The Filippov-Wazewski relaxation theorem revisited (Q1307420)

The authors deal with the Filippov-Wazewski relaxation theorem which states that the solution set to a Lipschitzian differential inclusion is dense in the set of relaxation solutions. Given a differential inclusion with convex-valued right-hand side the authors study the smaller set-valued map which yields the same attainable sets and, by examining the contingent derivative, they show that such a smaller set necessarily contains all the extremal points of the convex-valued map. This means that two differential inclusions have the same closure of their solution sets if and only if the right side have the same convex hull and therefore, in a certain sense, a converse of the previous relaxation theorem holds.