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 , we provide sufficient conditions to steer to every point of a suitable neighborhood of 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.
Keywords: Geometric control theory, small-time local attainability, state constraints
@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
Lipschitz continuity and local semiconcavity for exit time problems with state constraints. J. Differ. Eq. 245 (2008) 616–636. | DOI | MR | Zbl
, and ,Differentiability properties for a class of non-convex functions. Calc. Var. Partial Differ. Eq. 25 (2006) 1–31. | DOI | MR | Zbl
and ,The Clarke generalized gradient for functions whose epigraph has positive reach. Math. Oper. Res. 38 (2013) 451–468. | DOI | MR | Zbl
, and ,Higher order discrete controllability and the approximation of the minimum time function. Discr. Contin. Dyn. Syst. 35 (2015) 4293–4322. | DOI | MR | Zbl
and ,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
High-order variations and small-time local attainability. Control Cybernet. 38 (2009) 1411–1427. | MR | Zbl
,Local small time controllability and attainability of a set for nonlinear control system. ESAIM: COCV 6 (2001) 499–516. | Numdam | MR | Zbl
and ,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
Some regularity results for a class of upper semicontinuous functions. Indianaa Univ. Math. J. 62 (2013) 45–89. | DOI | MR | Zbl
, and ,Controllability of some nonlinear systems with drift via generalized curvature properties. SIAM J. Control Optimization 53 (2015) 434–474. | DOI | MR | Zbl
andLipschitz continuity of the value function in optimal control. J. Optim. Theory Appl. 94 (1997) 335–363. | DOI | MR | Zbl
.Cited by Sources: