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

1. Cost of observability inequalities for elliptic equations in 2-d with potentials and applications to control theory

Author

Ervedoza, Sylvain, Le Balc’h, Kévin, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-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), and ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020)

Subjects
Quantitative unique continuation, Applied Mathematics, Control, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Elliptic equations, Analysis
Abstract

The goal of this article is to obtain observability estimates for non-homogeneous elliptic equations in the presence of a potential, posed on a smooth bounded domain Ω in 2-d and observed from a non-empty open subset ω ⊂ Ω. More precisely, for every real-valued bounded potential V, our main result shows that, when Ω has a finite number of holes, the observability constant of the elliptic operator −∆ + V, with domain H^2 ∩ H^1_0(Ω), is of the form C exp (C |V|^{1/2} \log^{1/2}(|V|)) where C is a positive constant depending only on Ω and ω. Our methodology of proof is crucially based on the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov, in the context of the Landis conjecture on exponential decay of solutions to homogeneous elliptic equations in the plane. The main difference and additional difficulty is that the zero set of the solutions to elliptic equations with source term can be very intricate and should be dealt with carefully. As a consequence of these new observability estimates, we obtain new results concerning control of semi-linear elliptic equations in the spirit of Fernández-Cara, Zuazua's open problem concerning small-time global null-controllability of slightly super-linear heat equations.

Published
2023

2. Index theorems for graph-parametrized optimal control problems

Author

Andrei Agrachev, Stefano Baranzini, Ivan Beschastnyi, Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies (SISSA / ISAS), 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), Universidade de Aveiro, and The work of the third author was supported through the CIDMA Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology ('FCT - Fundacao para a Ciencia e a Tecnologia') within the project UIDP/04106/2020.

Subjects
Mathematics - Differential Geometry, Differential Geometry (math.DG), Optimization and Control (math.OC), Applied Mathematics, FOS: Mathematics, General Physics and Astronomy, Statistical and Nonlinear Physics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Mathematical Physics
Abstract

In this paper we prove Morse index theorems for a big class of constrained variational problems on graphs. Such theorems are useful in various physical and geometric applications. Our formulas compute the difference of Morse indices of two Hessians related to two different graphs or two different sets of boundary conditions. Several applications such as the iteration formulas or lower bounds for the index are proved., 30 pages, 4 figures

Published
2023

3. 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

4. Consensus formation in first-order graphon models with time-varying topologies

Author

Bonnet, Benoît, Duteil, Nastassia Pouradier, Sigalotti, Mario, 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), Modelling and Analysis for Medical and Biological Applications (MAMBA), 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 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
Consensus Formation, Algebraic Connectivity, MSC2020 Subject Classification : 05C63, 05C90, 37L15, 93A16, Applied Mathematics, Multi-Agent Systems, Functional Analysis (math.FA), Mathematics - Functional Analysis, Optimization and Control (math.OC), Modeling and Simulation, FOS: Mathematics, 05C63, 05C90, 37L15, 93A16, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH]Mathematics [math], Mathematics - Optimization and Control, Scrambling Coefficient, Graphon Dynamics, Persistence Conditions
Abstract

In this article, we investigate the asymptotic formation of consensus for several classes of time-dependent cooperative graphon dynamics. After motivating the use of this type of macroscopic models to describe multi-agent systems, we adapt the classical notion of scrambling coefficient to this setting, leverage it to establish sufficient conditions ensuring the exponential convergence to consensus with respect to the $L^{\infty}$-norm topology. We then shift our attention to consensus formation expressed in terms of the $L^2$-norm, and prove three different consensus result for symmetric, balanced and strongly connected topologies, which involve a suitable generalisation of the notion of algebraic connectivity to this infinite-dimensional framework. We then show that, just as in the finite-dimensional setting, the notion of algebraic connectivity that we propose encodes information about the connectivity properties of the underlying interaction topology. We finally use the corresponding results to shed some light on the relation between $L^2$- and $L^{\infty}$-consensus formation, and illustrate our contributions by a series of numerical simulations., Comment: 48 pages, 16 figures

Published
2022

5. Global stabilization of sterile insect technique model by feedback laws

Author

Bidi, Kala Agbo, Almeida, Luis, Coron, Jean-Michel, 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)), 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
[SDV.EE.SANT]Life Sciences [q-bio]/Ecology, environment/Health, Sterile Insect Technique, Mosquito population control, Dynamical control system, Pest control, Optimization and Control (math.OC), FOS: Mathematics, Lyapunov global stabilization, Vector borne disease, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 93D15, 93D20, 92D25, Feedback design, Mathematics - Optimization and Control, Backstepping feedback
Abstract

The Sterile Insect Technique or SIT is presently one of the most ecological methods for controlling insect pests responsible for disease transmission or crop destruction worldwide. This technique consists of releasing sterile males into the insect pest population. This approach aims at reducing fertility in the population and, consequently, reduce significantly the native insect population after a few generations. In this work, we study the global stabilization of a pest population at extinction equilibrium by the SIT method. We construct explicit feedback laws that stabilize the model and do numerical simulations to show the efficiency of our feedback laws. The different feedback laws are also compared taking into account their possible implementation in field interventions.

Published
2023

6. Existence of optimal domains for the helicity maximisation problem among domains satisfying a uniform ball condition

Author

Gerner, Wadim, 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), and This work has been supported by the Inria AEX StellaCage.

Subjects
Plasma fusion, FOS: Physical sciences, Mathematical Physics (math-ph), 35Q31, 35Q35, 35Q85, 49J35, 49Q10, 76W05, Mathematics - Spectral Theory, Magnetohydrodynamics, Helicity, 2020MSC: 35Q31, 35Q35, 35Q85, 49J35, 49Q10, 76W05, [PHYS.PHYS.PHYS-PLASM-PH]Physics [physics]/Physics [physics]/Plasma Physics [physics.plasm-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Spectral Theory (math.SP), Isoperimetric problems, Mathematical Physics
Abstract

In the present work we present a general framework which guarantees the existence of optimal domains for isoperimetric problems within the class of $C^{1,1}$-regular domains satisfying a uniform ball condition as long as the desired objective function satisfies certain properties. We then verify that the helicity isoperimetric problem studied by Cantarella, DeTurck, Gluck and Teytel in \cite{CDGT002} satisfies the conditions of our framework and hence establish the existence of optimal domains within the given class of domains. We additionally use the same framework to prove the existence of optimal domains among uniform $C^{1,1}$-domains for a first curl eigenvalue problem which has been studied recently for other classes of domains in \cite{EGPS23}., 11 pages, comments welcome!

Published
2023

7. Observability estimates for the Schr{\'o}dinger equation in the plane with periodic bounded potentials from measurable sets

Author

Balc'H, Kévin Le, Martin, Jérémy, 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)), 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
Strichartz estimates, Observability inequalities, Mathematics - Analysis of PDEs, Optimization and Control (math.OC), Semiclassical analysis, Floquet-Bloch theory, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Schrödinger equation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract

The goal of this article is to obtain observability estimates for Schr{\"o}dinger equations in the plane R 2. More precisely, considering a 2$\pi$Z 2-periodic potential V $\in$ L $\infty$ (R 2), we prove that the evolution equation i$\partial$tu = --$\Delta$u + V (x)u, is observable from any 2$\pi$Z 2-periodic measurable set, in any small time T > 0. We then extend Ta{\"u}ffer's recent result [T{\"a}u22] in the two-dimensional case to less regular observable sets and general bounded periodic potentials. The methodology of the proof is based on the use of the Floquet-Bloch transform, Strichartz estimates and semiclassical defect measures for the obtention of observability inequalities for a family of Schr{\"o}dinger equations posed on the torus R 2 /2$\pi$Z 2 .

