Nash Equilibrium Seeking with Players Acting through Heat PDE Dynamics

Tiago Roux Oliveira, Victor Hugo Pereira Rodrigues, Miroslav Krstic, Tamer Basar

Research output: Chapter in Book/Report/Conference proceedingConference contribution


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.

Original languageEnglish (US)
Title of host publication2021 American Control Conference, ACC 2021
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages6
ISBN (Electronic)9781665441971
StatePublished - May 25 2021
Externally publishedYes
Event2021 American Control Conference, ACC 2021 - Virtual, New Orleans, United States
Duration: May 25 2021May 28 2021

Publication series

NameProceedings of the American Control Conference
ISSN (Print)0743-1619


Conference2021 American Control Conference, ACC 2021
Country/TerritoryUnited States
CityVirtual, New Orleans

ASJC Scopus subject areas

  • Electrical and Electronic Engineering


Dive into the research topics of 'Nash Equilibrium Seeking with Players Acting through Heat PDE Dynamics'. Together they form a unique fingerprint.

Cite this