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Tiago Roux Oliveira
Universidade do Estado do Rio de Janeiro (UERJ), Rio de Janeiro, RJ
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Victor Hugo Pereira Rodrigues
Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, RJ
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Miroslav Krstic
University of California at San Diego (UCSD), San Diego – CA
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Tamer Basar
University of Illinois at Urbana-Champaign, Urbana – IL
Keywords:
Extremum seeking, Nash equilibrium, (Non)cooperative games, Transport equations, Heat equations, Predictor feedback, Averaging theory in infinite dimensions
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. 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.
