Differential inclusions on closed sets in Banach spaces with application to sweeping process

Abstract

This paper deals with the existence of absolutely continuous solutions of a differential inclusion with state constraint in a separable Banach space $$ x( 0) =x_{0}, \quad x( t) \in C( t) ,\quad \dot{x}( t) \in F( t,x( t) ) $$ where $C\colon [ 0,a] \rightarrow X$ is a multifunction with closed graph $G$ and $F\colon G\rightarrow X$ is a convex compact valued multifunction which is separately measurable in $t\in[ 0,a] $ and separately upper semicontinuous in $x\in X$. Application to a non convex sweeping process is also considered.