Sustainable Optimal Control for Switched Pollution-Control Problem with Random Duration
Considering the uncertainty of game duration and periodic seasonal fluctuation, an <i>n</i>-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 considere...
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MDPI AG
2023-10-01
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| Online Access: | https://www.mdpi.com/1099-4300/25/10/1426 |
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| author | Yilun Wu Anna Tur Hongbo Wang |
| author_facet | Yilun Wu Anna Tur Hongbo Wang |
| author_sort | Yilun Wu |
| collection | DOAJ |
| description | Considering the uncertainty of game duration and periodic seasonal fluctuation, an <i>n</i>-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, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>f</mi></msub></semantics></math></inline-formula>, described by the shifted exponential distribution. In the second case, we assumed that players’ equipment is heterogeneous, and the <i>i</i>-th player’s equipment failure time, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mrow><mi>f</mi><mi>i</mi></mrow></msub></semantics></math></inline-formula>, 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>f</mi></msub><mo>=</mo><mi>m</mi><mi>i</mi><mi>n</mi><mrow><mo>{</mo><msub><mi>T</mi><mrow><mi>f</mi><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>T</mi><mrow><mi>f</mi><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. 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 <i>n</i>-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. |
| first_indexed | 2024-03-10T21:16:39Z |
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| id | doaj.art-6ba883166ef7446fa58695017660209d |
| institution | Directory Open Access Journal |
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| language | English |
| last_indexed | 2024-03-10T21:16:39Z |
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| spelling | doaj.art-6ba883166ef7446fa58695017660209d2023-11-19T16:24:43ZengMDPI AGEntropy1099-43002023-10-012510142610.3390/e25101426Sustainable Optimal Control for Switched Pollution-Control Problem with Random DurationYilun Wu0Anna Tur1Hongbo Wang2College of Electronic Science and Engineering, Jilin University, Jilin 130012, ChinaFaculty of Applied Mathematics and Control Processes, Saint Petersburg University, Saint Petersburg 199034, RussiaCollege of Electronic Science and Engineering, Jilin University, Jilin 130012, ChinaConsidering the uncertainty of game duration and periodic seasonal fluctuation, an <i>n</i>-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, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>f</mi></msub></semantics></math></inline-formula>, described by the shifted exponential distribution. In the second case, we assumed that players’ equipment is heterogeneous, and the <i>i</i>-th player’s equipment failure time, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mrow><mi>f</mi><mi>i</mi></mrow></msub></semantics></math></inline-formula>, 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>f</mi></msub><mo>=</mo><mi>m</mi><mi>i</mi><mi>n</mi><mrow><mo>{</mo><msub><mi>T</mi><mrow><mi>f</mi><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>T</mi><mrow><mi>f</mi><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. 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 <i>n</i>-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.https://www.mdpi.com/1099-4300/25/10/1426differential gamecooperative gameswitched systemrandom durationsustainability |
| spellingShingle | Yilun Wu Anna Tur Hongbo Wang Sustainable Optimal Control for Switched Pollution-Control Problem with Random Duration Entropy differential game cooperative game switched system random duration sustainability |
| title | Sustainable Optimal Control for Switched Pollution-Control Problem with Random Duration |
| title_full | Sustainable Optimal Control for Switched Pollution-Control Problem with Random Duration |
| title_fullStr | Sustainable Optimal Control for Switched Pollution-Control Problem with Random Duration |
| title_full_unstemmed | Sustainable Optimal Control for Switched Pollution-Control Problem with Random Duration |
| title_short | Sustainable Optimal Control for Switched Pollution-Control Problem with Random Duration |
| title_sort | sustainable optimal control for switched pollution control problem with random duration |
| topic | differential game cooperative game switched system random duration sustainability |
| url | https://www.mdpi.com/1099-4300/25/10/1426 |
| work_keys_str_mv | AT yilunwu sustainableoptimalcontrolforswitchedpollutioncontrolproblemwithrandomduration AT annatur sustainableoptimalcontrolforswitchedpollutioncontrolproblemwithrandomduration AT hongbowang sustainableoptimalcontrolforswitchedpollutioncontrolproblemwithrandomduration |