Distributed Nash equilibrium seeking for multi‐agent systems with uncertainty in the cost function

Abstract In this paper, the Nash equilibrium (NE) seeking problem is investigated for a class of ‐integrator multi‐agent system with uncertain parameter in the cost function. In the game setup, each agent only accesses its neighbors' information. Without knowing the probability distribution of the uncertain parameter, the robust counterpart of the studied game is considered. Due to the nonsmoothness of the robust game, a smooth game that consists of real agents as well as fictive agents dealing with uncertainty is formulated. In contrast to conventional robust convex‐concave games, the assumption on the uncertain parameter is relaxed to Polyak‐Łojasiewicz (PL) inequality. Based on the relationship between the robust game and its reformulated version, a distributed NE seeking strategy is developed such that the states of the non‐cooperative agent can converge to the NE solution. At the end, numerical examples are given to verify the validity of the proposed method.