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

DOI: http://dx.doi.org/10.12775/TMNA.2004.006

#### 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.

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.

#### Keywords

Differential inclusions; Bouligand cone; Scorza-Dragoni theorem

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