Published
2023

8. Embedding the Grushin Cylinder in ${\bf R}^3$and Schroedinger evolution

Author

Beschastnyi, Ivan, Boscain, Ugo, Cannarsa, Daniele, Pozzoli, Eugenio, Centro de Investigaçao e Desenvolvimento em Matematica e Aplicaçoes (CIDMA), Universidade de Aveiro, 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), Department of Mathematics and Statistics [Jyväskylä Univ] (JYU), University of Jyväskylä (JYU), Università degli studi di Bari Aldo Moro = University of Bari Aldo Moro (UNIBA), Academy of Finland, grant 322898 'Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory', STARS Consolidator Grant 2021 'NewSRG' of the University of Padova, PNRR MUR project PE0000023-NQSTI, UIDP/04106/2020, UIDB/04106/2020, and ANR-22-CE92-0077,CoRoMo,Contrôle quantique performante des rotations moléculaires -- temps et controllabilité(2022)

Subjects
Mathematics - Functional Analysis, Mathematics - Differential Geometry, Grushin, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, FOS: Physical sciences, Mathematical Physics (math-ph), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Mathematical Physics, Functional Analysis (math.FA)
Abstract

We consider the evolution of a free quantum particle on the Grushin cylinder, under different type of quantizations. In particular we are interested to understand if the particle can cross the singular set, i.e., the set where the structure is not Riemannian. We consider intrinsic and extrinsic quantizations, where the latter are obtained by embedding the Grushin structure isometrically in ${\bf R}^3$ (with singularities). As a byproduct we provide formulas to embed the Grushin cylinder in ${\bf R^3}$ that could be useful for other purposes. Such formulas are not global, but permit to study the embedding arbitrarily close to the singular set. We extend these results to the case of $\alpha $-Grushin cylinders., Comment: 16 pages, 8 figures

Published
2023

9. 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

10. Geometric and probabilistic results for the observability of the wave equation

Author

Emmanuel Humbert, Yannick Privat, Emmanuel Trélat, Institut Denis Poisson (IDP), Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO), Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), TOkamaks and NUmerical Simulations (TONUS), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est, 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), 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)-Université de Tours (UT)-Université d'Orléans (UO), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Université d'Orléans (UO)-Université de Tours (UT)-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.), 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 d'Orléans (UO)-Université de Tours-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
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], observability, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], General Mathematics, random set, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], wave equation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Riemannian geometry
Abstract

Given any measurable subset $\omega$ of a closed Riemannian manifold $(M,g)$ and given any $T>0$, we define $\ell^T(\omega)\in[0,1]$ as the smallest average time over $[0,T]$ spent by all geodesic rays in $\omega$. This quantity appears naturally when studying observability properties for the wave equation on $M$, with $\omega$ as an observation subset: the condition $\ell^T(\omega)>0$ is the well known \emph{Geometric Control Condition}.In this article we establish two properties of the functional $\ell^T$, one is geometric and the other is probabilistic.The first geometric property is on the maximal discrepancy of $\ell^T$ when taking the closure. We may have $\ell^T(\mathring{\omega})1/2$ then the Geometric Control Condition is satisfied and thus the wave equation is observable on $\omega$ in time $T$. The second property is of probabilistic nature. We take $M=\mathbb{T}^2$, the flat two-dimensional torus, and we consider a regular grid on it, a regular checkerboard made of $n^2$ square white cells. We construct random subsets $\omega_\varepsilon^n$ by darkening each cell in this grid with a probability $\varepsilon$. We prove that the random law $\ell^T(\omega_\varepsilon^n)$ converges in probability to $\varepsilon$ as $n\rightarrow+\infty$.As a consequence, if $n$ is large enough then the Geometric Control Condition is satisfied almost surely and thus the wave equation is observable on $\omega_\varepsilon^n$ in time $T$.

Published
2022

11. Convergence in nonlinear optimal sampled-data control problems

Author

Bourdin, Loïc, Trélat, Emmanuel, XLIM (XLIM), 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 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)), 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
Optimization and Control (math.OC), Sampled-data control, Pontryagin maximum principle, FOS: Mathematics, Filippov approach, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Convergence, Mathematics - Optimization and Control
Abstract

Consider, on the one part, a general nonlinear finite-dimensional optimal control problem and assume that it has a unique solution whose state is denoted by $x^*$. On the other part, consider the sampled-data control version of it. Under appropriate assumptions, we prove that the optimal state of the sampled-data problem converges uniformly to $x^*$ as the norm of the corresponding partition tends to zero. Moreover, applying the Pontryagin maximum principle to both problems, we prove that, if $x^*$ has a unique weak extremal lift with a costate $p$ that is normal, then the costate of the sampled-data problem converges uniformly to $p$. In other words, under a nondegeneracy assumption, control sampling commutes, at the limit of small partitions, with the application of the Pontryagin maximum principle.

Published
2023

12. An algorithmic guide for finite-dimensional optimal control problems

Author

Caillau, Jean-Baptiste, Ferretti, Roberto, Trélat, Emmanuel, Zidani, Hasnaa, Caillau, J. -B., Ferretti, R., Trelat, E., Zidani, H., Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), 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), Dipartimento di Matematica e Fisica [Roma], Università degli Studi Roma Tre = Roma Tre University (ROMA TRE), 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), Laboratoire de Mathématiques de l'INSA de Rouen Normandie (LMI), Institut national des sciences appliquées Rouen Normandie (INSA Rouen Normandie), and Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)-Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)

Subjects
Direct and indirect methods, Optimistic planning algorithms, Pontryagin maximum principle, Dynamical systems, Direct and indirect method, Optimistic planning, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Dynamical system, Hamilton-Jacobi-Bellman equation, Optimal control
Abstract

International audience; We survey the main numerical techniques for finite-dimensional nonlinear optimal control. The chapter is written as a guide to practitioners who wish to get rapidly acquainted with the main numerical methods used to efficiently solve an optimal control problem. We consider two classical examples, simple but significant enough to be enriched and generalized to other settings: Zermelo and Goddard problems. We provide sample of the codes used to solve them and make these codes available online. We discuss direct and indirect methods, Hamilton-Jacobi approach, ending with optimistic planning. The examples illustrate the pros and cons of each method, and we show how these approaches can be combined into powerful tools for the numerical solution of optimal control problems for ordinary differential equations.

Published
2023

13. Quantum Limits on product manifolds

Author

Humbert, Emmanuel, Privat, Yannick, Trélat, Emmanuel, Institut Denis Poisson (IDP), Université d'Orléans (UO)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-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), and The first author is supported by the project THESPEGE (APR IA), Région Centre-Val de Loire, France, 2018-2020.

Subjects
Mathematics - Spectral Theory, General Mathematics, FOS: Mathematics, Mathematics::Differential Geometry, Mathematics::Geometric Topology, Mathematics::Symplectic Geometry, Spectral Theory (math.SP), [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

International audience; We establish some properties of quantum limits on a product manifold, proving for instance that, under appropriate assumptions, the quantum limits on the product of manifolds are absolutely continuous if the quantum limits on each manifolds are absolutely continuous. On a product of Riemannian manifolds satisfying the minimal multiplicity property, we prove that a periodic geodesic can never be charged by a quantum limit.

Published
2023

14. Catching all geodesics of a manifold with moving balls and application to controllability of the wave equation

Author

Cyril Letrouit, 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), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU), 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), École normale supérieure - Paris (ENS Paris)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Universités, 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), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), 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), Sorbonne Université, and École normale supérieure - Paris (ENS Paris)

Subjects
Geodesic, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Regular polygon, Dynamical Systems (math.DS), Riemannian manifold, Absolute continuity, Wave equation, Domain (mathematical analysis), Manifold, Theoretical Computer Science, Combinatorics, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), Optimization and Control (math.OC), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Ball (mathematics), Mathematics::Differential Geometry, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Dynamical Systems, Mathematics - Optimization and Control, Analysis of PDEs (math.AP), Mathematics
Abstract

We address the problem of catching all speed $1$ geodesics of a Riemannian manifold with a moving ball: given a compact Riemannian manifold $(M,g)$ and small parameters $\varepsilon>0$ and $v>0$, is it possible to find $T>0$ and an absolutely continuous map $x:[0,T]\rightarrow M, t\mapsto x(t)$ satisfying $\|\dot{x}\|_{\infty}\leq v$ and such that any geodesic of $(M,g)$ traveled at speed $1$ meets the open ball $B_g(x(t),\varepsilon)\subset M$ within time $T$? Our main motivation comes from the control of the wave equation: our results show that the controllability of the wave equation can sometimes be improved by allowing the domain of control to move adequately, even very slowly. We first prove that, in any Riemannian manifold $(M,g)$ satisfying a geodesic recurrence condition (GRC), our problem has a positive answer for any $\varepsilon>0$ and $v>0$, and we give examples of Riemannian manifolds $(M,g)$ for which (GRC) is satisfied. Then, we build an explicit example of a domain $X\subset\mathbb{R}^2$ (with flat metric) containing convex obstacles, not satisfying (GRC), for which our problem has a negative answer if $��$ and $v$ are small enough, i.e., no sufficiently small ball moving sufficiently slowly can catch all geodesics of $X$., Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore In press

Published
2023

15. 'Good Lie Brackets' for Control Affine Systems

Author

Agrachev, Andrei, Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies (SISSA / ISAS), 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), and Chaire Internationale Inria

Subjects
Optimization and Control (math.OC), FOS: Mathematics, [MATH]Mathematics [math], Mathematics - Optimization and Control
Abstract

We consider a smooth system of the form $\dot q=f_0(q)+\sum\limits_{i=1}^k u_i f_i(q)$, $q\in M,\ u_i\in\mathbb R,$ and study controllability issues on the group of diffeomorphisms of $M$. It is well-known that the system can arbitrarily well approximate the movement in the direction of any Lie bracket polynomial of $f_1,\ldots,f_k$. Any Lie bracket polynomial of $f_1,\ldots,f_k$ is good in this sense. Moreover, some combinations of Lie brackets which involve the drift term $f_0$ are also good but surely not all of them. In this paper we try to characterize good ones and, in particular, all universal good combinations, which are good for any nilpotent truncation of any system.

