Complete stability analysis of a control law for walking robots with non-permanent contacts

We're interested here in the stability analysis of a simple control law, for walking robots without any assumption on the state of the contacts. We propose to use the framework of nonsmooth dynamical systems since it provides a general formulation of the dynamics that does not depend on the contact state. Classical stability results cannot be applied in this framework, so we needed to derive a Lyapunov stability theorem for Lagrangian dynamical systems with non-permanent contacts. Based on this theorem, we will then prove, here, the stability of this control law.