The Hamilton–Jacobi–Bellman Equation for Differential Games with Composite Distribution of Random Time Horizon

Balas, Tatyana; Tur, Anna

doi:10.3390/math11020462

Cited by 2 publications

(2 citation statements)

References 15 publications

(20 reference statements)

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“…The duration of the game is a random variable with composite distribution function [23]. Thus, the cumulative distribution function, F(t), with different shifted exponential distributions is denoted as…”

Section: Cooperative Game With Different Shifted Exponential Distribu...mentioning

confidence: 99%

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

Wu,

Tur,

Wang

2023

Entropy

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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.

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Section: Cooperative Game With Different Shifted Exponential Distribu...mentioning

confidence: 99%

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

Wu,

Tur,

Wang

2023

Entropy

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“…In these works, the solutions were found in the class of open-loop strategies using the maximum Pontryagin principle. In (Balas and Tur, 2023), the cooperative games with a composite distribution function for the terminal time were studied using the methods of dynamic programming.…”

Section: Introductionmentioning

confidence: 99%

Public Good Differential Game with Composite Distribution of Random Time Horizon

Balas,

Tur

2023

Contributions to Game Theory and Management

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Differential games with random duration are considered. In some cases, the probability density function of the terminal time can change depending on different conditions and we cannot use the standard distribution. The purpose of this work is studying of games with a composite distribution function for terminal time using the dynamic programming methods. The solutions of the cooperative and non-cooperative public good differential game with random duration are considered.

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The Hamilton–Jacobi–Bellman Equation for Differential Games with Composite Distribution of Random Time Horizon

Cited by 2 publications

References 15 publications

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

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

Public Good Differential Game with Composite Distribution of Random Time Horizon

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