Published
2023

16. Null-controllability of cascade reaction-diffusion systems with odd coupling terms

Author

Le Balc'H, Kévin, Takahashi, Takéo, 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), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Parabolic system, Nonlinear coupling, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Carleman estimate, 2020 Mathematics Subject Classification. 35K45, 35K58, 93B05, 93C10, Null-controllability
Abstract

In this paper, we consider a nonlinear system of two parabolic equations, with a distributed control in the first equation and an odd coupling term in the second one. We prove that the nonlinear system is smalltime locally null-controllable. The main difficulty is that the linearized system is not null-controllable. To overcome this obstacle, we extend in a nonlinear setting the strategy introduced in [LB19] that consists in constructing odd controls for the linear heat equation. The proof relies on three main steps. First, we obtain from the classical L 2 parabolic Carleman estimate, conjugated with maximal regularity results, a weighted L p observability inequality for the nonhomogeneous heat equation. Secondly, we perform a duality argument, close to the well-known Hilbert Uniqueness Method in a reflexive Banach setting, to prove that the heat equation perturbed by a source term is null-controllable thanks to odd controls. Finally, the nonlinearity is handled with a Schauder fixed-point argument.

Published
2022

17. Approximate control of parabolic equations with on-off shape controls by Fenchel duality

Author

Pouchol, Camille, Trélat, Emmanuel, Zhang, Christophe, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-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), 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), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020)

Subjects
Bang-bang controls, Constrained control, Optimal control, Mathematics - Analysis of PDEs, Optimization and Control (math.OC), Fenchel-Rockafellar duality, FOS: Mathematics, [INFO.INFO-SY]Computer Science [cs]/Systems and Control [cs.SY], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Linear parabolic equations, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Shapes, Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract

We consider the internal control of linear parabolic equations through on-off shape controls, i.e., controls of the form $M(t)\chi_{\omega(t)}$ with $M(t) \geq 0$ and $\omega(t)$ with a prescribed maximal measure. We establish small-time approximate controllability towards all possible final states allowed by the comparison principle with nonnegative controls. We manage to build controls with constant amplitude $M(t) \equiv M$. In contrast, if the moving control set $\omega(t)$ is confined to evolve in some region of the whole domain, we prove that approximate controllability fails to hold for small times. The method of proof is constructive. Using Fenchel-Rockafellar duality and the bathtub principle, the on-off shape control is obtained as the bang-bang solution of an optimal control problem, which we design by relaxing the constraints. Our optimal control approach is outlined in a rather general form for linear constrained control problems, paving the way for generalisations and applications to other PDEs and constraints.

Published
2022

18. Spectral asymptotics for sub-Riemannian Laplacians

Author

de Verdìère, Yves Colin, Hillairet, Luc, Trélat, Emmanuel, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Institut Denis Poisson (IDP), Université d'Orléans (UO)-Université de Tours (UT)-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)), 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
Mathematics - Differential Geometry, Metric Geometry (math.MG), Sub-Riemannian Laplacian, Sub-Riemannian geometry, Mathematics - Spectral Theory, Differential Geometry (math.DG), Mathematics - Metric Geometry, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], FOS: Mathematics, Weyl laws, Spectral Theory (math.SP), Spectral theory, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

We study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. In the equiregular case, we infer the local and microlocal Weyl law, putting in light the Weyl measure in sR geometry. This measure coincides with the Popp measure in low dimension but differs from it in general. We prove that spectral concentration occurs on the shief generated by Lie brackets of length r-1, where r is the degree of nonholonomy. In the singular case, like Martinet or Grushin, the situation is more involved but we obtain small-time asymptotic expansions of the heat kernel and the Weyl law in some cases. Finally, we give the Weyl law in the general singular case, under the assumption that the singular set is stratifiable.

Published
2022

19. Small-time global null controllability of generalized Burgers' equations

Author

Rémi Robin, 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)), 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
Computational Mathematics, Control and Optimization, Mathematics - Analysis of PDEs, Optimization and Control (math.OC), Control and Systems Engineering, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract

In this paper, we study the small-time global null controllability of the generalized Burgers’ equations yt + γ|y|γ-1 yx — yxx = u(t) on the segment [0, 1]. The scalar control u(t) is uniform in space and plays a role similar to the pressure in higher dimension. We set a right Dirichlet boundary condition y(t, 1) = 0, and allow a left boundary control y(t, 0) = v(t). Under the assumption γ > 3/2 we prove that the system is small-time globally null controllable. Our proof relies on the return method and a careful analysis of the shape and dissipation of a boundary layer.

Published
2022

20. 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

21. Stability criteria for singularly perturbed linear switching systems

Author

Chitour, Yacine, Haidar, Ihab, Mason, Paolo, Sigalotti, Mario, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Université Paris-Saclay, Laboratoire QUARTZ (QUARTZ ), Université Paris 8 Vincennes-Saint-Denis (UP8)-Ecole Nationale Supérieure de l'Electronique et de ses Applications (ENSEA)-ISAE-Supméca Institut Supérieur de Mécanique de Paris (ISAE-Supméca), 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)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)

Subjects
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

International audience; We present some results on the uniform exponential stability of singularly perturbed linear systems undergoing switching. In absence of dwell-time constraints, the switching parameter can evolve on the same time scale as the fast variables, or even faster. We investigate the effect of switching laws evolving at a time scale comparable with the fast variables, describing the corresponding asymptotic effect on the slow variable. Based on this analysis, we propose stability criteria for the overall system, uniform for small values of the parameter of singular perturbation.

Published
2022

22. Reachability results for perturbed heat equations

Author

Sylvain Ervedoza, Kévin Le Balc'h, Marius Tucsnak, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-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), ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020), ANR-10-IDEX-0003,IDEX BORDEAUX,Initiative d'excellence de l'Université de Bordeaux(2010), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), 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
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis
Abstract

International audience; This work studies the reachable space of infinite dimensional control systems which are null controllable in any positive time, the typical example being the heat equation controlled from the boundary or from an arbitrary open set. The focus is on the robustness of the reachable space with respect to linear or nonlinear perturbations of the generator. More precisely, our first main results asserts that this space is invariant under perturbations which are small (in an appropriate sense). A second main result asserts the invariance of the reachable space with respect to perturbations which are compact (again in an appropriate sense), provided that a Hautus type condition is satisfied. Moreover, our methods give precise information on the behavior of the reachable space when the generator is perturbed by a class of nonlinear operators. When applied to the classical heat equation, our results provide detailed information on the reachable space when the generator is perturbed by a small potential or by a class of non local operators, and in particular in one space dimension, we deduce from our analysis that the reachable space for perturbations of the 1-d heat equation is a space of holomorphic functions. We also show how our approach leads to reachability results for a class of semi-linear parabolic equations.

