Differential inclusions and abstract control problems

We prove an existence theorem for differential inclusions x ' is an element of A(t)x + F(t,x),x(0) = x(0) in Banach spaces. Here {A(t) : t is an element of [O, T]} is a family of linear operators generating a continuous evolution operator K(t, s). We concentrate on maps F with F(t,.) weakly sequentially hemi-continuous. Moreover, we show a compactness of the set of all integral solutions of the above problem. These results are also applied to a semilinear optimal control problem. Some corollaries, important in the theory of optimal control, are given too. We extend in several ways theorems existing in the literature.