Sensitivity relations for the Mayer problem with differential inclusions^∗

¹ Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy
cannarsa@axp.mat.uniroma2.it; scarinci@axp.mat.uniroma2.it
² CNRS, IMJ-PRG, UMR 7586, Sorbonne Universités, UPMC Univ Paris 06, Univ Paris Diderot, Sorbonne Paris Cité, Case 247, 4 Place Jussieu, 75252 Paris, France
helene.frankowska@imj-prg.fr; teresa.scarinci@imj-prg.fr

Received: 12 February 2014
Revised: 7 August 2014

Abstract

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 ẋ ∈ 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.