On periodic solution to control problem with time-driven switching

Gromov, Dmitry; Bondarev, Anton; Gromova, Ekaterina

doi:10.1007/s11590-021-01749-6

Cited by 6 publications

(7 citation statements)

References 17 publications

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“…Some of these papers also considered cyclic or fluctuating optimal dynamics. However, to the best of our knowledge, the optimal policy in the form of hybrid limit cycle (HLC) was first reported in (Gromov, Bondarev and Gromova, 2021). We contribute to the line of research pioneered by that paper by proving the uniqueness of such a cyclical policy and further expanding it within the context of applied environmental problems.…”

Section: Introductionmentioning

confidence: 82%

Optimality and sustainability of hybrid limit cycles in the pollution control problem with regime shifts

Громов¹,

Shigoka²,

Bondarev³

2022

Preprint

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In this paper, we consider the problem of pollution control in a system that undergoes regular regime shifts. We first show that the optimal policy of pollution abatement is periodic as well, and is described by the unique hybrid limit cycle. We next introduce the notion of an environmentally sustainable solution, and demonstrate that such a policy is the only one that yields the best possible trade-off between steadily achieving profit and ensuring environmental preservation. In contrast to that, the policy that is not environmentally sustainable eventually enters stagnation. To further illustrate our findings, we compare the optimal periodic solution with a myopic one. Interestingly enough, the myopic solution yields higher overall payoff in the short-run, but completely fails in the long-run, while the environmentally sustainable policy yields maximal payoff and preserves the environment over the infinite time interval.

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Section: Introductionmentioning

confidence: 82%

Optimality and sustainability of hybrid limit cycles in the pollution control problem with regime shifts

Громов¹,

Shigoka²,

Bondarev³

2022

Preprint

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“…. r − 1 can be represented in the following form: (11) and the payoff of the player i at the whole game ( 12) can be written as…”

Section: T L(t)mentioning

confidence: 99%

“…We mention the works [7,8] for an overview of the main results on the hybrid optimal control. Some more examples of optimal control and-to some extent-differential games with regime switching can be found in [9][10][11][12][13]. It was only recently that the theory of hybrid control was formally extended to game-theoretic problems.…”

Section: Introductionmentioning

confidence: 99%

Time-Consistency of an Imputation in a Cooperative Hybrid Differential Game

Gromova

Malakhova

2021

Mathematics

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This work is aimed at studying the problem of maintaining the sustainability of a cooperative solution in an n-person hybrid differential game. Specifically, we consider a differential game whose payoff function is discounted with a discounting function that changes its structure with time. We solve the problem of time-inconsistency of the cooperative solution using a so-called imputation distribution procedure, which was adjusted for this general class of differential games. The obtained results are illustrated with a specific example of a differential game with random duration and a hybrid cumulative distribution function (CDF). We completely solved the presented example to demonstrate the application of the developed scheme in detail. All results were obtained in analytical form and illustrated by numerical simulations.

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“…In addition, changed systems, in the form of time-driven and state-driven switches, are increasingly common. The up-to-date contributions to those fields are [6][7][8][9][10]. In [9,11], an optimal solution in the form of a hybrid limit cycle (HLC) was introduced as the best possible candidate for the infinite-horizon optimization problem.…”

Section: Introductionmentioning

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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On periodic solution to control problem with time-driven switching

Cited by 6 publications

References 17 publications

Optimality and sustainability of hybrid limit cycles in the pollution control problem with regime shifts

Optimality and sustainability of hybrid limit cycles in the pollution control problem with regime shifts

Time-Consistency of an Imputation in a Cooperative Hybrid Differential Game

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

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