Local controllability does imply global controllability

Ugo Boscain; Daniele Cannarsa; Valentina Franceschi; Mario Sigalotti

doi:10.5802/crmath.538

Théorie du contrôle

Local controllability does imply global controllability
[La contrôlabilité locale implique la contrôlabilité globale]

Ugo Boscain ¹ ; Daniele Cannarsa ² ; Valentina Franceschi ³ ; Mario Sigalotti ¹

¹ Sorbonne Université, CNRS, Inria, Laboratoire Jacques-Louis Lions (LJLL), Paris, France
² Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland
³ Dipartimento di Matematica Tullio Levi-Civita, Università di Padova, Italy

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1813-1822.

Résumé
Abstract

Nous disons qu’un système de contrôle est localement controllable si les ensembles atteignables à partir de tout état $x$ sont un voisinage de $x$ , tandis que le système est contrôlable si les ensembles atteignables à partir de tout état coïncident avec la variété entière. Nous montrons qu’un système qui est localement controllable est contrôlable. Notre preuve est une alternative à la combinaison de deux résultats précédents par Kevin Grasse.

We say that a control system is locally controllable if the attainable set from any state $x$ contains an open neighborhood of $x$ , while it is controllable if the attainable set from any state is the entire state manifold. We show in this note that a control system satisfying local controllability is controllable. Our self-contained proof is alternative to the combination of two previous results by Kevin Grasse.

Reçu le : 2023-03-25
Révisé le : 2023-07-07
Accepté le : 2023-07-07
Publié le : 2023-12-21

DOI : 10.5802/crmath.538

Classification : 57R27, 34H05, 93B05

Affiliations des auteurs :

Ugo Boscain ¹ ; Daniele Cannarsa ² ; Valentina Franceschi ³ ; Mario Sigalotti ¹

¹ Sorbonne Université, CNRS, Inria, Laboratoire Jacques-Louis Lions (LJLL), Paris, France
² Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland
³ Dipartimento di Matematica Tullio Levi-Civita, Università di Padova, Italy

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2023__361_G11_1813_0,
     author = {Ugo Boscain and Daniele Cannarsa and Valentina Franceschi and Mario Sigalotti},
     title = {Local controllability does imply global controllability},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1813--1822},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.538},
     language = {en},
}

TY  - JOUR
AU  - Ugo Boscain
AU  - Daniele Cannarsa
AU  - Valentina Franceschi
AU  - Mario Sigalotti
TI  - Local controllability does imply global controllability
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1813
EP  - 1822
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.538
LA  - en
ID  - CRMATH_2023__361_G11_1813_0
ER  -

%0 Journal Article
%A Ugo Boscain
%A Daniele Cannarsa
%A Valentina Franceschi
%A Mario Sigalotti
%T Local controllability does imply global controllability
%J Comptes Rendus. Mathématique
%D 2023
%P 1813-1822
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.538
%G en
%F CRMATH_2023__361_G11_1813_0

Ugo Boscain; Daniele Cannarsa; Valentina Franceschi; Mario Sigalotti. Local controllability does imply global controllability. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1813-1822. doi : 10.5802/crmath.538. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.538/

Bibliographie
Cité par

[1] Andrea Bacciotti; Gianna Stefani On the relationship between global and local controllability, Math. Syst. Theory, Volume 16 (1983) no. 1, pp. 79-91 | DOI | MR | Zbl

[2] Alberto Bressan; Benedetto Piccoli Introduction to the mathematical theory of control, AIMS Series on Applied Mathematics, 2, American Institute of Mathematical Sciences, 2007, xiv+312 pages | MR

[3] Howie Choset; Kevin M. Lynch; Seth Hutchinson; George Kantor; Wolfram Burgard; Lydia E. Kavraki; Sebastian Thrun Principles of Robot Motion: Theory, Algorithms and Implementations, Intelligent Robotics and Autonomous Agents, 22, Cambridge University Press, 2007 no. 2, pp. 209-211 | DOI

[4] Jean-Michel Coron Control and nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007, xiv+426 pages | DOI | MR

[5] Kevin A. Grasse A condition equivalent to global controllability in systems of vector fields, J. Differ. Equations, Volume 56 (1985) no. 2, pp. 263-269 | DOI | MR | Zbl

[6] Kevin A. Grasse On the relation between small-time local controllability and normal self-reachability, Math. Control Signals Syst., Volume 5 (1992) no. 1, pp. 41-66 | DOI | MR | Zbl

[7] Kevin A. Grasse Reachability of interior states by piecewise constant controls, Forum Math., Volume 7 (1995) no. 5, pp. 607-628 | DOI | MR | Zbl

[8] Frédéric Jean Stabilité et commande des systèmes dynamiques, Presses de l’ENSTA, 2017

[9] Mikhail Ivanov Krastanov; Margarita Nikolaeva Nikolova A necessary condition for small-time local controllability, Automatica, Volume 124 (2021), 109258, 5 pages | DOI | MR | Zbl

[10] Arthur J Krener A generalization of Chow’s theorem and the bang-bang theorem to nonlinear control problems, SIAM J. Control, Volume 12 (1974) no. 1, pp. 43-52 | DOI | Zbl

[11] Ivan Kupka; Gauthier Sallet A sufficient condition for the transitivity of pseudosemigroups: application to system theory, J. Differ. Equations, Volume 47 (1983) no. 3, pp. 462-470 | DOI | MR | Zbl

[12] Jérôme Lohéac; Emmanuel Trélat; Enrique Zuazua Minimal controllability time for finite-dimensional control systems under state constraints, Automatica, Volume 96 (2018), pp. 380-392 | DOI | MR | Zbl

[13] Eduardo D. Sontag Mathematical control theory, Texts in Applied Mathematics, 6, Springer, 1998, xvi+531 pages (Deterministic finite-dimensional systems) | DOI | MR

[14] Hector J. Sussmann A general theorem on local controllability, SIAM J. Control Optim., Volume 25 (1987) no. 1, pp. 158-194 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique