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

Mikhail Krastanov; Marc Quincampoix

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • Volume: 6, page 499-516
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Krastanov, Mikhail, and Quincampoix, Marc. "Local small time controllability and attainability of a set for nonlinear control system." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 499-516. <http://eudml.org/doc/90605>.

@article{Krastanov2001,
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.},
author = {Krastanov, Mikhail, Quincampoix, Marc},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {attainability; controlability; local variations; polynomial control; linear controls; controllability},
language = {eng},
pages = {499-516},
publisher = {EDP-Sciences},
title = {Local small time controllability and attainability of a set for nonlinear control system},
url = {http://eudml.org/doc/90605},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Krastanov, Mikhail
AU - Quincampoix, Marc
TI - Local small time controllability and attainability of a set for nonlinear control system
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 499
EP - 516
AB - 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.
LA - eng
KW - attainability; controlability; local variations; polynomial control; linear controls; controllability
UR - http://eudml.org/doc/90605
ER -

References

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  1. [1] A. Agrachev and R. Gamkrelidze, The exponential representation of flows and the chronological calculus. Math. USSR Sbornik 35 (1978) 727-785. Zbl0429.34044
  2. [2] A. Bacciotti and G. Stefani, Self-accessibility of a set with respect to a multivalued field. JOTA 31 (1980) 535-552. Zbl0417.49048MR600203
  3. [3] R. Bianchini and G. Stefani, Time optimal problem and time optimal map. Rend. Sem. Mat. Univ. Politec. Torino 48 (1990) 401-429. Zbl0776.49003MR1199082
  4. [4] J.M. Bony, 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. Zbl0176.09703MR262881
  5. [5] P. Brunovsky, Local controllability of odd systems. Banach Center Publications,Warsaw, Poland 1 (1974) 39-45. Zbl0344.93016
  6. [6] P. Cardaliaguet, M. Quincampoix and P. Saint Pierre, Minimal time for constrained nonlinear control problems without controllability. Appl. Math. Optim. 36 (1997) 21-42. Zbl0884.49002MR1446790
  7. [7] K. Chen, Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula. Ann. Math. 65 (1957) 163-178. Zbl0077.25301
  8. [8] F.H. Clarke and P.R. Wolenski, Control of systems to sets and their interiors. JOTA 88 (1996) 3-23. Zbl0843.93009MR1367031
  9. [9] M. Fliess, Fonctionnelles causales nonlinéaires et indéterminées non commutatives. Bull. Soc. Math. France 109 (1981) 3-40. Zbl0476.93021MR613847
  10. [10] H. Frankowska, Local controllability of control systems with feedback. JOTA 60 (1989) 277-296. Zbl0633.93013MR984985
  11. [11] H. Hermes, Lie algebras of vector fields and local approximation of attainable sets. SIAM J. Control Optim. 16 (1978) 715-727. Zbl0388.49025MR493664
  12. [12] R. Hirshorn, Strong controllability of nonlinear systems. SIAM J. Control Optim. 16 (1989) 264-275. Zbl0594.93012MR826516
  13. [13] V. Jurdjevic and I. Kupka, Polynomial Control Systems. Math. Ann. 272 (1985) 361-368. Zbl0554.93033MR799667
  14. [14] A. Krener, The high order maximal principle and its applications to singular extremals. SIAM J. Control Optim. 15 (1977) 256-293. Zbl0354.49008MR433288
  15. [15] H. Kunita, On the controllability of nonlinear systems with application to polynomial systems. Appl. Math. Optim. 5 (1979) 89-99. Zbl0406.93011MR533613
  16. [16] G. Lebourg, Valeur moyenne pour gradient généralisé. C. R. Acad. Sci. Paris Sér. I Math. 281 (1975) 795-797. Zbl0317.46034MR388097
  17. [17] P. Soravia, Hölder Continuity of the Minimum-Time Function for C 1 -Manifold Targets. JOTA 75 (1992) 2. Zbl0792.93058MR1191596
  18. [18] H. Sussmann, A sufficient condition for local controllability. SIAM J. Control Optim. 16 (1978) 790-802. Zbl0391.93004MR527718
  19. [19] H. Sussmann, Lie brackets and local controllability – A sufficient condition for scalar-input control systems. SIAM J. Control Optim. 21 (1983) 683-713. Zbl0523.49026
  20. [20] H. Sussmann, A general theorem on local controllability. SIAM J. Control Optim. 25 (1987) 158-194. Zbl0629.93012MR872457
  21. [21] V. Veliov, On the controllability of control constrained systems. Mathematica Balkanica (N.S.) 2 (1988) 2-3, 147-155. Zbl0681.93009MR981273
  22. [22] V. Veliov and M. Krastanov, Controllability of piece-wise linear systems. Systems Control Lett. 7 (1986) 335-341. Zbl0609.93006MR859019
  23. [23] V. Veliov, Attractiveness and invariance: The case of uncertain measurement, edited by Kurzhanski and Veliov, Modeling Techniques for uncertain Systems. PSCT 18, Birkhauser (1994). Zbl0833.34015MR1287663
  24. [24] V. Veliov, On the Lipschitz continuity of the value function in optimal control. J. Optim. Theory Appl. 94 (1997) 335-361. Zbl0901.49022MR1460669

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