Filippov lemma for measure differential inclusion

Abstract

AbstractIn this work we propose a Filippov‐type lemma for the differential inclusion where is a given multifunction and μ is a finite Borel signed measure on [0, T] (possibly atomic). By a solution of (0.1) we mean a function such that and where is a μ‐integrable function such that for μ‐almost every and stands for either (0, t] for each or [0, t). Such setting leads to at least two nonequivalent notions of a solution to (0.1) and therefore we formulate two different Filippov‐type inequalities (Theorems 2.1 and 2.2). These two concepts coincide in case of the Lebesgue measure. The purpose of our considerations is to cover a class of impulsive control systems, a class of stochastic systems and differential systems on time scales.