Published
2022

23. Contrôle et optimisation de systèmes physiques : application à la mécanique quantique et au confinement magnétique dans les stellarators

Author

Robin, Rémi, 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), Sorbonne Université (SU), Sorbone Université, Mario Sigalotti, and Ugo Boscain

Subjects
Couche limite, Quantum control, Optimisation de forme, Physique des plasmas, Equation de Burgers, Optimal control, Burgers equation, Pasma physics, Boundary layer, Shape optimization, Ensemble controlability, Contrôle optimal, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], qubit, Stellarator, Contrôle quantique, Contrôlabilité d’ensemble
Abstract

This PhD manuscript deals with the optimization and control of several physical systems. It is divided into three parts.The first part is devoted to stellarators. This type of nuclear fusion reactor poses many challenges related to optimization. We focus on an inverse problem well known to physicists, modeling the optimal design of superconducting coils generating a given magnetic field. We conduct both a theoretical and a numerical study of an extension of this problem, involving shape optimization. Then, we develop a new method to prove the existence of optimal shapes in the case of hypersurface optimization problems. Finally, we study and optimize the Laplace forces acting on a current surface density. The second part of this manuscript deals with the control of finite dimensional quantum systems. We rigorously study the combination of the rotating wave approximation with the adiabatic approximation. First, we obtain the robustness of a population transfer method on qubits. The latter then allows to extend results of Li and Khaneja on the ensemble control of qubits by restricting to the use of a single control. We also present a second contribution, devoted to the analysis of a chattering phenomenon for an optimal control problem of a quantum system. Finally, the third part is dedicated to the proof of a small-time global null controllability result for generalized Burgers' equations using a boundary layer.; Cette thèse porte sur l’optimisation et le contrôle de plusieurs systèmes physiques : elle est composée de trois parties.La première partie est consacrée aux stellarators. Ce type de réacteur à fusion nucléaire pose de nombreux défis reliés à l’optimisation. Nous nous sommes concentrés sur un problème inverse bien connu des physiciens, modélisant la conception optimale de bobines supraconductrices générant un champ magnétique donné. Nous avons conduit une étude théorique et numérique d’une extension de ce problème, portant sur une optimisation de forme. Nous avons ensuite développé une nouvelle méthode afin de prouver l’existence de formes optimales dans le cas de problèmes d’optimisation d’hypersurfaces. Nous avons enfin effectué l’étude et l’optimisation des forces de Laplace s’exerçant sur une densité surfaciquede courant.La deuxième partie porte ensuite sur l’étude du contrôle de systèmes quantiques de dimension finie. Nous avons étudié rigoureusement la combinaison de l’approximation de l’onde tournante avec l’approximation adiabatique. Dans un premier temps, nous avons obtenu la robustesse des méthodes de transfert de population sur les qubits. Cette dernière permet alors d’étendre des résultats de Li et Khaneja sur le contrôle d’ensemble des qubits en se restreignant à l’utilisation d’un seul contrôle. Nous présentons égallement une seconde contribution, consacrée à l’analyse d’un phénomène de chattering pour un problème de contrôle optimal d’un système quantique.Enfin, la troisième partie est dédiée à la preuve d’un résultat de contrôlabilité à zéro en temps petit pour des équations de Burgers généralisées grâce à l’utilisation d’une couche limite.

Published
2022

24. Application of optimal control techniques to natural systems management

Author

Molina, Emilio, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Universidad de Chile = University of Chile [Santiago] (UCHILE), 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), This thesis was partially supported by ANID-PFCHA/Doctorado Nacional/2018-21180348, FONDECYTgrant 1201982 and Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for center of excellence, all of them from ANID (Chile), and Inria Cage team., Sorbonne University , UPMC, Universidad de Chile. Facultad de ciencias físicas y matemáticas, Mario Sigalotti, and Héctor Ramírez

Subjects
Epidemiology, Gestion du système naturel, Contrôle optimal, Exploitation minière, Epidémiologie, [MATH]Mathematics [math], Natural system management, Mining, Optimal control
Abstract

Optimal control techniques have numerous applications in engineering and real world problems. This thesis is devoted to using these techniques in two contexts, mining and epidemiology, dividing this document in two respective parts. In the first part related to mining, we work with the continuous formulation of the Open Pit Problem consisting of finding the optimal shape of an opencast mine representing its profile by a continuous function. Optimality in this context corresponds to maximizing the profit of mineral extraction. We introduce for the first time optimal control models of this problem. We present optimality conditions of solutions along with numerical experiments using local and global methods.Another relevant problem in this context corresponds to the Sequential version of the Open Pit Problem, which consists of scheduling an extraction program over consecutive time frames (for example, a profile each 6 months), finding nested profiles maximizing a discounted profit. We proposed a novel semi-continuous model to obtain solutions of the sequential problem and we use it to present for the first time, to the best of our knowledge, numerical solutions of a three dimensional case (a possible real world mine) including the original Open Pit Problem (case with a single time-frame).In the second part we deal with optimal control problems minimizing the maximum value of a state. This problematic was inspired by Covid-19, where hospitals and ICU beds were overcrowded due to a high amount of simultaneous infections. We present four different reformulations of this kind of optimal control problem as a Mayer one, each one having its pros and cons. We present the numerical performance of each formulation in an academic example and in a more realistic SIR model where the problem corresponds to minimizing the peak of infectious compartment with integral constraint in the control. With respect to the latter problem, we prove analytically that the structure of the optimal control is null-singular-null and we used it to assess numerical solutions.; Les techniques de contrôle optimale ont plusieurs applications dans le domaines de l'engenierie et les problèmes de la vie pratique. Cette thèse est consacrée a l'utilitation de ces techniques dans deux contextes, l’exploitation minière et l’épidémiologie, divisant ce document en deux parties respectives.Dans la première partie relative à l’exploitation minière, nous travaillons avec la formulation continue du problème Final Open Pit consistant à trouver la forme optimale d’une mine à ciel ouvert représentant son profil par une fonction continue. L’optimalité dans ce contexte correspond à maximiser le bénéfice de l’extraction minérale. Nous présentons pour la première fois des modèles de contrôle optimal pour ce problème. Nous présentons des conditions optimales de solutions avec des simulations numériques utilisant des méthodes locales et globales.La version séquentielle du Final Open Pit est également très pertinent dans le contexte de cette thèse. Le problème consiste à planifier un programme d’extraction sur des périodes de temps consécutives (par exemple, un profil tous les 6 mois), en trouvant des profils imbriqués maximisant un bénéfice actualisé. Nous avons formulé un nouveau modèle semi-continu permettant d'obtenir des solutions du problème séquentiel et que nous utilisons pour proposer des solutions numériques dans un cas tridimensionnelle ( une mine possible du monde réel) y compris le problème Open Pit original (cas avec une seule période de temps). Cela, à notre connaissance, n'avait jamais été fait auparavant.Dans la deuxième partie, nous traitons les problèmes de contrôle optimal minimisant la valeur maximale d’un état. Ce problème a été inspiré par le contexte sanitaire très difficile suite a la covid-19, où avoir de beaucoup infectés en même temps sature les hôpitaux et les lits de soin intensif. Nous présentons quatre reformulations différentes de ce type de problème comme un Mayer de contrôle optimal, ayant chacun ses avantages et ses inconvénients. Nous évaluons la performance numérique de chaque formulation dans un exemple académique et dans un modèle SIR plus réaliste où le problème correspond à minimiser le pic du compartiment infecté avec une contrainte intégrale dans le contrôle. En ce qui concerne ce dernier problème, nous prouvons analytiquement que la structure du contrôle optimal est null-singular-null et nous l’avons utilisé pour évaluer les solutions numériques.

Published
2022

25. Small-time asymptotics of hypoelliptic heat kernels near the diagonal, nilpotentization and related results

Author

Yves Colin de Verdière, Emmanuel Trélat, Luc Hillairet, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Institut Denis Poisson (IDP), Centre National de la Recherche Scientifique (CNRS)-Université de Tours (UT)-Université d'Orléans (UO), 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), ANR-15-CE40-0018,SRGI,Géométrie sous-Riemannienne et Interactions(2015), Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO), Université d'Orléans (UO)-Université de Tours (UT)-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
Nilpotentization, Pure mathematics, Mathematics::Complex Variables, 010102 general mathematics, Diagonal, Mathematics::Analysis of PDEs, Structure (category theory), Boundary (topology), Ocean Engineering, Sub-Riemannian Laplacian, Mathematics::Spectral Theory, 01 natural sciences, Mathematics - Analysis of PDEs, Small-time asymptotics, Hypoelliptic heat kernel, Hypoelliptic operator, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 010307 mathematical physics, 0101 mathematics, Smoothing, Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; We establish small-time asymptotic expansions for heat kernels of hypoelliptic Hörmander operators in a neighborhood of the diagonal, generalizing former results obtained in particular by Métivier and by Ben Arous. The coefficients of our expansions are identified in terms of the nilpotentization of the underlying sub-Riemannian structure. Our approach is purely analytic and relies in particular on local and global subelliptic estimates as well as on the local nature of small-time asymptotics of heat kernels. The fact that our expansions are valid not only along the diagonal but in an asymptotic neighborhood of the diagonal is the main novelty, useful in view of deriving Weyl laws for subelliptic Laplacians. In turn, we establish a number of other results on hypoelliptic heat kernels that are interesting in themselves, such as Kac's principle of not feeling the boundary, asymptotic results for singular perturbations of hypoelliptic operators, global smoothing properties for selfadjoint heat semigroups.

