Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-27T01:08:59.864Z Has data issue: false hasContentIssue false
  • English
  • Français

Local small time controllability and attainability of a set for nonlinear control system

Published online by Cambridge University Press:  15 August 2002

Mikhail Krastanov  and
Marc Quincampoix
Mikhail Krastanov
Affiliation:
Institute of Mathematics and Informatics, Acad. G. Bonchev Str., Bl. 8, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria; [email protected].
Marc Quincampoix
Affiliation:
Département de Mathématiques, Université de Bretagne Occidentale, 6 avenue Victor Le Gorgeu, BP. 809, 29285 Brest Cedex, France; [email protected].
Get access

Abstract

In the present paper, we study the problem of small-time local attainability (STLA) of a closed set. For doing this, we introduce a new concept of variations of the reachable set well adapted to a given closed set and prove a new attainability result for a general dynamical system. This provide our main result for nonlinear control systems. Some applications to linear and polynomial systems are discussed and STLA necessary and sufficient conditions are obtained when the considered set is a hyperplane.

Keywords

Attainabilitycontrolabilitylocal variationspolynomial controllinear controls.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 499 - 516
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agrachev, A. and Gamkrelidze, R., The exponential representation of flows and the chronological calculus. Math. USSR Sbornik 35 (1978) 727-785. CrossRef
Bacciotti, A. and Stefani, G., Self-accessibility of a set with respect to a multivalued field. JOTA 31 (1980) 535-552. CrossRef
Bianchini, R. and Stefani, G., Time optimal problem and time optimal map. Rend. Sem. Mat. Univ. Politec. Torino 48 (1990) 401-429.
Bony, J.M., Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble) 19 (1969) 277-304. CrossRef
P. Brunovsky, Local controllability of odd systems. Banach Center Publications,Warsaw, Poland 1 (1974) 39-45.
Cardaliaguet, P., Quincampoix, M. and Saint Pierre, P., Minimal time for constrained nonlinear control problems without controllability. Appl. Math. Optim. 36 (1997) 21-42. CrossRef
Chen, K., Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math. 65 (1957) 163-178. CrossRef
Clarke, F.H. and Wolenski, P.R., Control of systems to sets and their interiors. JOTA 88 (1996) 3-23. CrossRef
Fliess, M., Fonctionnelles causales nonlinéaires et indéterminées non commutatives. Bull. Soc. Math. France 109 (1981) 3-40. CrossRef
Frankowska, H., Local controllability of control systems with feedback. JOTA 60 (1989) 277-296. CrossRef
Hermes, H., Lie algebras of vector fields and local approximation of attainable sets. SIAM J. Control Optim. 16 (1978) 715-727. CrossRef
Hirshorn, R., Strong controllability of nonlinear systems. SIAM J. Control Optim. 16 (1989) 264-275.
Jurdjevic, V. and Kupka, I., Polynomial Control Systems. Math. Ann. 272 (1985) 361-368. CrossRef
Krener, A., The high order maximal principle and its applications to singular extremals. SIAM J. Control Optim. 15 (1977) 256-293. CrossRef
Kunita, H., On the controllability of nonlinear systems with application to polynomial systems. Appl. Math. Optim. 5 (1979) 89-99. CrossRef
Lebourg, G., Valeur moyenne pour gradient généralisé. C. R. Acad. Sci. Paris Sér. I Math. 281 (1975) 795-797.
Soravia, P., Hölder Continuity of the Minimum-Time Function for C 1-Manifold Targets. JOTA 75 (1992) 2. CrossRef
Sussmann, H., A sufficient condition for local controllability. SIAM J. Control Optim. 16 (1978) 790-802. CrossRef
Sussmann, H., Lie brackets and local controllability - A sufficient condition for scalar-input control systems. SIAM J. Control Optim. 21 (1983) 683-713. CrossRef
Sussmann, H., A general theorem on local controllability. SIAM J. Control Optim. 25 (1987) 158-194. CrossRef
Veliov, V., On the controllability of control constrained systems. Mathematica Balkanica (N.S.) 2 (1988) 2-3, 147-155.
Veliov, V. and Krastanov, M., Controllability of piece-wise linear systems. Systems Control Lett. 7 (1986) 335-341. CrossRef
V. Veliov, Attractiveness and invariance: The case of uncertain measurement, edited by Kurzhanski and Veliov, Modeling Techniques for uncertain Systems. PSCT 18, Birkhauser (1994).
Veliov, V., On the Lipschitz continuity of the value function in optimal control. J. Optim. Theory Appl. 94 (1997) 335-361. CrossRef