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A general Hamilton-Jacobi framework for non-linear state-constrained control problems∗

Published online by Cambridge University Press:  21 June 2012

Albert Altarovici ,
Olivier Bokanowski  and
Hasnaa Zidani
Albert Altarovici
Affiliation:
Projet Commands, INRIA Saclay & ENSTA ParisTech, 32 Bd. Victor, 75739 Paris Cedex 15, France. Albert.Altarovici@inria.fr; Hasnaa.Zidani@ensta.fr
Olivier Bokanowski
Affiliation:
Projet Commands, INRIA Saclay & ENSTA ParisTech, 32 Bd. Victor, 75739 Paris Cedex 15, France. Albert.Altarovici@inria.fr; Hasnaa.Zidani@ensta.fr Univ. Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, 75205 Paris, France; boka@math.jussieu.fr
Hasnaa Zidani
Affiliation:
Projet Commands, INRIA Saclay & ENSTA ParisTech, 32 Bd. Victor, 75739 Paris Cedex 15, France. Albert.Altarovici@inria.fr; Hasnaa.Zidani@ensta.fr

Abstract

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The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypasses the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.

Keywords

State constraintsoptimal control problemsnonlinear controlled systemsHamilton-Jacobi equationsviscosity solutions

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 337 - 357
Copyright
© EDP Sciences, SMAI, 2012

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