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

Abstract

For a Banach space $E$, a given pair $(\overline p, \overline x)\in[0,a]\times E$, and control system governed by a semilinear functional differential includion of the form $$ x'(t)\in Ax(t) +F(t, x(t), Tx) $$ the existence of a mild trajectory of $x(t)$ satisfying the condition $x(\overline p)=\overline x$ is considered. Using topological methods we develop an unified approach to the cases when a multivalued nonlinearity $F$ is Carathéodory upper semicontinuous or almost lower semicontinuous and an abstract extension operator $T$ allows to deal with variable and infinite delay. For the Carathéodory case, the compactness of the solutions set and, as a corollary, an optimization result are obtained.