Robust Dynamic Games With State Constraints

ABSTRACT The goal of this study is to construct a robust tracking controller based on an Average Sub‐Gradient (ASG) strategy to find the Nash ‐equilibrium for a class of non‐cooperative multiplayer games. We consider a class of controlled nonlinear representations for each player, the dynamics of which are governed by a vector system of ordinary differential equations with a right‐hand side that is partially known. The dynamics of each player in the game have certain constraints on the state variables, assuming that the state variables and their time derivatives can be measured. Applying Tanaka–Yokohama's formula that characterizes the Nash equilibrium leads to finding the strategy for each player in the dynamic game. The Legendre–Fenchel transform and a chosen proxy function are utilized to develop this mathematical development using the mirror descent approach. ASG and the Integral Sliding‐Mode methodology for continuous‐time systems are extended to enforce the existence of the Nash ‐equilibrium. The primary findings demonstrate that the desired regime—obtained from a non‐stationary analogous of the sliding surface—can be achieved from the beginning of the process and get an upper bound on the Tanaka–Yokohama's function's decrement. A numerical example shows the application of the proposed controller on a co‐culture of two microorganisms with mutual interaction.