Fixed-Time Nash Equilibrium Seeking in Time-Varying Networks

Abstract

In this paper we introduce first-order and zeroth-order Nash equilibrium seeking dynamics with fixed-time convergence certificates for non-cooperative games having finitely many players. The first-order algorithms achieve exact convergence to the Nash equilibrium of the game in a finite time that can be additionally upper bounded by a constant that is independent of the initial conditions of the actions of the players. Moreover, these fixed-time bounds can be prescribed a priori by the system designer under an appropriate tuning of the parameters of the algorithms. When players have access only to measurements of their cost functions, we consider a class of distributed zeroth-order model-free adaptive dynamics that achieve semi-global practical fixed-time stability, qualitatively preserving the fixed-time bounds of the first-order dynamics. Moreover, by leveraging the property of fixed-time input-to-state stability, further results are obtained for mixed games where some of the players implement different seeking dynamics. Fast and slow switching communication graphs are also incorporated using tools from hybrid dynamical systems theory. We consider potential games as well as gener