# Characterization of the Value Function of Optimal Control Problems with State Constraints, and Feedback Optimal Control

Olivier Bokanowski^{*}

We consider deterministic optimal control problems of nonlinear controlled systems in presence of state constraints. For such problems the continuity of the value function requires, in general, some restrictive controllability assumptions involving the dynamics and the set of state constraints. Here, we are mainly interested by the characterization of the epigraph of the value function. In particular, 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. This framework allows to bypass the regularity issues related to the qualification of the state constraints and leads to a constructive way for compute the epigraph by a large panel of numerical schemes. It allows also to obtain an optimal control law in feedback form. This is a joint work with H. Zidani.

Mathematics Subject Classification: 49J15 49L25

Keywords: State constraints; optimal control problems; nonlinear controlled systems; Hamilton-Jacobi equations; epigraph; viscosity solutions

Minisymposion: On Optimal Feedback Control for Partial Differential Equations: Theory and Numerical Methods