A relaxation result for state constrained inclusions in infinite dimension

In this paper we consider a state constrained differential inclusion$\dot x\in \mathbb A x+ F(t,x)$, with $\mathbb A$ generator of a strongly continuous semigroup inan infinite dimensional separable Banach space. Under an``inward pointing condition'' we prove a relaxation result stating thatthe set of trajectories lying in the interior of the constraint is dense in the set ofconstrained trajectories of the convexified inclusion$\dot x\in \mathbb A x+ \overline{\textrm{co}}F(t,x)$.Some applications to control problems involving PDEs are given.

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