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26 results on '"Caponigro, Marco"'

8. Sparse stabilization and control of the Cucker-Smale model

Author

Caponigro, Marco, Fornasier, Massimo, Piccoli, Benedetto, Trélat, Emmanuel, Trélat, Emmanuel, Modélisation mathématique et numérique (M2N), Conservatoire National des Arts et Métiers [CNAM] (CNAM), Fakultät Mathematik, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Rutgers University, Department of Mathematics, Department of Mathematics - Rutgers School of Arts and Sciences, 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), and 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)

Subjects
Cucker-Smale model, optimal complexity, 34D45, 35B36, 37D35, 49J15, 65K10, 93D15, 93B05, sparse stabilization, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], $\ell_1$-norm minimization, sparse optimal control, consensus emergence
Abstract

From a mathematical point of view self-organization can be described as patterns to which certain dynamical systems modeling social dynamics tend spontaneously to be attracted. In this paper we explore situations beyond self-organization, in particular how to externally control such dynamical systems in order to eventually enforce pattern formation also in those situations where this wished phenomenon does not result from spontaneous convergence. Our focus is on dynamical systems of Cucker-Smale type, modeling consensus emergence, and we question the existence of stabilization and optimal control strategies which require the minimal amount of external intervention for nevertheless inducing consensus in a group of interacting agents. First we follow a greedy approach, by designing instantaneous feedback controls with two different sparsity properties: componentwise sparsity, meaning that the controls have at most one nonzero component at every instant of time and their implementation is based on a variational criterion involving $\ell_1$-norm penalization terms; time sparsity, meaning that the number of switchings is bounded on every compact interval of time, and such controls are realized by means of a sample-and-hold procedure. Controls sharing these two sparsity features are very realistic and convenient for practical issues. Moreover we show that among the controls built out of the mentioned variational principle, the maximally sparse ones are instantaneously optimal in terms of the decay rate of a suitably designed Lyapunov functional, measuring the distance from consensus. As a consequence we provide a mathematical justification to the general principle according to which ''sparse is better'' in the sense that a policy maker, who is not allowed to predict future developments, should always consider more favorable to intervene with stronger action on the fewest possible instantaneous optimal leaders rather than trying to control more agents with minor strength in order to achieve group consensus. We then establish local and global sparse controllability properties to consensus. Finally, we analyze the sparsity of solutions of the finite time optimal control problem where the minimization criterion is a combination of the distance from consensus and of the $\ell_1$-norm of the control. Such an optimization models the situation where the policy maker is actually allowed to observe future developments. We show that the lacunarity of sparsity is related to the codimension of certain manifolds in the space of cotangent vectors.

Published
2015

9. Efficient finite dimensional approximations for the bilinear Schrodinger equation with bounded variation controls

Author

Boussaid, Nabile, Caponigro, Marco, Chambrion, Thomas, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique et numérique (M2N), Conservatoire National des Arts et Métiers [CNAM] (CNAM), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), 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), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM)

Subjects
Mathematics - Analysis of PDEs, bilinear control, FOS: Mathematics, Shrodinger, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Galerkin, Analysis of PDEs (math.AP)
Abstract

International audience; This the text of a proceeding accepted for the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014). We present some results of an ongoing research on the controllability problem of an abstract bilinear Schrodinger equation. We are interested by approximation of this equation by finite dimensional systems. Assuming that the uncontrolled term $A$ has a pure discrete spectrum and the control potential $B$ is in some sense regular with respect to $A$ we show that such an approximation is possible. More precisely the solutions are approximated by their projections on finite dimensional subspaces spanned by the eigenvectors of $A$. This approximation is uniform in time and in the control, if this control has bounded variation with a priori bounded total variation. Hence if these finite dimensional systems are controllable with a fixed bound on the total variation of the control then the system is approximatively controllable. The main outcome of our analysis is that we can build solutions for low regular controls such as bounded variation ones and even Radon measures.