Published
2021

26. Point interactions for 3D sub-Laplacians

Author

Riccardo Adami, Valentina Franceschi, Dario Prandi, Ugo Boscain, Politecnico di Torino = Polytechnic of Turin (Polito), 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), Dipartimento di Matematica 'Tullio Levi-Civita', Universita degli Studi di Padova, Université Paris-Saclay, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), The authors acknowledge that the present research is partially supported by: MIUR Grant Dipartimenti di Eccellenza (2018-2022) E11G18000350001, ANR-15-CE40-0018 projectSRGI - Sub-Riemannian Geometry and Interactions, ANR-17-CE40-0007 pojectQUACO - Contrôle quantique: systèmes d’EDPs et applications à l’IRM, G.N.A.M.P.A. projectProblemi isoperi-metrici in spazi Euclidei e non. The third author acknowledges the support received from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant No 794592., ANR-15-CE40-0018,SRGI,Géométrie sous-Riemannienne et Interactions(2015), Politecnico di Torino [Torino] (Polito), 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), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Sud - Paris 11 (UP11), 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 Università degli Studi di Padova = University of Padua (Unipd)

Subjects
Mathematics - Differential Geometry, Pure mathematics, Structure (category theory), Heisenberg group, 01 natural sciences, 010104 statistics & probability, symbols.namesake, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Essential self-adjointness, Point interactions, Rotation of molecules, Sub-Laplacian, Sub-Riemannian geometry, Point (geometry), 0101 mathematics, Mathematical Physics, Physics, Applied Mathematics, Operator (physics), 010102 general mathematics, Mathematics::Spectral Theory, Riemannian manifold, 16. Peace & justice, Manifold, Differential Geometry (math.DG), symbols, Center of mass, Analysis, Schrödinger's cat, Analysis of PDEs (math.AP)
Abstract

In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point q 0 ∈ M exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on C 0 ∞ ( M ∖ { q 0 } ) is essentially self-adjoint in L 2 ( M ) . A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold N, whose associated Laplace-Beltrami operator acting on C 0 ∞ ( N ∖ { q 0 } ) is never essentially self-adjoint in L 2 ( N ) , if dim ⁡ N ≤ 3 . We then apply this result to the Schrodinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.

Published
2021

27. On universal classes of Lyapunov functions for linear switched systems

Author

Mason, Paolo, Chitour, Yacine, Sigalotti, Mario, 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), and This research was partially supported by the iCODE institute, research project of theIdex Paris-Saclay.

Subjects
Optimization and Control (math.OC), FOS: Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control
Abstract

In this paper we discuss the notion of universality for classes of candidate common Lyapunov functions of linear switched systems. On the one hand, we prove that a family of absolutely homogeneous functions is universal as soon as it approximates arbitrarily well every convex absolutely homogeneous function for the $C^0$ topology of the unit sphere. On the other hand, we prove several obstructions for a class to be universal, showing, in particular, that families of piecewise-polynomial continuous functions whose construction involves at most $l$ polynomials of degree at most $m$ (for given positive integers $l,m$) cannot be universal.

Published
2022

28. ct: control toolbox - Numerical tools and examples in optimal control

Author

Jean-Baptiste Caillau, Olivier Cots, Pierre Martinon, Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), 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), Université Côte d'Azur (UCA), Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), 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)), 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
Differential continuation, Direct transcription method, Control and Systems Engineering, Automatic differentiation, Bocop, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Shooting, Nutopy
Abstract

International audience; Combining direct and indirect methods to have the best of both worlds is an efficient method to solve numerically optimal control problems. A direct solver will typically provide information on the structure of the optimal control, allowing an educated guess for indirect shooting. The control toolbox ct offers such possibilities and is presented on two examples. The first example has a bang-singular solution and is solved by chaining direct and indirect solvers. The second one consists in computing conjugate and cut loci on an ellipsoid of revolution, which is performed using a more advanced combination of indirect methods with differential continuation.

Published
2022

29. Numerics for finite-dimensional optimal control problems

Author

Caillau, Jean-Baptiste, Ferretti, Roberto, Trélat, Emmanuel, Zidani, Hasnaa, Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Université Côte d'Azur (UCA), 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), Università degli Studi Roma Tre = Roma Tre University (ROMA TRE), 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), Sorbonne Université (SU), Laboratoire de Mathématiques de l'INSA de Rouen Normandie (LMI), Institut national des sciences appliquées Rouen Normandie (INSA Rouen Normandie), and Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)-Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)

Subjects
Direct and indirect methods, Pontryagin maximum principle, Optimistic planning, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Hamilton-Jacobi-Bellman equation, Optimal control
Abstract

We survey on numerics for finite-dimensional nonlinear optimal control. The chapter is written as a guide to practitioners who wish to get rapidly acquainted with the main numerical methods used to efficiently solve an optimal control problem. We consider throughout two classical examples, quite simple but representative enough to be complexified and generalized to other problems: the Zermelo and the Goddard problems. We provide their solving codes that are available on the web and make the point on the most up-to-date and efficient methods existing nowadays. We range on direct and indirect methods, on Hamilton-Jacobi approaches and we end with optimistic planning. Our examples illustrate the pros and cons of the methods and we also show how those various approaches can be combined in view of augmenting the efficiency of the numerical solving.

Published
2022

30. Lie algebra for rotational subsystems of a driven asymmetric top

Author

E Pozzoli, M Leibscher, M Sigalotti, U Boscain, C P Koch, 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), Institut für Theoretische Physik, Universität Kassel [Kassel], Fachbereich Physik [Freie Univeristät Berlin] | Department of Physics [Freie Univeristät Berlin], Freie Universität Berlin, Dahlem Center for Complex Quantum Systems & Fachbereich Physik, We gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft through CRC 1319 ELCH and from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement Nr. 765267 (QuSCo). MS and UB also thank the ANR projects SRGI ANR-15-CE40-0018 and Quaco ANR-17-CE40-0007-01., 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), European Project: 765267,H2020-EU.1.3.1.,QuSCo(2017), 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 Fachbereich Physik [Berlin]

Subjects
Statistics and Probability, Quantum Physics, molecular rotation, Lie algebra, Molekülrotation, FOS: Physical sciences, General Physics and Astronomy, Statistical and Nonlinear Physics, controllability, Lie-Algebra, Modeling and Simulation, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Steuerbarkeit, Quantum Physics (quant-ph), Mathematical Physics
Abstract

We present an analytical approach to construct the Lie algebra of finite-dimensional subsystems of the driven asymmetric top rotor. Each rotational level is degenerate due to the isotropy of space, and the degeneracy increases with rotational excitation. For a given rotational excitation, we determine the nested commutators between drift and drive Hamiltonians using a graph representation. We then generate the Lie algebra for subsystems with arbitrary rotational excitation using an inductive argument., 10 pages, 7 figures

Published
2022
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31. Equivalent formulations of the minmum peak problem and applications

Author

Molina, Emilio, Rapaport, Alain, Ramirez Cabrera, Hector, Center for Mathematical Modeling (CMM), Universidad de Chile = University of Chile [Santiago] (UCHILE), Departamento de Ingeniería Matemática [Santiago] (DIM), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut Agro Montpellier, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), 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), ANID-PFCHA/Doctorado Nacional/2018-21180348 (Chile), FONDECYT grant 1201982 (Chile), ACE210010 and FB210005, BASAL funds for center of excellence, ANID Chile, and Rapaport, Alain

Subjects
[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

International audience

Published
2022

32. On the pole placement of scalar linear delay systems with two delays

Author

Sébastien Fueyo, Guilherme Mazanti, Islam Boussaada, Yacine Chitour, Silviu-Iulian Niculescu, 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), Dynamical Interconnected Systems in COmplex Environments (DISCO), 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), Institut Polytechnique des Sciences Avancées (IPSA), Laboratoire des signaux et systèmes (L2S), and CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)

Subjects
Control and Optimization, Control and Systems Engineering, Applied Mathematics, FOS: Mathematics, Multiplicity-induced-dominancy (MID), Dynamical Systems (math.DS), Delay dynamics, [MATH]Mathematics [math], Mathematics - Dynamical Systems, Pole placement
Abstract

This paper concerns some spectral properties of the scalar dynamical system defined by a linear delay-differential equation with two positive delays. More precisely, the existing links between the delays and the maximal multiplicity of the characteristic roots are explored, as well as the dominance of such roots compared with the spectrum localization. As a by-product of the analysis, the pole placement issue is revisited with more emphasis on the role of the delays as control parameters in defining a partial pole placement guaranteeing the closed-loop stability with an appropriate decay rate of the corresponding dynamical system.

