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BV prox-regular sweeping process with bounded truncated variation. (English) Zbl 1454.49009

This paper is devoted to the existence and uniqueness of solutions for perturbed sweeping process measure differential inclusions in infinite dimensional setting, i.e. \[-du\in N(C(t);u(t))+f(t,u(t)),\,\,\, u(T_0)=u_0,\] where \(C:I\rightrightarrows \mathcal{H}\) be an \(r\)-prox-regular valued multimapping for some extended real \(r\in ]0,+\infty]\), \(u_0\in C(T_0),\) \(f:I\times \mathcal{H}\to \mathcal{H}\) be a mapping with \(f\neq 0,\) \(\mu\) be a positive Radon measure on \(I\) with \(\sup_{\tau\in ]T_0,T]}\mu(\{\tau\})<r/2,\) \(\mathcal{H}\) be a Hilbert space.
Building on work started by Moreau, existence and uniqueness of solutions to perturbed sweeping process measure differential inclusions in infinitely dimensional setting are investigated, when the moving set is prox-regular, being controlled by means of the truncated Hausdorff-Pompeiu distance. A Carathéodory type mapping satisfying a time-dependent hypomonotonicity assumption on bounded sets is employed for perturbing the involved normal cone. Various additional results, some of which of interest of their own, are provided in order to lead to the main results.

MSC:

49J21 Existence theories for optimal control problems involving relations other than differential equations
34A60 Ordinary differential inclusions
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