A Fully Distributed Nash Equilibrium Seeking Algorithm for N-Coalition Games of Euler–Lagrange Players

This article investigates the N-coalition games of Euler–Lagrange (EL) systems. In the games, every coalition consists of a group of players, whose dynamics are formulated as EL equations. The objective of each coalition is to minimize its cost function, which is the sum of local cost functions of the players in the corresponding coalition. That is, the players within the coalition collaboratively minimize their coalition’s cost function. The cost function of each player is only available to itself and depends on not only its decision, but also the aggregate of all decisions. Based on gradient descent, consensus technique, and state feedback, a fully distributed Nash equilibrium seeking algorithm for N-coalition games is proposed. The exponential convergence of the algorithm is analyzed by the Lyapunov stability theorem. A numerical example in smart grids is given to illustrate the effectiveness of the proposed algorithm.