Filippov lemma for measure differential inclusion

In this work we propose a Filippov‐type lemma for the differential inclusion 0.1 ddμx(t)∈F(t,x(t)),x(0)=x0,where F:[0,T]×Rd⇝Rd 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 x:[0,T]⟶Rd such that x(0)=x0 and x(t)=x0+∫S(t)v(s)dμ(s)fort>0,where v(·) is a μ‐integrable function such that v(t)∈F(t,x(t)) for μ‐almost every t∈[0,T] and S(t) stands for either (0, t] for each t∈J 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.