Published
2014

10. Approximate controllability of the Schrödinger Equation with a polarizability term in higher Sobolev norms

Author

Boussaid, Nabile, Caponigro, Marco, Chambrion, Thomas, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique et numérique (M2N), Conservatoire National des Arts et Métiers [CNAM] (CNAM), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), 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), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM)

Subjects
Mathematics - Analysis of PDEs, Optimization and Control (math.OC), bilinear control, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Approximate controllability, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Schrödinger Equation, Mathematics - Optimization and Control, polarizability, Analysis of PDEs (math.AP)
Abstract

This analysis is concerned with the controllability of quantum systems in the case where the standard dipolar approximation, involving the permanent dipole moment of the system, is corrected with a polarizability term, involving the field induced dipole moment. Sufficient conditions for approximate controllability are given. For transfers between eigenstates of the free Hamiltonian, the control laws are explicitly given. The results apply also for unbounded or non-regular potentials.

Published
2014

11. EXACT CONTROLLABILITY IN PROJECTIONS OF THE BILINEAR SCHRÖDINGER EQUATION.

Author

CAPONIGRO, MARCO and SIGALOTTI, MARIO

Subjects
*NUMERICAL analysis, *GALERKIN methods, *APPROXIMATION theory, *CONTROLLABILITY in systems engineering, *PARTIAL differential equations
Abstract

We consider the bilinear Schrödinger equation with discrete-spectrum drift. We show, for n ∈ N arbitrary, exact controllability in projections on the first n given eigenstates. The controllability result relies on a generic controllability hypothesis on some associated finite-dimensional approximations. The method is based on Lie-algebraic control techniques applied to the finite-dimensional approximations coupled with classical topological arguments issuing from degree theory. [ABSTRACT FROM AUTHOR]

Published
2018
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12. Approximate controllability of the Schrödinger equation with a polarizability term

Author

Boussaid, Nabile, Caponigro, Marco, Chambrion, Thomas, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique et numérique (M2N), Conservatoire National des Arts et Métiers [CNAM] (CNAM), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), This work has been partially supported by Inria Nancy Grand Est., European Project: 239748,EC:FP7:ERC,ERC-2009-StG,GECOMETHODS(2010), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), 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), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Chambrion, Thomas, and Geometric control methods for heat and Schroedinger equations - GECOMETHODS - - EC:FP7:ERC2010-05-01 - 2016-04-30 - 239748 - VALID

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

International audience; This paper is concerned with the controllability of quantum systems in the case where the standard dipolar approximation, involving the permanent dipole moment of the system, has to be corrected by a so-called polarizability term, involving the field induced dipole moment. Sufficient conditions for controllability between eigenstates of the free Hamiltonian are derived and control laws are explicitly given. As an illustration, the results are applied to the planar rotation of the HCN molecule.

Published
2012

13. Controllability of the bilinear Schr\'odinger equation with several controls and application to a 3D molecule

Author

Boscain, Ugo, Caponigro, Marco, Sigalotti, Mario, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Geometric Control Design (GECO), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Center for Computational and Integrative Biology [Camden] (CCIB), Rutgers University [Camden], Rutgers University System (Rutgers)-Rutgers University System (Rutgers), and Department of Mathematical Sciences [Camden]

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

International audience; We show the approximate rotational controllability of a polar linear molecule by means of three nonresonant linear polarized laser fields. The result is based on a general approximate controllability result for the bilinear Schrödinger equation, with wavefunction varying in the unit sphere of an infinite-dimensional Hilbert space and with several control potentials, under the assumption that the internal Hamiltonian has discrete spectrum.

Published
2012

15. Mean-field sparse Jurdjevic-Quinn control.

Author

Caponigro, Marco, Piccoli, Benedetto, Rossi, Francesco, and Trélat, Emmanuel

Subjects
*MULTIAGENT systems, *CROWD control, *DISTRIBUTED parameter systems, *PARTIAL differential equations, *LYAPUNOV functions
Abstract

We consider nonlinear transport equations with non-local velocity describing the time-evolution of a measure. Such equations often appear when considering the mean-field limit of finite-dimensional systems modeling collective dynamics. We address the problem of controlling these equations by means of a time-varying bounded control action localized on a time-varying control subset of small Lebesgue measure. We first define dissipativity for nonlinear transport equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, assuming that the uncontrolled system is dissipative, we provide an explicit construction of a control law steering the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function. In this sense the construction can be seen as an infinite-dimensional analogue of the well-known Jurdjevic-Quinn procedure. Moreover, the control law presents sparsity properties in the sense that the support of the control is small. Finally, we show that our result applies to a large class of kinetic equations modeling multi-agent dynamics. [ABSTRACT FROM AUTHOR]

Published
2017
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17. KEEP RIGHT OR LEFT? TOWARDS A COGNITIVE-MATHEMATICAL MODEL FOR PEDESTRIANS.

