Sensitivity analysis for relaxed optimal control problems with final-state constraints

In this article, we compute a second-order expansion of the value function of a family of relaxed optimal control problems with final-state constraints, parameterized by a perturbation variable. In this framework, relaxation with Young measures enables us to consider a wide class of perturbations and therefore to derive sharp estimates of the value function. The sensitivity analysis is performed in a neighborhood of a local optimal solution of a reference problem. The local solution ū is assumed to be optimal with respect to the set of feasible relaxed controls having their support in a ball of a given radius R>‖ū‖∞ and having an associated trajectory very close to the reference trajectory, for the L∞-norm. We call such a solution a relaxed R-strong solution.