Sustainable Optimal Control for Switched Pollution-Control Problem with Random Duration

Abstract

Considering the uncertainty of game duration and periodic seasonal fluctuation, an n-player switched pollution-control differential game is modeled to investigate a sustainable and adaptive strategy for players. Based on the randomness of game duration, two scenarios are considered in this study. In the first case, the game duration is a random variable, Tf, described by the shifted exponential distribution. In the second case, we assumed that players’ equipment is heterogeneous, and the i-th player’s equipment failure time, Tfi, is described according to the shifted exponential distribution. The game continues until a player’s equipment breaks down. Thus, the game duration is defined as Tf=min{Tf1,…,Tfn}. To achieve the goal of sustainable development, an environmentally sustainable strategy and its corresponding condition are defined. By using Pontryagin’s maximum principle, a unique control solution is obtained in the form of a hybrid limit cycle, the state variable converges to a stable hybrid limit cycle, and the total payoff of all players increases and then converges. The results indicate that the environmentally sustainable strategy in the n-player pollution-control cooperative differential game with switches and random duration is a unique strategy that not only ensures profit growth but also considers environmental protection.