Nash ε-equilibrium in state constraint online games: Tanaka-Yokoyama function analysis and inertial mirror descent in continuous time Nash ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-equilibrium in state constraint

• Published in: International journal of dynamics and control Vol. 13; no. 9
• Main Authors: Nazin, Alexander, Chairez, Isaac, Poznyak, Alexander
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 28.08.2025
• Subjects: Complexity; Control; Control and Systems Theory; Dynamical Systems; Engineering; Vibration; Inertial mirror descent; State restrictions; Legendre–Fenchel transformation; Dynamic games; Nash equilibrium
• ISSN: 2195-268X; 2195-2698
• DOI: 10.1007/s40435-025-01817-0

This study focuses on the development of an online inertial mirror descent algorithm for characterizing a novel class of averaged ε -Nash equilibrium in a class of non-cooperative multiplayer games with dynamic strategies in continuous time. Each player’s dynamic strategy is subject to constraints defined on a compact and convex set. Using the Tanaka-Yokohama formula, which characterizes the Nash ε -equilibrium, the strategy is determined for each player in the dynamic game. The Min–Max property of this function confirms the existence of the Nash ε -equilibrium. The algorithm design employs the Legendre–Fenchel transform and a selected proxy function to facilitate the inertial mirror descent approach for the averaged trajectories of the game dynamics. This transformation is instrumental in proving the convergence to the Nash ε -equilibrium with a rate of O ( t - 1 ) . In addition, various proxy functions are proposed and analyzed for their effectiveness in constructing the online inertial mirror descent algorithm. A numerical example contributes to evidence of the application of the mirror descent algorithm presented in this study, considering two players with a state of three components each. A particular selection of a proxy function characterizes the existence of the Nash ϵ -equilibrium.