Abstract
The paper is devoted to the study of the measure-driven differential inclusions , for arbitrary finite Borel measure μ. This type of results allows one to treat in a similar manner differential and difference inclusions, as well as impulsive problems and therefore to study the evolution of hybrid systems with very complex (including Zeno) behavior. Our method is based on viewing the Borel measures as Lebesgue-Stieltjes measures. We thus obtain, under very general assumptions, the existence of regulated or bounded variation solutions of the considered problem and we indicate some advantages of our approach. MSC:49N25, 34A60, 93C30, 49J53, 37N35, 34A37.
Highlights
Let us consider the following problem: dx(t) ∈ G t, x(t) dμ(t), ( )x( ) = x, where G : [, ] × Rd → P(Rd) is a closed convex-valued multifunction and μ is a positive regular Borel measure
Where G : [, ] × Rd → P(Rd) is a closed convex-valued multifunction and μ is a positive regular Borel measure. This kind of problems covers some well-known cases like usual differential inclusions, difference inclusions and some impulsive multivalued problems
We concentrate on two aspects of mentioned papers: we relax the assumptions on μ and on G by imposing some conditions related to the optimal control theory and we reduce the general problem by working with Lebesgue-Stieltjes integral equations and utilizing their methods and results
Summary
It should be noted that our results lead in a natural way to some existence theorems for differential and difference inclusions or impulsive problems (for particular measures). The following property of the indefinite Kurzweil-Stieltjes integral implies that solutions of measure differential equations are regulated functions.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
