Nash Equilibrium Seeking for Games in Hybrid Systems

Nash equilibrium seeking for games in a class of hybrid systems is investigated in this paper. Different from the existing works, the players in the present game framework are composite of a set of continuous-time players and a set of discrete-time players. Nash equilibrium seeking for such a hybrid game is challenging as some players update their actions in continuous time while the remainders update their actions in discrete time. To accommodate the hybrid games, we firstly consider a case in which the players update their actions according to the hybrid gradient play (i.e., the continuous-time players update their actions according to the continuous-time gradient play with sampled information flow while the discrete-time players update their actions according to the discrete-time gradient play). Then, we consider the case in which the players have restricted access to their opponents' actions. A hybrid consensus-based strategy is proposed for this case. The stability of the Nash equilibrium under the proposed hybrid seeking strategies is theoretically proven by utilizing Lyapunov stability analysis. Lastly, the hybrid seeking strategies are validated through a numerical example.