On a controllability problem for systems governed by semilinear functional differential inclusions in Banach spaces

In the present paper we study a problem of attaining a given point at a prescribed time for a system governed by a semilinear functional differential inclusion in a Banach space. The frame of our functional argument has been introducedby Ragni and Rubbioni in 2000 for the study of terminal value problems for functional differential equations with unbounded delay. We use the technique of condensing multivalued maps to develop an unified approach to the cases when multivalued nonlinearity is Carathèodory upper semicontinuous or almost lower semicontinuous. We obtain the existence result basing, in the first case, on the Bohnenblust-Karlin fixed point theorem, and, in the second case, on the Fryszkowski continuous selection result and the Schauder fixed point theorem. For the case of Carathèodory nonlinearity, we also prove the compactness of the solutions set and, as a corollary, we obtain an optimization result.