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Deterministic state-constrained optimal control problems without controllability assumptions

Published online by Cambridge University Press:  06 August 2010

Olivier Bokanowski ,
Nicolas Forcadel  and
Hasnaa Zidani
Olivier Bokanowski
Affiliation:
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie 75252 Paris Cedex 05, France. boka@math.jussieu.fr UFR de Mathématiques, Site Chevaleret, Université Paris-Diderot, 75205 Paris Cedex, France.
Nicolas Forcadel
Affiliation:
CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France. forcadel@ceremade.dauphine.fr
Hasnaa Zidani
Affiliation:
Projet Commands, INRIA Saclay & ENSTA, 32 Bd Victor, 75739 Paris Cedex 15, France. Hasnaa.Zidani@ensta.fr
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Abstract

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.

Keywords

Optimal control problemstate constraintsHamilton-Jacobi equation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 995 - 1015
Copyright
© EDP Sciences, SMAI, 2010

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References

J.-P. Aubin and A. Cellina, Differential inclusions, Comprehensive studies in mathematics 264. Springer, Berlin, Heidelberg, New York, Tokyo (1984).
J.-P. Aubin and H. Frankowska, Set-valued analysis, Systems and Control: Foundations and Applications 2. Birkhäuser Boston Inc., Boston (1990).
M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems and Control: Foundations and Applications. Birkhäuser, Boston (1997).
Bardi, M., Goatin, P. and Ishii, H., Dirichlet, A type problem for nonlinear degenerate elliptic equations arising in time-optimal stochastic control. Adv. Math. Sci. Appl. 10 (2000) 329–352.
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques et Applications 17. Springer, Paris (1994).
Barles, G. and Perthame, B., Discontinuous solutions of deterministic optimal stopping time problems. RAIRO: Modél. Math. Anal. Numér. 21 (1987) 557–579.
Barles, G. and Perthame, B., Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133–1148. CrossRef
Barles, G. and Perthame, B., Comparaison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations. Appl. Math. Optim. 21 (1990) 21–44. CrossRef
E.N. Barron, Viscosity solutions and analysis in L ∞, in Proceedings of the NATO advanced Study Institute (1999) 1–60.
Barron, E.N. and Jensen, R., Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Diff. Equ. 15 (1990) 1713–1742. CrossRef
Blanc, A., Deterministic exit time problems with discontinuous exit cost. SIAM J. Control Optim. 35 (1997) 399–434. CrossRef
Bokanowski, O., Forcadel, N. and Zidani, H., Reachability and minimal times for state constrained nonlinear problems without any controllability assumption. SIAM J. Control Optim. 48 (2010) 4292–4316. CrossRef
Capuzzo-Dolcetta, I. and Lions, P.-L., Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318 (1990) 643–683. CrossRef
Cardaliaguet, P., Quincampoix, M. and Saint-Pierre, P., Optimal times for constrained nonlinear control problems without local controllability. Appl. Math. Optim. 36 (1997) 21–42. CrossRef
F. Clarke, Y.S. Ledyaev, R. Stern and P. Wolenski, Nonsmooth analysis and control theory. Springer (1998).
Frankowska, H., Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257–272. CrossRef
Frankowska, H. and Plaskacz, S., Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints. J. Math. Anal. Appl. 251 (2000) 818–838. CrossRef
Frankowska, H. and Vinter, R.B., Existence of neighboring feasible trajectories: applications to dynamic programming for state-constrained optimal control problems. J. Optim. Theory Appl. 104 (2000) 21–40. CrossRef
Ishii, H. and Koike, S., A new formulation of state constraint problems for first-order PDEs. SIAM J. Control Optim. 34 (1996) 554–571. CrossRef
Motta, M., On nonlinear optimal control problems with state constraints. SIAM J. Control Optim. 33 (1995) 1411–1424. CrossRef
Soner, H.M., Optimal control with state-space constraint, I. SIAM J. Control Optim. 24 (1986) 552–561. CrossRef
Soner, H.M., Optimal control with state-space constraint, II. SIAM J. Control Optim. 24 (1986) 1110–1122. CrossRef
Soravia, P., 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.