Continuous dependence results for set-valued measure differential problems

Bianca Satco

doi:10.14232/ejqtde.2015.1.79

Abstract

We discuss existence and continuous dependence properties of the solutions set of measure differential inclusions dx(t)2 G(t, x(t))dm(t), x(0) = x0.

Highlights

We focus on the problem (1.1), where G : [0, 1] × Rd → P (Rd) is a compact convex-valued multifunction and μ is a positive regular Borel measure
The motivation for studying such a kind of problems comes from the fact that it contains, as special cases, differential and difference inclusions and hybrid problems and it allows to describe systems with state discontinuities
We are interested in studying measure differential problems with bounded variation or, even more generally, regulated functions on the right-hand side, especially from the point of view of the possibility to obtain the solutions by the solutions of similar problem driven by approximating measures

Summary

Introduction

We focus on the problem (1.1), where G : [0, 1] × Rd → P (Rd) is a compact convex-valued multifunction and μ is a positive regular Borel measure. We want to obtain the existence of solutions with special properties for the case when the multifunction on the right-hand side is regulated, respectively of bounded variation and, for the family of these solutions, to prove continuous dependence results via usual convergence notions for measures.

Results

Conclusion