ϵ$$ \epsilon $$‐Nash equilibrium of non‐cooperative Lagrangian dynamic games based on the average sub‐gradient robust integral sliding mode control

Abstract

SummaryIn non‐cooperative multi‐player games, an average sub‐gradient (ASG) strategy for finding an ‐Nash equilibrium. For convex (but not necessarily strictly convex) individual cost functions, the function convergence of the multi‐player game is demonstrated. Applying Tanaka's formula that characterizes the Nash equilibrium leads to finding the strategy for each player. A min‐max formulation defines the corresponding solution for each player. The main result is proving the reachability of the desired regime, obtaining an explicit upper bound for Tanaka's function decrement. Detailed analyses for two‐player and multi‐player games are developed. For a class of two‐player games with a convex performance function, a set of numerical studies shows the applicability of the proposed method. So, the tracking of the arm in 3‐links robot is considered as an illustrative example.