A priori estimates for operational differential inclusions

Abstract

We prove a set-valued Gronwall lemma and a relaxation theorem for the semilinear differential inclusion x′ ϵ Ax + F( t, x), x(0) = x 0, where A is the infinitesimal generator of a C 0-semigroup on a separable Banach space X and F: [0, T] × X ↦ X is a set-valued map. This allows us to investigate infinitesimal generators of reachable sets and variational inclusions. The results are applied to a semilinear optimal control problem with end point constraints.