Characterization of Filippov representable maps and Clarke subdifferentials

Abstract

The ordinary differential equation $\dot{x}(t)=f(x(t)),t\ge 0$, 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 $F_f$ and consider the differential inclusion $\dot{x}(t)\in F_f(x(t))$ which always has a solution. It is interesting to know, inversely, when a set-valued map $\Phi$ 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.