On contraction of piecewise smooth systems

In this paper we extend to a class of piecewise-smooth dynamical systems, a fundamental property of dynamical systems which has been used in a number of different applications in the case of smooth dynamical systems: contraction theory. We give analytical conditions under which trajectories of discontinuous vector fields, satisfying Caratheodory conditions for the existence and unicity of a solution, converge towards each other. In particular, we prove that if each mode of the vector field is contracting then the dynamical systems of interest is contracting. We apply our results to the problem of synchronizing a network of piecewise-linear dynamical systems.