Characterization of Filippov representable maps and Clarke subdifferentials

The ordinary differential equation \(\dot{x}(t)=f(x(t)), \; t \geq 0 \), 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.