Quasitrajectories and trajectories 4 Relaxation of differential inclusions 11 Infinitesimal generator of reachable map 16 Variational inclusion 19 Application: semilinear optimal control problem with end point constraints 22 Quasitraj ect ories and trajectories Consider a separable Banach space X and a strongly continuous semigroup G(t) E L (X , X) , t 2 0 of bounded linear operators from X to X having the infinitesimal generator A. Let 0 5 to < T be given and F be a set-valued map from [to, TI x X into closed nonempty subsets of X. We associate with
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