Wu Yilun, Tur Anna, Wang Hongbo
College of Electronic Science and Engineering, Jilin University, Jilin 130012, China.
Faculty of Applied Mathematics and Control Processes, Saint Petersburg University, Saint Petersburg 199034, Russia.
Entropy (Basel). 2023 Oct 8;25(10):1426. doi: 10.3390/e25101426.
Considering the uncertainty of game duration and periodic seasonal fluctuation, an -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 -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 -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.
考虑到博弈持续时间的不确定性和周期性季节波动，建立了一个n人切换污染控制微分博弈模型，以研究参与者的可持续和自适应策略。基于博弈持续时间的随机性，本研究考虑了两种情况。在第一种情况下，博弈持续时间是一个随机变量Tf，由移位指数分布描述。在第二种情况下，我们假设参与者的设备是异构的，第i个参与者的设备故障时间Tfi根据移位指数分布描述。博弈持续到某个参与者的设备发生故障。因此，博弈持续时间定义为Tf = min{Tf1,…, Tfn}。为实现可持续发展目标，定义了一种环境可持续策略及其相应条件。通过使用庞特里亚金极大值原理，获得了一个以混合极限环形式存在的唯一控制解，状态变量收敛到一个稳定的混合极限环，并且所有参与者的总收益增加然后收敛。结果表明，具有切换和随机持续时间的n人污染控制合作微分博弈中的环境可持续策略是一种独特的策略，它不仅确保利润增长，还考虑环境保护。