Author

BRAVO, MARY J., CAPONIGRO, MARCO, LEIBOWITZ, EMILY, and PICCOLI, BENEDETTO

Subjects
PEDESTRIANS, MATHEMATICAL models, COGNITION, PHYSIOLOGICAL aspects of walking, APPLIED mathematics, MATHEMATICAL optimization, ROBOTICS
Abstract

In this paper we discuss the necessity of insight in the cognitive processes involved in environment navigation into mathematical models for pedestrian motion. We first provide a review of psychological literature on the cognitive processes involved in walking and on the quantitative one coming from applied mathematics, physics, and engineering. Then, we present a critical analysis of the experimental setting for model testing and we show experimental results given by observation. Finally we propose a cognitive model making use of psychological insight as well as optimization models from robotics. [ABSTRACT FROM AUTHOR]

Published
2015
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18. On the control through leadership of the Hegselmann-Krause opinion formation model.

Author

Wongkaew, Suttida, Caponigro, Marco, and Borzì, Alfio

Subjects
*CONTROL theory (Engineering), *FEEDBACK control systems, *PROBLEM solving, *OPTIMAL control theory, *LEADERSHIP, *PREDICTIVE control systems, *SCHEME programming language
Abstract

This paper deals with control strategies for the Hegselmann-Krause opinion formation model with leadership. In this system, the control mechanism is included in the leader dynamics and the feedback control functions are determined via a stabilization procedure and with a model predictive optimal control process. Correspondingly, the issues of global stabilization, controllability, and tracking are investigated. The model predictive control scheme requires to solve a sequence of open-loop optimality systems discretized by an appropriate Runge-Kutta scheme and solved by a nonlinear conjugate gradient method. Results of numerical experiments demonstrate the validity of the proposed control strategies and their ability to drive the system to attain consensus. [ABSTRACT FROM AUTHOR]

Published
2015
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19. Sparse stabilization and control of alignment models.

Author

Caponigro, Marco, Fornasier, Massimo, Piccoli, Benedetto, and Trélat, Emmanuel

Subjects
*CONTROL theory (Engineering), *SELF-organizing systems, *OPTIMAL control theory, *MANIFOLDS (Mathematics), *COMPUTATIONAL complexity, *COTANGENT function
Abstract

Starting with the seminal papers of Reynolds (1987), Vicsek et al. (1995), Cucker-Smale (2007), there has been a lot of recent works on models of self-alignment and consensus dynamics. Self-organization has so far been the main driving concept of this research direction. However, the evidence that in practice self-organization does not necessarily occur (for instance, the achievement of unanimous consensus in government decisions) leads to the natural question of whether it is possible to externally influence the dynamics in order to promote the formation of certain desired patterns. Once this fundamental question is posed, one is also faced with the issue of defining the best way of obtaining the result, seeking for the most "economical" way to achieve a certain outcome. Our paper precisely addressed the issue of finding the sparsest control strategy in order to lead us optimally towards a given outcome, in this case the achievement of a state where the group will be able by self-organization to reach an alignment consensus. As a consequence, we provide a mathematical justification to the general principle according to which "sparse is better": in order to achieve group consensus, a policy maker not allowed to predict future developments should decide to control with stronger action the fewest possible leaders rather than trying to act on more agents with minor strength. We then establish local and global sparse controllability properties to consensus. Finally, we analyze the sparsity of solutions of the finite time optimal control problem where the minimization criterion is a combination of the distance from consensus and of the ℓ1-norm of the control. Such an optimization models the situation where the policy maker is actually allowed to observe future developments. We show that the lacunarity of sparsity is related to the codimension of certain manifolds in the space of cotangent vectors. [ABSTRACT FROM AUTHOR]

Published
2015
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21. Small time reachable set of bilinear quantum systems.

