optimization and Control (math.OC), Control and Systems Engineering, Guidance of vehicle systems, Trajectory, Euler's formula, symbols, Gravitational singularity, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: International audience; In this paper we develop a geometric analysis and a numerical algorithm, based on indirect methods, to solve optimal guidance of endo-atmospheric launch vehicle systems under mixed control-state constraints. Two main difficulties are addressed. First, we tackle the presence of Euler singularities by introducing a representation of the configuration manifold in appropriate local charts. In these local coordinates, not only the problem is free from Euler singularities but also it can be recast as an optimal control problem with only pure control constraints. The second issue concerns the initialization of the shooting method. We introduce a strategy which combines indirect methods with homotopies, thus providing high accuracy. We illustrate the efficiency of our approach by numerical simulations on missile interception problems under challenging scenarios.
Published: 2019

40. Continuity of Pontryagin Extremals with Respect to Delays in Nonlinear Optimal Control

Author: Bruno Hérissé, Riccardo Bonalli, Emmanuel Trélat, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), DTIS, ONERA, Université Paris Saclay (COmUE) [Palaiseau], ONERA-Université Paris Saclay (COmUE), and Université Paris Saclay (COmUE)-ONERA
Subjects: Time-delayed systems, 0209 industrial biotechnology, Control and Optimization, 02 engineering and technology, Network topology, 01 natural sciences, Projection (linear algebra), 020901 industrial engineering & automation, Shooting method, Dimension (vector space), Pontryagin extremals, Nonlinear optimal control, Shooting method for problems with delays, FOS: Mathematics, Applied mathematics, 0101 mathematics, Mathematics - Optimization and Control, Mathematics, Continuity with respect to delays, Applied Mathematics, 010102 general mathematics, State (functional analysis), Optimal control, AMS subject classifications. 49J15, 49K15, 49K40, Optimization and Control (math.OC), Trajectory, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Constant (mathematics)
Abstract: Consider a general nonlinear optimal control problem in finite dimension, with constant state and/or control delays. By the Pontryagin Maximum Principle, any optimal trajectory is the projection of a Pontryagin extremal. We establish that, under appropriate assumptions, Pontryagin extremals depend continuously on the parameter delays, for adequate topologies. The proof of the continuity of the trajectory and of the control is quite easy, however, for the adjoint vector, the proof requires a much finer analysis. The continuity property of the adjoint with respect to the parameter delay opens a new perspective for the numerical implementation of indirect methods, such as the shooting method. We also discuss the sharpness of our assumptions., arXiv admin note: text overlap with arXiv:1709.04383
Published: 2019

41. Bounds on time-optimal concatenations of arcs for two-input driftless 3D systems

Author: Mario Sigalotti, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), The project leading to this publication has received funding fromthe ANR project SRGI ANR-15-CE40-0018, ANR-15-CE40-0018,SRGI,Géométrie sous-Riemannienne et Interactions(2015), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: 0209 industrial biotechnology, Singular arc, Scalar (mathematics), Second-order optimality conditions, 02 engineering and technology, Geometric control, 020901 industrial engineering & automation, Lie algebra, 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, Applied mathematics, Point (geometry), Mathematics - Optimization and Control, Mathematics, Time-optimal, 020208 electrical & electronic engineering, 16. Peace & justice, Time optimal, Optimal control, Manifold, Control and Systems Engineering, Optimization and Control (math.OC), Trajectory, Bang arc, Vector field, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: International audience; We study a driftless system on a three-dimensional manifold driven by two scalar controls. We assume that each scalar control has an independent bound on its modulus and we prove that, locally around every point where the controlled vector fields satisfy some suitable nondegeneracy Lie bracket condition, every time-optimal trajectory has at most five bang or singular arcs. The result is obtained using first-and second-order necessary conditions for optimality.
Published: 2019
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42. Incorporation of a deformation prior in image reconstruction

