Sensitivity relations for the Mayer problem of optimal control

Abstract

Sensitivity relations in optimal control refer to the interpretation of the gradients of the value function in terms of the costate arc and the Hamiltonian evaluated along an extremal. In general, the value function is not differentiable and for this reason its gradients have to be replaced by generalized differentials. In this paper we prove such sensitivity relations for the Mayer optimal control problem with dynamics described by a differential inclusion. If the associated Hamiltonian is semiconvex with respect to the state variable, then we show that sensitivity relations hold true for any dual arc associated to an optimal solution, instead of more traditional statements about the existence of a dual arc satisfying such relations. Furthermore, several applications are provided.