Algorithms and the calculation of Nash equilibria for multi-objective control of time-discrete systems and polynomial-time algorithms for dynamic c-games on networks

We consider a multi-objective control problem of time-discrete systems with given starting and final states. The dynamics of the system are controlled by p actors (players). Each of the players intends to minimize his own integral-time cost of the system’s transitions using a certain admissible trajectory. Nash Equilibria conditions are derived and algorithms for solving dynamic games in positional form are proposed in this paper. The existence theorem for Nash equilibria is related to the introduction of an auxiliary dynamic c-game. Stationary and non-stationary cases are described. The paper concludes with a complexity analysis for that decision process.