Nash Equilibrium Seeking with Players Acting through Heat PDE Dynamics

We propose a non-model based strategy for locally stable convergence to Nash equilibria in quadratic noncooperative (duopoly) games with player actions subject to diffusion (heat) PDE dynamics with distinct diffusion coefficients and each player having access only to his own payoff value. The proposed approach employs extremum seeking, with sinusoidal perturbation signals employed to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown quadratic functions. In our previous work, we solved Nash equilibrium seeking problems with input delays. This is the first instance of noncooperative games being tackled in a model-free fashion in the presence of heat PDE dynamics. In order to compensate distinct diffusion processes in the inputs of the two players, we employ boundary control with averaging-based estimates. We apply a small-gain analysis for the resulting Input-to-State Stable (ISS) parabolic PDE-ODE loop as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the heat PDEs, in order to obtain local convergence results to a small neighborhood of the Nash equilibrium. We quantify the size of these residual sets and illustrate the theoretical results numerically on an example of a two-player game under heat PDEs.