Author: Barbara Gris, Gris, Barbara, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), The work by Barbara Gris was supported by the Swedish Foundation for Strategic Research grant AM13-0049., Centre de Mathématiques et de Leurs Applications ( CMLA ), and École normale supérieure - Cachan ( ENS Cachan ) -Centre National de la Recherche Scientifique ( CNRS )
Subjects: [ MATH ] Mathematics [math], Statistics and Probability, [ INFO.INFO-TS ] Computer Science [cs]/Signal and Image Processing, Computer science, [INFO.INFO-TS] Computer Science [cs]/Signal and Image Processing, [INFO.INFO-IM] Computer Science [cs]/Medical Imaging, 02 engineering and technology, Iterative reconstruction, [MATH] Mathematics [math], [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, 0202 electrical engineering, electronic engineering, information engineering, [INFO.INFO-IM]Computer Science [cs]/Medical Imaging, [MATH]Mathematics [math], [ INFO.INFO-IM ] Computer Science [cs]/Medical Imaging, business.industry, Applied Mathematics, Modular design, Condensed Matter Physics, Reconstruction method, Formalism (philosophy of mathematics), Modeling and Simulation, 020201 artificial intelligence & image processing, Geometry and Topology, Computer Vision and Pattern Recognition, Diffeomorphism, business, Algorithm
Abstract: International audience; This article presents a method to incorporate a deformation prior in image reconstruction via the formalism of deformation modules. The framework of deformation modules allows to build diffeomorphic deformations that satisfy a given structure. The idea is to register a template image against the indirectly observed data via a modular deformation, incorporating this way the deformation prior in the reconstruction method. We show that this is a well-defined regularization method (proving existence, stability and convergence) and present numerical examples of reconstruction from 2-D tomographic simulations and partially-observed images.
Published: 2019

43. Semi-conical eigenvalue intersections and the ensemble controllability problem for quantum systems

Author: Ugo Boscain, Nicolas Augier, Mario Sigalotti, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), ANR-17-CE40-0007,QUACO,Contrôle quantique : systèmes d'EDPs et applications à l'IRM(2017), ANR-15-CE40-0018,SRGI,Géométrie sous-Riemannienne et Interactions(2015), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0007,QUACO,Contrôle quantique : systèmes d’EDPs et applications à l’IRM(2017), and ANR-15-CE40-0018,SRGI,Sub-Riemannian Geometry and Interactions(2015)
Subjects: 0209 industrial biotechnology, Control and Optimization, Quantum control, FOS: Physical sciences, 02 engineering and technology, conical intersections of eigenvalues, Adiabatic theorem, symbols.namesake, 020901 industrial engineering & automation, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0202 electrical engineering, electronic engineering, information engineering, FOS: Mathematics, Mathematics - Optimization and Control, Quantum, Mathematical Physics, Eigenvalues and eigenvectors, Mathematical physics, Physics, Applied Mathematics, Ensemble control, 020208 electrical & electronic engineering, Degenerate energy levels, Conical surface, Mathematical Physics (math-ph), Controllability, adiabatic approximation, Optimization and Control (math.OC), symbols, normal forms, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Hamiltonian (quantum mechanics), quantum control
Abstract: International audience; We study one-parametric perturbations of finite dimensional real Hamiltonians depending on two controls, and we show that generically in the space of Hamiltonians, conical intersections of eigenvalues can degenerate into semi-conical intersections of eigenvalues. Then, through the use of normal forms, we study the problem of ensemble controllability between the eigenstates of a generic Hamiltonian.
Published: 2019
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44. PI controllers for 1-D nonlinear transport equation

Author: Jean-Michel Coron, Amaury Hayat, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), ANR Finite4SoS, Corps des IPEF, ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), Hayat, Amaury, and Fondements du numérique - Commande et estimation en temps fini pour les Systèmes de Systèmes - - Finite4SoS2015 - ANR-15-CE23-0007 - AAPG2015 - VALID
Subjects: Lyapunov function, Distributed parameter system, 0209 industrial biotechnology, 02 engineering and technology, Exponential stability, symbols.namesake, 020901 industrial engineering & automation, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Electrical and Electronic Engineering, Mathematics, Lyapunov functions, Partial differential equation, Boundary control, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Computer Science Applications, Nonlinear system, Stability conditions, Control and Systems Engineering, Hyperbolic Equation, symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Convection–diffusion equation, Hyperbolic partial differential equation, Numerical stability
Abstract: International audience; In this paper, we introduce a method to get necessary and sufficient stability conditions for systems governed by 1-D nonlinear hyperbolic partial-differential equations with closed-loop integral controllers, when the linear frequency analysis cannot be used anymore. We study the stability of a general nonlinear transport equation where the control input and the measured output are both located on the boundaries. The principle of the method is to extract the limiting part of the stability from the solution using a projector on a finite-dimensional space and then use a Lyapunov approach. This paper improves a result of Trinh, Andrieu and Xu, and gives an optimal condition for the design of the controller. The results are illustrated with numerical simulations where the predicted stable and unstable regions can be clearly identified.
Published: 2019