Author

Boussaid, Nabile, Caponigro, Marco, and Chambrion, Thomas

Abstract

This note presents an example of bilinear conservative system in an infinite dimensional Hilbert space for which approximate controllability in the Hilbert unit sphere holds for arbitrary small times. This situation is in contrast with the finite dimensional case and is due to the unboundedness of the drift operator. [ABSTRACT FROM PUBLISHER]

Published
2012
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22. Implementation of logical gates on infinite dimensional quantum oscillators.

Author

Boussaid, Nabile, Caponigro, Marco, and Chambrion, Thomas

Abstract

In this paper we study the error in the approximate simultaneous controllability of the bilinear Schro¨dinger equation. We provide estimates based on a tracking algorithm for general bilinear quantum systems and on the study of the finite dimensional Galerkin approximations for a particular class of quantum systems, weakly-coupled systems. We then present two physical examples: the perturbed quantum harmonic oscillator and the infinite potential well. [ABSTRACT FROM PUBLISHER]

Published
2012
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23. Periodic control laws for bilinear quantum systems with discrete spectrum.

Author

Boussaid, Nabile, Caponigro, Marco, and Chambrion, Thomas

Abstract

We provide bounds on the error between dynamics of an infinite dimensional bilinear Schro¨dinger equation and of its finite dimensional Galerkin approximations. Standard averaging methods are used on the finite dimensional approximations to obtain constructive controllability results. As an illustration, the methods are applied on a model of a 2D rotating molecule. [ABSTRACT FROM PUBLISHER]

Published
2012
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24. SPARSE STABILIZATION AND OPTIMAL CONTROL OF THE CUCKER-SMALE MODEL.

Author

CAPONIGRO, MARCO, FORNASIER, MASSIMO, PICCOLI, BENEDETTO, and TRÉLAT, EMMANUEL

Subjects
OPTIMAL control theory, STOCHASTIC convergence, VARIATIONAL principles, CONTROLLABILITY in systems engineering, CALCULUS of variations
Abstract

This article is mainly based on the work [7], and it is dedicated to the 60th anniversary of B. Bonnard, held in Dijon in June 2012. We focus on a controlled Cucker--Smale model in finite dimension. Such dynamics model self-organization and consensus emergence in a group of agents. We explore how it is possible to control this model in order to enforce or facilitate pattern formation or convergence to consensus. In particular, we are interested in designing control strategies that are componentwise sparse in the sense that they require a small amount of external intervention, and also time sparse in the sense that such strategies are not chattering in time. These sparsity features are desirable in view of practical issues. We first show how very simple sparse feedback strategies can be designed with the use of a variational principle, in order to steer the system to consensus. These feedbacks are moreover optimal in terms of decay rate of some functional, illustrating the general principle according to which \sparse is better". We then combine these results with local controllability properties to get global controllability results. Finally, we explore the sparsity properties of the optimal control minimizing a combination of the distance from consensus and of a norm of the control. [ABSTRACT FROM AUTHOR]

Published
2013
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25. Weakly Coupled Systems in Quantum Control.

Author

Boussaid, Nabile, Caponigro, Marco, and Chambrion, Thomas

Subjects
*COUPLED mode theory (Wave-motion), *QUANTUM theory, *SCHRODINGER equation, *APPROXIMATION theory, *GALERKIN methods, *DISCRETE systems
Abstract

Weakly coupled systems are a class of infinite dimensional conservative bilinear control systems with discrete spectrum. An important feature of these systems is that they can be precisely approached by finite dimensional Galerkin approximations. This property is of particular interest for the approximation of quantum system dynamics and the control of the bilinear Schrödinger equation. The present study provides rigorous definitions and analysis of the dynamics of weakly coupled systems and gives sufficient conditions for an infinite dimensional quantum control system to be weakly coupled. As an illustration we provide examples chosen among common physical systems. [ABSTRACT FROM AUTHOR]

Published
2013
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