Stability Criterion for Nonsmooth Systems and Nash Equilibrium Seeking via Projected Gradient Dynamics

We derive a novel stability criterion for nonsmooth dynamical systems by virtue of a new set-valued Lie derivative of nonsmooth Lyapunov functions. This set-valued Lie derivative requires no computation of generalized gradients. Instead, it only calculates the directional derivatives. Moreover, our criterion allows for Lyapunov function candidates to be locally Lipschitz continuous and not necessarily regular. These merits strengthen the existing stability criterion and simplify its checking process. As an application, we establish the stability of projected gradient dynamics for distributed Nash equilibrium seeking under very mild conditions, compared to the existing ones.