Sensitivity relations for the Mayer problem with differential inclusions

In optimal control, sensitivity relations are usually understood as inclusions that identify the pair formed by the dual arc and the Hamiltonian as a suitable generalized gradient of the value function, evaluated along a given minimizing trajectory. In this paper, sensitivity relations are obtained for the Mayer problem associated with the differential inclusion x ̇ ∈ F(x) and applied to express optimality conditions. The first application of our results concerns the maximum principle and consists in showing that a dual arc can be constructed for every element of the superdifferential of the final cost as a solution of an adjoint system. The second and last application we discuss in this paper concerns optimal design. We show that one can associate a family of optimal trajectories, starting at some point (t, x), with every nonzero reachable gradient of the value function at (t, x), in such a way that families corresponding to distinct reachable gradients have empty intersection.