Fixed-Time Seeking and Tracking of Time-Varying Nash Equilibria in Noncooperative Games

We study the solution of time-varying Nash equilibrium seeking and tracking problems in non-cooperative games via nonsmooth, model-based and model-free algorithms. Specifically, for potential and non-potential games, we derive tracking bounds for the actions of the players with respect to the Nash Equilibrium Trajectory (NET) of the game using the property of fixed-time input-to-state stability. We show that, in the model-based case, traditional pseudo-gradient flows achieve only exponential tracking with a residual error that is proportional to the time-variation of the NET. In contrast, exact and fixed-time tracking can be achieved by using nonsmooth dynamics with discontinuous vector fields. For continuous but non-Lipschitz dynamics, we show that the residual tracking error can be dramatically decreased whenever the learning gains of the dynamics exceed a particular threshold. In the model-free case, we derive similar semi-global practical input-to-state stability bounds using multi-time scale tools for nonsmooth systems.