45. Finite-time internal stabilization of a linear 1-D transport equation

Author: Christophe Zhang, Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Mines Paristech, ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), Zhang, Christophe, and Fondements du numérique - Commande et estimation en temps fini pour les Systèmes de Systèmes - - Finite4SoS2015 - ANR-15-CE23-0007 - AAPG2015 - VALID
Subjects: 0209 industrial biotechnology, General Computer Science, Scalar (mathematics), Fredholm transformations, 02 engineering and technology, Transport equation, 020901 industrial engineering & automation, 0202 electrical engineering, electronic engineering, information engineering, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Electrical and Electronic Engineering, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Finite-time, Physics, Feedback stabilization, Mechanical Engineering, 020208 electrical & electronic engineering, Mathematical analysis, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], 16. Peace & justice, Sobolev space, Linear transport equation, Control and Systems Engineering, Backstepping, Internal control, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Finite time, Convection–diffusion equation
Abstract: International audience; We consider a 1-D linear transport equation on the interval (0, L), with an internal scalar control. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized in finite time, and we give an explicit feedback law.
Published: 2019

46. Sparse control of Hegselmann-Krause models: Black hole and declustering

Author: Emmanuel Trélat, Nastassia Pouradier Duteil, Benedetto Piccoli, Department of Mathematical Sciences [Camden], Rutgers University [Camden], Rutgers University System (Rutgers)-Rutgers University System (Rutgers), Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Control And GEometry (CaGE )
Subjects: 0209 industrial biotechnology, Collective behavior, Mathematical optimization, Control and Optimization, 02 engineering and technology, 01 natural sciences, Active particles, Black swan, Set (abstract data type), 020901 industrial engineering & automation, Convergence (routing), Control, FOS: Mathematics, 0101 mathematics, [MATH]Mathematics [math], Cluster analysis, Mathematics - Optimization and Control, Declustering, Mathematics, Applied Mathematics, 010102 general mathematics, Function (mathematics), Variance (accounting), Kinetic model, Optimization and Control (math.OC), Pairwise comparison, AMS Subject Classifications: 93C15, 93C20, 91D10, 35B36, 34H05, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Focus (optics)
Abstract: International audience; This paper elaborates control strategies to prevent clustering effects in opinion formation models. This is the exact opposite of numerous situations encountered in the literature where, on the contrary, one seeks controls promoting consensus. In order to promote declustering, instead of using the classical variance that does not capture well the phenomenon of dispersion, we introduce an entropy-type functional that is adapted to measuring pairwise distances between agents. We then focus on a Hegselmann-Krause-type system and design declustering sparse controls both in finite-dimensional and kinetic models. We provide general conditions characterizing whether clustering can be avoided as function of the initial data. Such results include the description of black holes (where complete collapse to consensus is not avoidable), safety zones (where the control can keep the system far from clustering), basins of attraction (attractive zones around the clustering set) and collapse prevention (when convergence to the clustering set can be avoided).
Published: 2019

47. Exponential stability of PI control for Saint-Venant equations with a friction term

Author: Georges Bastin, Jean-Michel Coron, Centre for Systems Engineering and Applied Mechanics (CSAM), Université Catholique de Louvain = Catholic University of Louvain (UCL), Sorbonne Université (SU), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), Coron, Jean-Michel, Fondements du numérique - Commande et estimation en temps fini pour les Systèmes de Systèmes - - Finite4SoS2015 - ANR-15-CE23-0007 - AAPG2015 - VALID, Université Catholique de Louvain (UCL), and UCL - SST/ICTM/INMA - Pôle en ingénierie mathématique
Subjects: Lyapunov stability, 0209 industrial biotechnology, 010102 general mathematics, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], 02 engineering and technology, 01 natural sciences, Hyperbolic systems, Term (time), 020901 industrial engineering & automation, Exponential stability, Pi, Applied mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Shallow water equations, Mathematics
Abstract: International audience; We consider open channels represented by Saint-Venant equations that are monitored and controlled at the downstream boundary and subject to unmeasured flow disturbances at the upstream boundary. We address the issue of feedback stabilization and disturbance rejection under Proportional-Integral (PI) boundary control. For channels with uniform steady states, the analysis has been carried out previously in the literature with spectral methods as well as with Lyapunov functions in Riemann coordinates. In this article, our main contribution is to show how the analysis can be extended to channels with non-uniform steady states with a Lyapunov function in physical coordinates.
Published: 2019

