Nash Equilibrium Seeking in Heterogeneous Noncooperative Games with Players Acting Through Heat PDE Dynamics and Delays

Abstract

We propose a non-model based strategy for locally stable convergence to Nash equilibria in a quadratic noncooperative (duopoly) game with player actions subject to heterogeneous PDE dynamics. In this duopoly scenario, where different players use different types of PDEs, one player compensates for a delay (transport PDE) and the other a heat (diffusion) PDE, each player having access only to his own payoff value. The proposed approach employs extremum seeking, with sinusoidal perturbation signals applied to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown quadratic functions. In our previous works, we solved Nash equilibrium seeking problems with homogeneous games, where the PDE dynamics of distinct nature were not allowed. This is the first instance of noncooperative games being tackled in a model-free fashion in the presence of heat PDE dynamics AND delays. In order to compensate distinct PDE-modeled 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) coupled hyperbolic-parabolic PDE system as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the heat PDE and the delay, 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 combining hyperbolic and parabolic dynamics in a 2-player setting.