On the Existence of Solutions to Differential Inclusions with Nonconvex Right-Hand Sides

Abstract

We study the existence of solutions of differential inclusions with upper semicontinuous right-hand sides. The investigation was prompted by the well-known Filippov examples. We define a new concept, “colliding on a set.” In the case when the admissible velocities do not “collide” on the set of discontinuities of the right-hand side, we expect that at least one trajectory emanates from every point. If the velocities do “collide” on the set of discontinuities of the right-hand side, the existence of solutions is not guaranteed, as is seen from one of Filippov's examples. In this case we impose an additional condition in order to prove the existence of a solution starting at a point of the discontinuity set. For the right-hand sides under consideration, we assume the following: whenever the velocities “collide” on a set S there exist tangent velocities (belonging to the Clarke tangent cone to S) on a dense subset of S. Then we prove the existence of an $\varepsilon$-solution for every $\varepsilon >0$. Under additional assumptions we can pass to the limit as $\varepsilon \to 0$ and obtain a solution of the considered differential inclusion.