Published
2022

33. Optimization of vaccination for COVID-19 in the midst of a pandemic

Author

Qi Luo, Ryan Weightman, Sean T. McQuade, Mateo Díaz, Emmanuel Trélat, William Barbour, Dan Work, Samitha Samaranayake, Benedetto Piccoli, Clemson University, Center for Computational and Integrative Biology [Camden] (CCIB), Rutgers University [Camden], Rutgers University System (Rutgers)-Rutgers University System (Rutgers), Department of Computing and Mathematical sciences, California Institute of Technology (CALTECH), 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), Vanderbilt University [Nashville], Cornell University [New York], and The authors acknowledge the support of the NSF CMMI project # 2033580 'Managing pan-demic by managing mobility'. R.W., S.T.M. and B.P. acknowledge the support of the Joseph and Loretta Lopez Chair endowment.

Subjects
Statistics and Probability, SARS-CoV-2, Applied Mathematics, General Engineering, Populations and Evolution (q-bio.PE), COVID-19, 2020 Mathematics Subject Classification. Primary: 58F15, 58F17, Secondary: 53C35, Computer Science Applications, Optimal control, SEIR compartmental models, Optimization and Control (math.OC), FOS: Biological sciences, FOS: Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Quantitative Biology - Populations and Evolution, Vaccine, Mathematics - Optimization and Control
Abstract

During the Covid-19 pandemic a key role is played by vaccination to combat the virus. There are many possible policies for prioritizing vaccines, and different criteria for optimization: minimize death, time to herd immunity, functioning of the health system. Using an age-structured population compartmental finite-dimensional optimal control model, our results suggest that the eldest to youngest vaccination policy is optimal to minimize deaths. Our model includes the possible infection of vaccinated populations. We apply our model to real-life data from the US Census for New Jersey and Florida, which have a significantly different population structure. We also provide various estimates of the number of lives saved by optimizing the vaccine schedule and compared to no vaccination.

Published
2022

34. An optimal feedback control that minimizes the epidemic peak in the SIR model under a budget constraint

Author

Emilio Molina, Alain Rapaport, Rapaport, Alain, Departamento de Ingeniería Matemática [Santiago] (DIM), Universidad de Chile = University of Chile [Santiago] (UCHILE)-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 (UPC), 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 (UPC)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPC), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut Agro Montpellier, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANID-PFCHA/Doctorado Nacional/2018-21180348 (Chile), FONDECYT grant 1201982 (Chile), ACE210010 and FB210005, BASAL funds for center of excellence, ANID Chile, 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), Center for Mathematical Modeling (CMM), Universidad de Chile = University of Chile [Santiago] (UCHILE), and Departamento de Ingeniera Matematica [Santiago] (DIM)

Subjects
Maximum cost, Epidemiology, digestive, oral, and skin physiology, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Feedback control, Optimal control, [SDV.SPEE] Life Sciences [q-bio]/Santé publique et épidémiologie, Control and Systems Engineering, Optimization and Control (math.OC), FOS: Mathematics, Quantitative Biology::Populations and Evolution, [SDV.SPEE]Life Sciences [q-bio]/Santé publique et épidémiologie, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], SIR model, Electrical and Electronic Engineering, Mathematics - Optimization and Control
Abstract

International audience; We give the explicit solution of the optimal control problem which consists in minimizing the epidemic peak in the SIR model when the control is an attenuation factor of the infectious rate, subject to a L 1 constraint on the control which represents a budget constraint. The optimal strategy is given as a feedback control which consists in a singular arc maintaining the infected population at a constant level until the immunity threshold is reached, and no intervention outside the singular arc. We discuss and compare this strategy with the one that minimizes the peak when fixing the duration of a single intervention, as already proposed in the literature. Numerical simulations illustrate the benefits of the proposed control.

Published
2022

35. Constructive exact control of semilinear 1D wave equations by a least-squares approach

Author

Arnaud Münch, Emmanuel Trélat, Université Clermont Auvergne (UCA), Laboratoire de Mathématiques Blaise Pascal (LMBP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA), Sorbonne Université (SU), 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), ANR-16-IDEX-0001,CAP 20-25,CAP 20-25(2016), ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), 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
Least-squares approach, Control and Optimization, Mathematics - Analysis of PDEs, Optimization and Control (math.OC), Applied Mathematics, FOS: Mathematics, Semilinear wave equation, AMS Classifications:35Q30, 93E24, Exact controllability, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract

It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation $\partial_{tt}y-\partial_{xx}y + g(y)=f 1_{\omega}$, with Dirichlet boundary conditions, is exactly controllable in $H^1_0(0,1)\cap L^2(0,1)$ with controls $f\in L^2((0,1)\times(0,T))$, for any $T>0$ and any nonempty open subset $\omega$ of $(0,1)$, assuming that $g\in \mathcal{C}^1(\R)$ does not grow faster than $\beta\vert x\vert \ln^{2}\vert x\vert$ at infinity for some $\beta>0$ small enough. The proof, based on the Leray-Schauder fixed point theorem, is however not constructive. In this article, we design a constructive proof and algorithm for the exact controllability of semilinear 1D wave equations. Assuming that $g^\prime$ does not grow faster than $\beta \ln^{2}\vert x\vert$ at infinity for some $\beta>0$ small enough and that $g^\prime$ is uniformly H\"older continuous on $\R$ with exponent $s\in[0,1]$, we design a least-squares algorithm yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+s$ after a finite number of iterations., Comment: arXiv admin note: text overlap with arXiv:2010.14067

Published
2022

36. 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

37. Lack of local controllability for a water-tank system when the time is not large enough

Author

Coron, Jean-Michel, Nguyen, Hoai-Minh, Koenig, Armand, 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), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), ANR-10-IDEX-0001,PSL,Paris Sciences et Lettres(2010), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Koenig, Armand, and Initiative d'excellence - Paris Sciences et Lettres - - PSL2010 - ANR-10-IDEX-0001 - IDEX - VALID

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]
Abstract

We consider the small-time local controllability property of a water tank modeled by 1D Saint-Venant equations, where the control is the acceleration of the tank. It is known from the work of Dubois et al. that the linearized system is not controllable. Moreover, concerning the linearized system, they showed that a traveling time * is necessary to bring the tank from one position to another for which the water is still at the beginning and at the end. Concerning the nonlinear system, Coron showed that local controllability around equilibrium states holds for a time large enough. In this paper, we show that for the local controllability of the nonlinear system around the equilibrium states, the necessary time is at least 2 * even for the tank being still at the beginning and at the end. The key point of the proof is a coercivity property for the quadratic approximation of the water-tank system.

Published
2022

38. Hautus--Yamamoto criteria for approximate and exact controllability of linear difference delay equations

Author

Chitour, Yacine, Fueyo, Sébastien, Mazanti, Guilherme, Sigalotti, Mario, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université 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 Paris Cité (UPCité), Analyse fonctionnelle pour la conception et l'analyse de systèmes (FACTAS), 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), Dynamical Interconnected Systems in COmplex Environments (DISCO), 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), 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), Sorbonne Université (SU), and G.M. was partially supported by ANR PIA funding: ANR-20-IDEES-0002

Subjects
Realization theory, 2020 Mathematics Subject Classification: 39A06, 93B05, 93C05, Difference delay equations, Optimization and Control (math.OC), FOS: Mathematics, Exact controllability, Approximate controllability, Bézout's identity, [MATH]Mathematics [math], Mathematics - Optimization and Control
Abstract

International audience; The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time $t$ is obtained as a linear combination of the control evaluated at time $t$ and of the state evaluated at finitely many previous instants of time $t-\Lambda_1,\dots,t-\Lambda_N$. Based on the realization theory developed by Y.~Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in $L^q$ spaces, $q \in [1, +\infty)$. We also provide a necessary condition for $L^1$ exact controllability, which can be seen as the closure of the $L^1$ approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by $d\max\{\Lambda_1,\dots,\Lambda_N\}$, where $d$ is the dimension of the state space.

