Online non-cooperative zero-sum games for linear systems over wireless MIMO fading channels with uncountable state space

In this paper, we explore a non-cooperative zerosum game for a linear dynamic system, operating over wireless Multiple-Input Multiple-Output (MIMO) fading channels that feature an uncountable channel state space between the remote controllers and the actuator of the dynamic plant. We initiate our work by framing the stochastic zero-sum game as an ergodic optimization problem, from which we derive the structural properties of the coupled optimality equations. To overcome the “curse of dimensionality,” we develop equivalent structured reduced-state optimality equations. Leveraging these equations, we derive the necessary and sufficient conditions for the existence of a Nash equilibrium in the stochastic game. Furthermore, we propose an online structured learning algorithm based on Stochastic Approximation (SA) to compute the Nash equilibrium. Employing Ordinary Differential Equation (ODE) techniques and Lyapunov stability analysis, we demonstrate the asymptotic optimality of our learning algorithm. Numerical results show that the proposed method outperforms existing state-of-the-art baselines.