The paper is devoted to the study of the measure-driven differential inclusions dx(t)∈G(t,x(t))dμ(t), x(0)=x0 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.

We focus on a very general problem in the theory of dynamic systems, namely that of studying measure differential inclusions with varying measures. The multifunction on the right hand side has compact non-necessarily convex values in a real Euclidean space and satisfies bounded variation hypotheses with respect to the Pompeiu excess (and not to the Hausdorff-Pompeiu distance, as usually in literature). This is possible due to the use of interesting selection principles for excess bounded variation set-valued mappings. Conditions for the minimization of a generic functional with respect to a family of measures generated by equiregulated left-continuous, nondecreasing functions and to associated solutions of the differential inclusion driven by these measures are deduced, under constraints only on the initial point of the trajectory.