Published
2022
Full Text
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39. 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

40. Control of COVID-19 outbreak using an extended SEIR model

Author

Nathaniel J. Merrill, Sean T. McQuade, Ryan Weightman, Aayush Yadav, Sarah R. Allred, Emmanuel Trélat, Benedetto Piccoli, Rutgers University [Camden], Rutgers University System (Rutgers), Pacific Northwest National Laboratory (PNNL), 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), and The authors would like to acknowledge the support of the NSF CMMI project # 2033580 'Managing pandemic by managing mobility' in collaboration with Cornell University and Vanderbilt University, and the support of the Joseph and Loretta Lopez Chair endowment.

Subjects
2019-20 coronavirus outbreak, Coronavirus disease 2019 (COVID-19), Computer science, Applied Mathematics, Modeling and Simulation, Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), Outbreak, Biological Science, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Virology
Abstract

International audience; The outbreak of COVID-19 resulted in high death tolls all over the world. The aim of this paper is to show how a simple SEIR model was used to make quick predictions for New Jersey in early March 2020 and call for action based on data from China and Italy. A more refined model, which accounts for social distancing, testing, contact tracing and quarantining, is then proposed to identify containment measures to minimize the economic cost of the pandemic. The latter is obtained taking into account all the involved costs including reduced economic activities due to lockdown and quarantining as well as the cost for hospitalization and deaths. The proposed model allows one to find optimal strategies as combinations of implementing various non-pharmaceutical interventions and study different scenarios and likely initial conditions.

Published
2022
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41. Equivalent formulations of optimal control problems with maximum cost and applications

Author

Emilio Molina, Alain Rapaport, Héctor Ramírez, Departamento de Ingeniera Matematica [Santiago] (DIM), Center for Mathematical Modeling (CMM), Universidad de Chile = University of Chile [Santiago] (UCHILE), 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), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Institut Agro Montpellier, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANID-PFCHA/Doctorado Nacional/2018-21180348, ANID-FONDECYT grant 1201982, ANID-FB210005 BASAL fund, ANID-PFCHA/Doctorado Nacional/2018-21180348, FONDECYT grant 1201982, ANID (Chile), Centro de Modelamiento Matematico (CMM), ACE210010, ANID (Chile), and FB210005, BASAL funds for center of excellence, ANID (Chile)

Subjects
Differential inclusion, Control and Optimization, Maximum cost, Applied Mathematics, Management Science and Operations Research, Optimal control, Mathematics Subject Classification. 49N90, 49J35, 49J45, 65K10, 90-08, Numerical schemes, Optimization and Control (math.OC), SIR models, Mathematics Subject Classification ( 2000): 49N90 · 49J35 · 49J45 · 65K10 · 90-08, FOS: Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], SIR model, State constraint, Mathematics - Optimization and Control, Mayer problem
Abstract

International audience; We revisit the optimal control problem with maximum cost with the objective to provide different equivalent reformulations suitable to numerical methods. We propose two reformulations in terms of extended Mayer problems with state constraints, and another one in terms of a differential inclusion with upper-semi continuous right member without state constraint. For the latter we also propose a scheme that approximates from below the optimal value. These approaches are illustrated and discussed in several examples.

Published
2022
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42. Numerical issues and turnpike phenomenon in optimal shape design

Author

Lance, Gontran, Trélat, Emmanuel, Zuazua, Enrique, 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), 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-R COSNET of MINECO (Spain) and by the ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government, Transregio 154 Project *Mathematical Modelling, Simulation and Optimization using the Example of Gas Net-works* of the German DFG, the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement 765579-ConFlex., Roland Herzog, Matthias Heinkenschloss, Dante Kalise, Georg Stadler, Emmanuel Trélat, ANR-16-ACHN-0014,ICON,Interactions du Contrôle, les Équations aux Dérivées Partielles, et l'Analyse Numérique(2016), ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), and European Project: 694126,DyCon

Subjects
Turnpike, Optimal shape design, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Numerical analysis, Optimal control
Abstract

International audience; This article follows and complements where we have established the turnpike property for some optimal shape design problems. Considering linear parabolic partial differential equations where the shapes to be optimized acts as a source term, we want to minimize a quadratic criterion. Existence of optimal shapes is proved under some appropriate assumptions. We prove and provide numerical evidence of the turnpike phenomenon for those optimal shapes, meaning that the extremal time-varying optimal solution remains essentially stationary; actually, it remains essentially close to the optimal solution of an associated static problem.

Published
2022

43. Diffusion and robustness of boundary feedback stabilization of hyperbolic systems

Author

Bastin, Georges, Coron, Jean-Michel, Hayat, Amaury, UCL - SST/ICTM/INMA - Pôle en ingénierie mathématique, 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), and École des Ponts ParisTech (ENPC)

Subjects
93B52, 93C20, 93D20, Control and Optimization, Control and Systems Engineering, Optimization and Control (math.OC), Applied Mathematics, Signal Processing, FOS: Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control
Abstract

We consider the problem of boundary feedback control of single-input-single-output (SISO) one-dimensional linear hyperbolic systems when sensing and actuation are anti-located. The main issue of the output feedback stabilization is that it requires dynamic control laws that include delayed values of the output (directly or through state observers) which may not be robust to infinitesimal uncertainties on the characteristic velocities. The purpose of this paper is to highlight some features of this problem by addressing the feedback stabilization of an unstable open-loop system which is made up of two interconnected transport equations and provided with anti-located boundary sensing and actuation. The main contribution is to show that the robustness of the control against delay uncertainties is recovered as soon as an arbitrary small diffusion is present in the system. Our analysis also reveals that the effect of diffusion on stability is far from being an obvious issue by exhibiting an alternative simple example where the presence of diffusion has a destabilizing effect instead., Comment: 21 pages

Published
2022
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44. Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe

Author

Koch, Christiane, Boscain, Ugo, Calarco, Tommaso, Dirr, Gunther, Filipp, Stefan, Glaser, Steffen, Kosloff, Ronnie, Montangero, Simone, Schulte-Herbrüggen, Thomas, Sugny, Dominique, Wilhelm, Frank K., Freie Universität Berlin, Sorbonne Université (SU), 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), Peter Grünberg Institut (PGI), University of Würzburg = Universität Würzburg, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), The Hebrew University of Jerusalem (HUJ), Università degli Studi di Padova = University of Padua (Unipd), Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), Université de Technologie de Belfort-Montbeliard (UTBM)-Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Saarland University [Saarbrücken], Financial support from European Union’s Horizon 2020 research and innovation programme under the MarieSk lodowska-Curie grant agreement Nr. 765267 (QuSCo) is gratefully acknowledged. SF, SJG and TSH would also like to acknowledge the Munich Quantum Valley (MQV) e.V., München, Germany., and European Project: 765267,H2020-EU.1.3.1.,QuSCo(2017)

Subjects
Controllability, [PHYS]Physics [physics], Quantum Physics, Quantum sensing, Quantum technologies, FOS: Physical sciences, TheoryofComputation_GENERAL, Quantum control, Quantum computing, Optimal control, ComputerSystemsOrganization_MISCELLANEOUS, ddc:530, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Quantum simulation, [MATH]Mathematics [math], Quantum Physics (quant-ph)
Abstract

Quantum optimal control, a toolbox for devising and implementing the shapes of external fields that accomplish given tasks in the operation of a quantum device in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The last few years have seen a rapid evolution and expansion of the field. We review here recent progress in our understanding of the controllability of open quantum systems and in the development and application of quantum control techniques to quantum technologies. We also address key challenges and sketch a roadmap for future developments., this is a living document - we welcome feedback and discussion

Published
2022

45. Existence of surfaces optimizing geometric and PDE shape functionals under reach constraint

Author

Yannick Privat, Rémi Robin, Mario Sigalotti, ROBIN, Rémi, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), TOkamaks and NUmerical Simulations (TONUS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, 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), 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)), 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
Calculus of variation, [MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA], [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Optimal surface, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Reach constraint, Mathematics - Analysis of PDEs, Optimization and Control (math.OC), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract

This article deals with the existence of hypersurfaces minimizing general shape functionals undercertain geometric constraints. We consider as admissible shapes orientable hypersurfaces satis-fying a so-called reach condition, also known as the uniform ball property, which ensures C^{1,1}regularity of the hypersurface. In this paper, we revisit and generalise the results of Guo et al and,J. Dalphin. We provide a simpler framework and more concise proofs of some of the results con-tained in these references and extend them to a new class of problems involving PDEs. Indeed, byusing the signed distance introduced by Delfour and Zolesio, we avoid the intensive and technicaluse of local maps, as was the case in the above references. Our approach, originally developedto solve an existence problem in a recent work by the same authors dedicated to optimal shapeissues for Plasma Physics, can be easily extended to costs involving different mathematical objectsassociated with the domain, such as solutions of elliptic equations on the hypersurface.

