Local small time controllability and attainability of a set for nonlinear control system
ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 499-516.

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.

Classification : 93B05, 93B03, 93C05, 93C10, 49J53
Mots clés : attainability, controlability, local variations, polynomial control, linear controls 
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     title = {Local small time controllability and attainability of a set for nonlinear control system},
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     pages = {499--516},
     publisher = {EDP-Sciences},
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Krastanov, Mikhail; Quincampoix, Marc. Local small time controllability and attainability of a set for nonlinear control system. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 499-516. http://www.numdam.org/item/COCV_2001__6__499_0/

[1] A. Agrachev and R. Gamkrelidze, The exponential representation of flows and the chronological calculus. Math. USSR Sbornik 35 (1978) 727-785. | Zbl

[2] A. Bacciotti and G. Stefani, Self-accessibility of a set with respect to a multivalued field. JOTA 31 (1980) 535-552. | MR | Zbl

[3] R. Bianchini and G. Stefani, Time optimal problem and time optimal map. Rend. Sem. Mat. Univ. Politec. Torino 48 (1990) 401-429. | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[5] P. Brunovsky, Local controllability of odd systems. Banach Center Publications,Warsaw, Poland 1 (1974) 39-45. | EuDML | Zbl

[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. | MR | Zbl

[7] K. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math. 65 (1957) 163-178. | MR | Zbl

[8] F.H. Clarke and P.R. Wolenski, Control of systems to sets and their interiors. JOTA 88 (1996) 3-23. | MR | Zbl

[9] M. Fliess, Fonctionnelles causales nonlinéaires et indéterminées non commutatives. Bull. Soc. Math. France 109 (1981) 3-40. | EuDML | Numdam | MR | Zbl

[10] H. Frankowska, Local controllability of control systems with feedback. JOTA 60 (1989) 277-296. | MR | Zbl

[11] H. Hermes, Lie algebras of vector fields and local approximation of attainable sets. SIAM J. Control Optim. 16 (1978) 715-727. | MR | Zbl

[12] R. Hirshorn, Strong controllability of nonlinear systems. SIAM J. Control Optim. 16 (1989) 264-275. | MR | Zbl

[13] V. Jurdjevic and I. Kupka, Polynomial Control Systems. Math. Ann. 272 (1985) 361-368. | MR | Zbl

[14] A. Krener, The high order maximal principle and its applications to singular extremals. SIAM J. Control Optim. 15 (1977) 256-293. | MR | Zbl

[15] H. Kunita, On the controllability of nonlinear systems with application to polynomial systems. Appl. Math. Optim. 5 (1979) 89-99. | MR | Zbl

[16] G. Lebourg, Valeur moyenne pour gradient généralisé. C. R. Acad. Sci. Paris Sér. I Math. 281 (1975) 795-797. | MR | Zbl

[17] P. Soravia, Hölder Continuity of the Minimum-Time Function for C 1 -Manifold Targets. JOTA 75 (1992) 2. | MR | Zbl

[18] H. Sussmann, A sufficient condition for local controllability. SIAM J. Control Optim. 16 (1978) 790-802. | MR | Zbl

[19] H. Sussmann, Lie brackets and local controllability - A sufficient condition for scalar-input control systems. SIAM J. Control Optim. 21 (1983) 683-713. | Zbl

[20] H. Sussmann, A general theorem on local controllability. SIAM J. Control Optim. 25 (1987) 158-194. | MR | Zbl

[21] V. Veliov, On the controllability of control constrained systems. Mathematica Balkanica (N.S.) 2 (1988) 2-3, 147-155. | MR | Zbl

[22] V. Veliov and M. Krastanov, Controllability of piece-wise linear systems. Systems Control Lett. 7 (1986) 335-341. | MR | Zbl

[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). | MR | Zbl

[24] V. Veliov, On the Lipschitz continuity of the value function in optimal control. J. Optim. Theory Appl. 94 (1997) 335-361. | MR | Zbl