Distributed Nash Equilibrium Seeking With Dynamics Subject to Disturbance of Unknown Frequencies Over Jointly Strongly Connected Switching Networks

Abstract

In this paper, we study the problem of distributed Nash equilibrium seeking of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> -player games with single integrator dynamics subject to disturbances generated by an uncertain exosystem. Similar problems have been studied for games with single integrator dynamics subject to disturbances generated by a known exosystem. In addition, this paper offers some other new features. First, the existing results only apply to static, connected, and undirected graphs while the result of this paper applies to jointly strongly connected switching graphs, which can be disconnected at every time instant. Second, the existing results are asymptotic or ultimately uniformly bounded, while our result guarantees the exponential convergence when the exosystem is known. Third, the validity of existing results relies on an inequality involving some Lipschitz constant and the smallest nonzero eigenvalue of the Laplacian of the network graph. In contrast, the result of this paper is unconditional other than satisfying the same assumptions on the cost functions as those in the literature. Our design is illustrated by an application to velocity-actuated robots in sensor networks.