Published
2022
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46. Proportional Integral regulation control of a one-dimensional semilinear wave equation

Author

Hugo Lhachemi, Christophe Prieur, Emmanuel Trélat, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), CentraleSupélec, 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), 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), 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-19-P3IA-0003,MIAI,MIAI @ Grenoble Alpes(2019)

Subjects
Control and Optimization, Applied Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

International audience; This paper is concerned with the Proportional Integral (PI) regulation control of the left Neumann trace of a one-dimensional semilinear wave equation. The control input is selected as the right Neumann trace. The control design goes as follows. First, a preliminary (classical) velocity feedback is applied in order to shift all but a finite number of the eivenvalues of the underlying unbounded operator into the open left half-plane. We then leverage on the projection of the system trajectories into an adequate Riesz basis to obtain a truncated model of the system capturing the remaining unstable modes. Local stability of the resulting closed-loop infinite-dimensional system composed of the semilinear wave equation, the preliminary velocity feedback, and the PI controller, is obtained through the study of an adequate Lyapunov function. Finally, an estimate assessing the set point tracking performance of the left Neumann trace is derived.

Published
2022

47. Upper and lower bounds for the maximal Lyapunov exponent of singularly perturbed linear switching systems

Author

Chitour, Yacine, Haidar, Ihab, Mason, Paolo, Sigalotti, Mario, Université Paris-Saclay, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire QUARTZ (QUARTZ ), Université Paris 8 Vincennes-Saint-Denis (UP8)-Ecole Nationale Supérieure de l'Electronique et de ses Applications (ENSEA)-ISAE-Supméca Institut Supérieur de Mécanique de Paris (ISAE-Supméca), 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)), 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
Maximal Lyapunov exponent, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Differential inclusions, FOS: Mathematics, Switching systems, Dynamical Systems (math.DS), Mathematics - Dynamical Systems, Singular perturbation, Exponential stability, [SPI.AUTO]Engineering Sciences [physics]/Automatic
Abstract

In this paper we consider the problem of determining the stability properties, and in particular assessing the exponential stability, of a singularly perturbed linear switching system. One of the challenges of this problem arises from the intricate interplay between the small parameter of singular perturbation and the rate of switching, as both tend to zero. Our approach consists in characterizing suitable auxiliary linear systems that provide lower and upper bounds for the asymptotics of the maximal Lyapunov exponent of the linear switching system as the parameter of the singular perturbation tends to zero.

Published
2022
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48. Full quantum control of enantiomer-selective state transfer in chiral molecules despite degeneracy

Author

Leibscher, Monika, Pozzoli, Eugenio, Pérez, Cristobal, Schnell, Melanie, Sigalotti, Mario, Boscain, Ugo, Koch, Christiane P., Institut für Theoretische Physik, Universität Kassel [Kassel], 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), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Deutsches Elektronen-Synchrotron [Hamburg] (DESY), Institut für Physikalische Chemie [Kiel], Christian-Albrechts-Universität zu Kiel (CAU), Centre National de la Recherche Scientifique (CNRS), Dahlem Center for Complex Quantum Systems & Fachbereich Physik, Freie Universität Berlin, We gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft through CRC 1319ELCH and from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement Nr. 765267 (QuSCo).MS and UB also thank the ANR projects SRGI ANR-15-CE40-0018 and Quaco ANR-17-CE40-0007-01., ANR-15-CE40-0018,SRGI,Géométrie sous-Riemannienne et Interactions(2015), and ANR-17-CE40-0007,QUACO,Contrôle quantique : systèmes d'EDPs et applications à l'IRM(2017)

Subjects
[PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], 500 Naturwissenschaften und Mathematik::530 Physik::530 Physik, ddc:530, Atomic and molecular interactions with photons, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Circular dichroism, Applied mathematics
Abstract

Communications Physics 5(1), 110 (2022). doi:10.1038/s42005-022-00883-6, The driven quantum asymmetric top is an important paradigm in molecular physics with applications ranging from quantum information to chiral-sensitive spectroscopy. A key prerequisite for these applications is the ability to completely control the rotational dynamics. The inherent degeneracy of quantum rotors poses a challenge for quantum control since selecting a particular rotational state cannot be achieved by spectral selection alone. Here, we prove complete controllability for rotational states of an asymmetric top belonging to degenerate values of the orientational quantum number M. Based on this insight, we construct a pulse sequence that energetically separates population in degenerate M-states. Introducing the concept of enantio-selective controllability, we determine the conditions for complete enantiomer-specific population transfer in chiral molecules and construct pulse sequences for the example of propanediol and carvone molecules for population initially distributed over degenerate M-states. Our work shows how to leverage controllability analysis for the solution of practical quantum control problems., Published by Springer Nature, London

Published
2022
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49. Exponential stability of linear periodic difference-delay equations

Author

Baratchart, Laurent, Fueyo, Sébastien, Pomet, Jean-Baptiste, Analyse fonctionnelle pour la conception et l'analyse de systèmes (FACTAS), 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), Mathematics for Control, Transport and Applications (McTAO), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-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 Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)

Subjects
di↵erence delay systems, exponential stability, Dynamical Systems (math.DS), harmonic transfer function, Functional Analysis (math.FA), Mathematics - Functional Analysis, Optimization and Control (math.OC), FOS: Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Dynamical Systems, linear periodic systems, Mathematics - Optimization and Control, Dfference delay systems, Henry-Hale theorem
Abstract

This paper deals with stability of linear periodic time-varying difference delay systems, i.e. dynamical systems where a finite dimensional signal at a certain time is given as a linear time-varying function of its values at a finite number of delayed times. We give a necessary and sufficient condition for exponential stability, that is a generalisation of the one by Henry and Hale in the 1970s. It has a control theoretic interpretation in terms of the harmonic transfer function (HTF) of a corresponding linear control system., See also HAL: hal-03500720. Changes in version 2: improved introduction

Published
2021

50. On the gap between deterministic and probabilistic Lyapunov exponents for continuous-time linear systems

Author

Chitour, Yacine, Mazanti, Guilherme, Monmarché, Pierre, Sigalotti, Mario, Laboratoire des signaux et systèmes (L2S), CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Dynamical Interconnected Systems in COmplex Environments (DISCO), 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), 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), G. Mazanti was partially supported by ANR PIA funding number ANR-20-IDEES-0002, ANR-20-CE40-0022,SWIDIMS,Diffusions modulées : métastabilité et algorithmes stochastiques(2020), Université Paris-Saclay, 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
Statistics and Probability, convexified Markov processes, Probability (math.PR), [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Lyapunov exponents, 2020 Mathematics Subject Classification: 60J25, 34A38, 34D08, Dynamical Systems (math.DS), linear switched systems, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], continuous-time Markov processes, Optimization and Control (math.OC), 60J25, 34A38, 34D08, FOS: Mathematics, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Statistics, Probability and Uncertainty, Mathematics - Dynamical Systems, piecewise deterministic Markov processes, Mathematics - Optimization and Control, Mathematics - Probability
Abstract

International audience; Consider a non-autonomous continuous-time linear system in which the time-dependent matrix determining the dynamics is piecewise constant and takes finitely many values $A_1, \dotsc, A_N$. This paper studies the equality cases between the maximal Lyapunov exponent associated with the set of matrices $\{A_1, \dotsc, A_N\}$, on the one hand, and the corresponding ones for piecewise deterministic Markov processes with modes $A_1, \dotsc, A_N$, on the other hand. A fundamental step in this study consists in establishing a result of independent interest, namely, that any sequence of Markov processes associated with the matrices $A_1,\dotsc, A_N$ converges, up to extracting a subsequence, to a Markov process associated with a suitable convex combination of those matrices.

Published
2021

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