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

Mikhail Krastanov; Marc Quincampoix

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

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topKrastanov, 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 (2010): 499-516. <http://eudml.org/doc/197313>.

@article{Krastanov2010,

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.; attainability; controllability; linear controls},

language = {eng},

month = {3},

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/197313},

volume = {6},

year = {2010},

}

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

DA - 2010/3//

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.; attainability; controllability; linear controls

UR - http://eudml.org/doc/197313

ER -

## References

top- A. Agrachev and R. Gamkrelidze, The exponential representation of flows and the chronological calculus. Math. USSR Sbornik35 (1978) 727-785.
- A. Bacciotti and G. Stefani, Self-accessibility of a set with respect to a multivalued field. JOTA31 (1980) 535-552.
- R. Bianchini and G. Stefani, Time optimal problem and time optimal map. Rend. Sem. Mat. Univ. Politec. Torino48 (1990) 401-429.
- 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.
- P. Brunovsky, Local controllability of odd systems. Banach Center Publications,Warsaw, Poland 1 (1974) 39-45.
- P. Cardaliaguet, M. Quincampoix and P. Saint Pierre, Minimal time for constrained nonlinear control problems without controllability. Appl. Math. Optim.36 (1997) 21-42.
- K. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. Math.65 (1957) 163-178.
- F.H. Clarke and P.R. Wolenski, Control of systems to sets and their interiors. JOTA88 (1996) 3-23.
- M. Fliess, Fonctionnelles causales nonlinéaires et indéterminées non commutatives. Bull. Soc. Math. France109 (1981) 3-40.
- H. Frankowska, Local controllability of control systems with feedback. JOTA60 (1989) 277-296.
- H. Hermes, Lie algebras of vector fields and local approximation of attainable sets. SIAM J. Control Optim.16 (1978) 715-727.
- R. Hirshorn, Strong controllability of nonlinear systems. SIAM J. Control Optim.16 (1989) 264-275.
- V. Jurdjevic and I. Kupka, Polynomial Control Systems. Math. Ann.272 (1985) 361-368.
- A. Krener, The high order maximal principle and its applications to singular extremals. SIAM J. Control Optim.15 (1977) 256-293.
- H. Kunita, On the controllability of nonlinear systems with application to polynomial systems. Appl. Math. Optim.5 (1979) 89-99.
- G. Lebourg, Valeur moyenne pour gradient généralisé. C. R. Acad. Sci. Paris Sér. I Math.281 (1975) 795-797.
- P. Soravia, Hölder Continuity of the Minimum-Time Function for C1-Manifold Targets. JOTA75 (1992) 2.
- H. Sussmann, A sufficient condition for local controllability. SIAM J. Control Optim.16 (1978) 790-802.
- H. Sussmann, Lie brackets and local controllability - A sufficient condition for scalar-input control systems. SIAM J. Control Optim.21 (1983) 683-713.
- H. Sussmann, A general theorem on local controllability. SIAM J. Control Optim.25 (1987) 158-194.
- V. Veliov, On the controllability of control constrained systems. Mathematica Balkanica (N.S.)2 (1988) 2-3, 147-155.
- V. Veliov and M. Krastanov, Controllability of piece-wise linear systems. Systems Control Lett.7 (1986) 335-341.
- V. Veliov, Attractiveness and invariance: The case of uncertain measurement, edited by Kurzhanski and Veliov, Modeling Techniques for uncertain Systems. PSCT 18, Birkhauser (1994).
- V. Veliov, On the Lipschitz continuity of the value function in optimal control. J. Optim. Theory Appl.94 (1997) 335-361.

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