On the optimal value function of optimal control problems

Abstract

The optimal value of parameterized deterministic optimal control problems with state and mixed state-control constraints is studied. The main results are estimates for the effect of small variations of the parameter $p$ on the optimal value $V=V(p)$. In particular, upper bounds for several directional derivatives of $V$ as well as estimates for a generalized subgradient $\partial V(\cdot )$ are given. These estimates are given in terms of multipliers which satisfy first or higher order necessary optimality conditions along an optimal solution of the unperturbed problem. The theory is based on the separability or nonseparability of suitable approximating cones and requires no or only very weak regularity assumptions.