Filippov solutions on a Lipschitz continuous surface

In this paper, the further discussion on Filippov solutions is presented. New criterions are proposed to determine the relation between system trajectories and an arbitrary Lipschitz continuous surface. We show that the set-valued vector field of Filippovpsilas differential inclusion and the cone property of the hypersurface play an important role in the criterions. Though the hypersurface is only Lipschitz continuous, we prove that there exists the trajectory sliding along the surface. This result allows us to construct the sliding mode control for more general sense, because the sliding surface can be designed Lipschitz continuous. Finally, we provide some numerical examples to illustrate our designs.