Résumé (EN)

Derivation of necessary conditions for optimality in optimal control theory relies on variational calculus of control systems. We provide here such calculus for the semilinear differential inclusion 1 x′∈Ax+F(t,x),x(0)=x0 (1) where A is the infinitesimal generator of a C o-semigroup on a separable Banach space X and F: [0,T]×X∼ X is a set-valued map. Such inclusion encompass a number of control systems, including those with state dependent controls. We prove a relaxation theorem and investigate infinitesimal generators of reachable sets and variational inclusions. Results are applied to derive necessary conditions for optimality for some semilinear optimal control problems.