Small-time local attainability for a class of control systems with state constraints
ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 1003-1021.

In this paper we consider the problem of small time local attainability (STLA) for nonlinear finite-dimensional time-continuous control systems in presence of state constraints. More precisely, given a nonlinear control system subjected to state constraints and a closed set S, we provide sufficient conditions to steer to S every point of a suitable neighborhood of S along admissible trajectories of the system, respecting the constraints, and giving also an upper estimate of the minimum time needed for each point to reach the target. Methods of nonsmooth analysis are used.

DOI: 10.1051/cocv/2016022
Classification: 49J52, 90C56
Keywords: Geometric control theory, small-time local attainability, state constraints
Le Thuy, T.T. 1, 2; Marigonda, Antonio 3

1 Department of Mathematics, University of Padova, Via Trieste 63, 35121 Padova, Italy.
2 Faculty of Mathematics, Otto-von-Guericke University Magdeburg, Universitätsplatz 2, 02-222, 39106 Magdeburg, Germany.
3 Department of Computer Science, University of Verona, Strada Le Grazie 15, 37134 Verona, Italy.
@article{COCV_2017__23_3_1003_0,
     author = {Le Thuy, T.T. and Marigonda, Antonio},
     title = {Small-time local attainability for a class of control systems with state constraints},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1003--1021},
     publisher = {EDP-Sciences},
     volume = {23},
     number = {3},
     year = {2017},
     doi = {10.1051/cocv/2016022},
     zbl = {1369.49016},
     mrnumber = {3660457},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2016022/}
}
TY  - JOUR
AU  - Le Thuy, T.T.
AU  - Marigonda, Antonio
TI  - Small-time local attainability for a class of control systems with state constraints
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2017
SP  - 1003
EP  - 1021
VL  - 23
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2016022/
DO  - 10.1051/cocv/2016022
LA  - en
ID  - COCV_2017__23_3_1003_0
ER  - 
%0 Journal Article
%A Le Thuy, T.T.
%A Marigonda, Antonio
%T Small-time local attainability for a class of control systems with state constraints
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2017
%P 1003-1021
%V 23
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2016022/
%R 10.1051/cocv/2016022
%G en
%F COCV_2017__23_3_1003_0
Le Thuy, T.T.; Marigonda, Antonio. Small-time local attainability for a class of control systems with state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 1003-1021. doi : 10.1051/cocv/2016022. http://archive.numdam.org/articles/10.1051/cocv/2016022/

A.A. Agračëv and Y.L. Sachkov, Control theory from the geometric viewpoint. Control Theory and Optimization II. Vol. 87 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2004). | MR | Zbl

M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. With appendices by Maurizio Falcone and Pierpaolo Soravia. Systems & Control: Foundations & Applications. Birkhäuser Boston Inc. (1997). | MR | Zbl

P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Vol. 58 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston Inc. Boston. MA (2004). | MR | Zbl

P. Cannarsa, and M. Castelpietra, Lipschitz continuity and local semiconcavity for exit time problems with state constraints. J. Differ. Eq. 245 (2008) 616–636. | DOI | MR | Zbl

G. Colombo and A. Marigonda, Differentiability properties for a class of non-convex functions. Calc. Var. Partial Differ. Eq. 25 (2006) 1–31. | DOI | MR | Zbl

G. Colombo, A. Marigonda and P.R. Wolenski, The Clarke generalized gradient for functions whose epigraph has positive reach. Math. Oper. Res. 38 (2013) 451–468. | DOI | MR | Zbl

G. Colombo and T.T.T. Le, Higher order discrete controllability and the approximation of the minimum time function. Discr. Contin. Dyn. Syst. 35 (2015) 4293–4322. | DOI | MR | Zbl

H. Federer, Curvature measures. Trans. Amer. Math. Soc. 93 (1959) 418–491. | DOI | MR | Zbl

V. Jurdjevic, Geometric control theory. Vol. 52 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1997). | MR | Zbl

M.I. Krastanov, High-order variations and small-time local attainability. Control Cybernet. 38 (2009) 1411–1427. | MR | Zbl

M.I. Krastanov and M. Quincampoix, Local small time controllability and attainability of a set for nonlinear control system. ESAIM: COCV 6 (2001) 499–516. | Numdam | MR | Zbl

M. Kawski and H.J. Sussmann, Noncommutative power series and formal lie-algebraic techniques in nonlinear control theory. Operators, systems, and linear algebra (Kaiserlautern, 1997), European Consort. Math. Indust. Teubner, Stuttgart (1997) 111–128. | MR | Zbl

A. Marigonda, Second order conditions for the controllability of nonlinear systems with drift. Comm. Pure Appl. Anal. 5 861–885. | MR | Zbl

A. Marigonda, K.T. Nguyen and D. Vittone, Some regularity results for a class of upper semicontinuous functions. Indianaa Univ. Math. J. 62 (2013) 45–89. | DOI | MR | Zbl

Marigonda, A. and Rigo, S. Controllability of some nonlinear systems with drift via generalized curvature properties. SIAM J. Control Optimization 53 (2015) 434–474. | DOI | MR | Zbl

V.M. Veliov. Lipschitz continuity of the value function in optimal control. J. Optim. Theory Appl. 94 (1997) 335–363. | DOI | MR | Zbl

Cited by Sources: