no. G11
Théorie du contrôle
Local controllability does imply global controllability
[La contrôlabilité locale implique la contrôlabilité globale]
Boscain, Ugo1 ; Cannarsa, Daniele2 ; Franceschi, Valentina3 ; Sigalotti, Mario1 
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, Tome 361 (2023) no. G11, pp. 1813-1822

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.538
Classification : 57R27, 34H05, 93B05

Boscain, Ugo 1 ; Cannarsa, Daniele 2 ; Franceschi, Valentina 3 ; Sigalotti, Mario 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 = {Boscain, Ugo and Cannarsa, Daniele and Franceschi, Valentina and Sigalotti, Mario},
     title = {Local controllability does imply global controllability},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1813--1822},
     year = {2023},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     number = {G11},
     doi = {10.5802/crmath.538},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/crmath.538/}
}
TY  - JOUR
AU  - Boscain, Ugo
AU  - Cannarsa, Daniele
AU  - Franceschi, Valentina
AU  - Sigalotti, Mario
TI  - Local controllability does imply global controllability
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1813
EP  - 1822
VL  - 361
IS  - G11
PB  - Académie des sciences, Paris
UR  - https://www.numdam.org/articles/10.5802/crmath.538/
DO  - 10.5802/crmath.538
LA  - en
ID  - CRMATH_2023__361_G11_1813_0
ER  - 
%0 Journal Article
%A Boscain, Ugo
%A Cannarsa, Daniele
%A Franceschi, Valentina
%A Sigalotti, Mario
%T Local controllability does imply global controllability
%J Comptes Rendus. Mathématique
%D 2023
%P 1813-1822
%V 361
%N G11
%I Académie des sciences, Paris
%U https://www.numdam.org/articles/10.5802/crmath.538/
%R 10.5802/crmath.538
%G en
%F CRMATH_2023__361_G11_1813_0
Boscain, Ugo; Cannarsa, Daniele; Franceschi, Valentina; Sigalotti, Mario. Local controllability does imply global controllability. Comptes Rendus. Mathématique, Tome 361 (2023) no. G11, pp. 1813-1822. doi: 10.5802/crmath.538

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

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

[3] Choset, Howie; Lynch, Kevin M.; Hutchinson, Seth; Kantor, George; Burgard, Wolfram; Kavraki, Lydia E.; Thrun, Sebastian 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] Coron, Jean-Michel Control and nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007, xiv+426 pages | DOI | MR

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

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

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

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

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

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

Cité par Sources :