On Filippov and Utkin sliding solutions of discontinuous systems

Discontinuous dynamical systems with sliding modes are often used in Control Theory to model differential equations with discontinuous control. Filippov and Utkin (see(2,7)) have proposed two different approaches to define the solution of these dynamical systems. In case of linear systems, these two approaches are equivalent, but in case of nonlinear systems, the ways to extend the vector field on the sliding surface is generally different. In this note, we obtain a seemingly new approach, which lends support to Utkin's approach, but it is somewhat more general.