Characterization of Filippov representable maps and Clarke subdifferentials

Abstract

The ordinary differential equation x˙ (t) = f(x(t)), t ≥ 0, for f measurable, is not sufficiently regular to guarantee existence of solutions. To remedy this we may relax the problem by replacing the function f with its Filippov regularization Ff and consider the differential inclusion x˙ (t) ∈ Ff (x(t)) which always has a solution. It is interesting to know, inversely, when a setvalued map can be obtained as the Filippov regularization of a (single-valued, measurable) function. In this work we give a full characterization of such set-valued maps, hereby called Filippov representable. This characterization also yields an elegant description of those maps that are Clarke subdifferentials of a Lipschitz function.