48. Boundary feedback stabilization of hydraulic jumps

Author: Georges Bastin, Jean-Michel Coron, Amaury Hayat, Peipei Shang, Centre for Systems Engineering and Applied Mechanics (CSAM), Université Catholique de Louvain = Catholic University of Louvain (UCL), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Tongji University, Laboratoire International Associé Sino-Français de Mathématiques Appliquées (LIASFMA), French Corps des IPEF, ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), Université Catholique de Louvain (UCL), UCL - SST/ICTM/INMA - Pôle en ingénierie mathématique, Hayat, Amaury, and Fondements du numérique - Commande et estimation en temps fini pour les Systèmes de Systèmes - - Finite4SoS2015 - ANR-15-CE23-0007 - AAPG2015 - VALID
Subjects: Lyapunov function, 0209 industrial biotechnology, Lyapunov approach, 02 engineering and technology, Hydraulic jump, Physics::Geophysics, Physics::Fluid Dynamics, symbols.namesake, 020901 industrial engineering & automation, Computer Science::Computational Engineering, Finance, and Science, 0202 electrical engineering, electronic engineering, information engineering, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Shallow water equations, Mathematics, AMS Subject Classification. 93D15, 35L40, 35L67, 020208 electrical & electronic engineering, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Mechanics, Nonlinear EDP, Open-channel flow, Lyapunov Functions, Boundary feedback controls, Norm (mathematics), symbols, Saint-Venant equations, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: International audience; In an open channel, a hydraulic jump is an abrupt transition between a torrential (super-critical) flow and a fluvial (subcritical) flow. In this article hydraulic jumps are represented by discontinuous shock solutions of hyperbolic Saint-Venant equations. Using a Lyapunov approach, we prove that we can stabilize the state of the system in $H^2$-norm as well as the hydraulic jump location, with simple feedback boundary controls and an arbitrary decay rate, by appropriately choosing the gains of the feedback boundary controls.
Published: 2019

49. Adaptive scheme for pathological oscillations disruption in a delayed neuronal population model

Author: Mario Sigalotti, Antoine Chaillet, Alain Destexhe, Jakub Orlowski, Laboratoire des signaux et systèmes (L2S), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Institut des Neurosciences Paris-Saclay (NeuroPSI), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Sud - Paris 11 (UP11), and Institut des Neurosciences de Paris-Saclay (Neuro-PSI)
Subjects: 0209 industrial biotechnology, Computer science, Oscillation, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], 02 engineering and technology, 03 medical and health sciences, Nonlinear system, Formalism (philosophy of mathematics), 020901 industrial engineering & automation, 0302 clinical medicine, Robustness (computer science), Control theory, [SDV.NEU]Life Sciences [q-bio]/Neurons and Cognition [q-bio.NC], Neuronal population, 030217 neurology & neurosurgery
Abstract: International audience; Motivated by improved ways to disrupt brain oscillations linked to Parkinson's disease, we propose an adaptive output feedback strategy for the stabilization of nonlinear time-delay systems evolving on a bounded set. To that aim, using the formalism of input-to-output stability (IOS), we first show that, for such systems, internal stability guarantees robustness to exogenous disturbances. We then use this feature to establish a general result on scalar adaptive output feedback of time-delay systems inspired by the "σ-modification" strategy. We finally apply this result to a delayed neuronal population model and assess numerically the performance of the adaptive stimulation.
Published: 2018

50. Minimal controllability time for finite-dimensional control systems under state constraints

Author: Emmanuel Trélat, Enrique Zuazua, Jérôme Lohéac, Centre de Recherche en Automatique de Nancy (CRAN), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Département Automatique, Productique et Informatique (IMT Atlantique - DAPI), IMT Atlantique (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT), Laboratoire des Sciences du Numérique de Nantes (LS2N), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique (IMT Atlantique), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Basque Center for Applied Mathematics (BCAM), Basque Center for Applied Mathematics, IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Université de Nantes - Faculté des Sciences et des Techniques, Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)
Subjects: 0209 industrial biotechnology, Computer science, 010102 general mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], 02 engineering and technology, Kalman filter, State (functional analysis), Nonlinear control, 01 natural sciences, Constraint (information theory), Controllability, 020901 industrial engineering & automation, Control and Systems Engineering, Control theory, Control system, Autonomous control, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Electrical and Electronic Engineering, Control (linguistics)
Abstract: International audience; We consider the controllability problem for finite-dimensional linear autonomous control systems, under state constraints but without imposing any control constraint. It is well known that, under the classical Kalman condition, in the absence of constraints on the state and the control, one can drive the system from any initial state to any final one in an arbitrarily small time. Furthermore, it is also well known that there is a positive minimal time in the presence of compact control constraints. We prove that, surprisingly, a positive minimal time may be required as well under state constraints, even if one does not impose any restriction on the control. This may even occur when the state constraints are unilateral, like the nonnegativity of some components of the state, for instance. Using the Brunovsky normal forms of controllable systems, we analyze this phenomenon in detail, that we illustrate by several examples. We discuss some extensions to nonlinear control systems and formulate some challenging open problems.
Published: 2018

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