Differential inclusions on closed sets in Banach spaces with application to sweeping process. (English) Zbl 1069.34086

Benabdellah, Houcine

Topol. Methods Nonlinear Anal. 23, No. 1, 115-148 (2004).

The author considers the following differential inclusion with state constraint \[ x(0)=x_0,\;x(t)\in C(t), \dot{x}(t) \in F(t,x(t)), \] where \(C:[0,a] \to X\) is a multifunction with closed graph \(G\), \(F:G \to X\) is a multifunction with nonempty, convex and compact values and \(X\) is a separable Banach space. Under suitable assumptions on the set-valued functions \(F\) and \(C\), using a new extension of Scorza-Dragoni’s theorem for upper semicontinuous multifunctions, the author proves that for every \(x_0 \in C(0)\) there exists an absolutely continuous function \(x:[0,a] \to X\) such that \(x(t) \in C(t)\) on \([0,a]\) and \(\dot{x}(t) \in F(t,x(t))\) a.e. on \([0,a]\). An application to a nonconvex sweeping process is also considered.

Reviewer: Francesca Papalini (Ancona)

MSC:

34G25

Evolution inclusions

Keywords:

differential inclusions; Bouligand cone; Scorza-Dragoni theorem