In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position x. In the constrained case, when no controllability assumption is made, the HJB equation may have several solutions. Our first result aims to give the precise information that should be added to the HJB equation in order to obtain a characterization of the value function. This result is very general and holds even when the dynamics is not continuous and the state constraints set is not smooth. On the other hand we study also some stability results for relaxed or penalized control problems.
Mots clés : optimal control problem, state constraints, Hamilton-Jacobi equation
@article{COCV_2011__17_4_995_0, author = {Bokanowski, Olivier and Forcadel, Nicolas and Zidani, Hasnaa}, title = {Deterministic state-constrained optimal control problems without controllability assumptions}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {995--1015}, publisher = {EDP-Sciences}, volume = {17}, number = {4}, year = {2011}, doi = {10.1051/cocv/2010030}, mrnumber = {2859862}, zbl = {1237.35030}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2010030/} }
TY - JOUR AU - Bokanowski, Olivier AU - Forcadel, Nicolas AU - Zidani, Hasnaa TI - Deterministic state-constrained optimal control problems without controllability assumptions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 995 EP - 1015 VL - 17 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010030/ DO - 10.1051/cocv/2010030 LA - en ID - COCV_2011__17_4_995_0 ER -
%0 Journal Article %A Bokanowski, Olivier %A Forcadel, Nicolas %A Zidani, Hasnaa %T Deterministic state-constrained optimal control problems without controllability assumptions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 995-1015 %V 17 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010030/ %R 10.1051/cocv/2010030 %G en %F COCV_2011__17_4_995_0
Bokanowski, Olivier; Forcadel, Nicolas; Zidani, Hasnaa. Deterministic state-constrained optimal control problems without controllability assumptions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 995-1015. doi : 10.1051/cocv/2010030. http://www.numdam.org/articles/10.1051/cocv/2010030/
[1] Differential inclusions, Comprehensive studies in mathematics 264. Springer, Berlin, Heidelberg, New York, Tokyo (1984). | MR | Zbl
and ,[2] Set-valued analysis, Systems and Control: Foundations and Applications 2. Birkhäuser Boston Inc., Boston (1990). | MR | Zbl
and ,[3] Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems and Control: Foundations and Applications. Birkhäuser, Boston (1997). | MR | Zbl
and ,[4] A Dirichlet type problem for nonlinear degenerate elliptic equations arising in time-optimal stochastic control. Adv. Math. Sci. Appl. 10 (2000) 329-352. | MR | Zbl
, and ,[5] Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques et Applications 17. Springer, Paris (1994). | MR | Zbl
,[6] Discontinuous solutions of deterministic optimal stopping time problems. RAIRO: Modél. Math. Anal. Numér. 21 (1987) 557-579. | Numdam | MR | Zbl
and ,[7] Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133-1148. | MR | Zbl
and ,[8] Comparaison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations. Appl. Math. Optim. 21 (1990) 21-44. | MR | Zbl
and ,[9] Viscosity solutions and analysis in L∞, in Proceedings of the NATO advanced Study Institute (1999) 1-60. | MR | Zbl
,[10] Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Diff. Equ. 15 (1990) 1713-1742. | MR | Zbl
and ,[11] Deterministic exit time problems with discontinuous exit cost. SIAM J. Control Optim. 35 (1997) 399-434. | MR | Zbl
,[12] Reachability and minimal times for state constrained nonlinear problems without any controllability assumption. SIAM J. Control Optim. 48 (2010) 4292-4316. | MR | Zbl
, and ,[13] Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318 (1990) 643-683. | MR | Zbl
and ,[14] Optimal times for constrained nonlinear control problems without local controllability. Appl. Math. Optim. 36 (1997) 21-42. | MR | Zbl
, and ,[15] Nonsmooth analysis and control theory. Springer (1998). | MR | Zbl
, , and ,[16] Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257-272. | MR | Zbl
,[17] Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints. J. Math. Anal. Appl. 251 (2000) 818-838. | MR | Zbl
and ,[18] Existence of neighboring feasible trajectories: applications to dynamic programming for state-constrained optimal control problems. J. Optim. Theory Appl. 104 (2000) 21-40. | MR | Zbl
and ,[19] A new formulation of state constraint problems for first-order PDEs. SIAM J. Control Optim. 34 (1996) 554-571. | MR | Zbl
and ,[20] On nonlinear optimal control problems with state constraints. SIAM J. Control Optim. 33 (1995) 1411-1424. | MR | Zbl
,[21] Optimal control with state-space constraint, I. SIAM J. Control Optim. 24 (1986) 552-561. | MR | Zbl
,[22] Optimal control with state-space constraint, II. SIAM J. Control Optim. 24 (1986) 1110-1122. | MR | Zbl
,[23] Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints. Diff. Int. Equ. 12 (1999) 275-293. | MR | Zbl
,Cité par Sources :