Sustainable solution for hybrid differential game with regime shifts and random duration

Switching phenomena are ubiquitous in real-world applications. An $n$-player hybrid pollution-control problem is considered with switching behavior and uncertain game duration. The payoff functional with a random duration is converted to a payoff functional in the infinite horizon, which is discounted by a heterogeneous discounting function. By applying Pontryagin’s maximum principle and analyzing the structure of the adjoint variable, the sustainable equilibria in cooperative and noncooperative games are uniquely determined. The convergence of the corresponding state variable in cooperative and noncooperative games is proved. Furthermore, a unique state trajectory represented by a hybrid limit cycle is found to compute the players’ payoffs in cooperative and noncooperative scenarios. Finally, a reasonable, cooperative solution that employs the Shapley value as a single-point solution is proposed. All results are derived analytically and demonstrated with a numerical example.