Optimal control of stochastic system with Fractional Brownian Motion

Chaofeng Zhao; Zhibo Zhai; Qinghui Du; Chaofeng Zhao; Zhibo Zhai; Qinghui Du

doi:10.3934/mbe.2021284

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 5: 5625-5634. doi: 10.3934/mbe.2021284

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Optimal control of stochastic system with Fractional Brownian Motion

1.
School of Information Technology, Luoyang Normal University, Luoyang, 471022, China
2.
College of Mechanical and Equipment Engineering, Hebei University of Engineering, Handan, 056038, China
3.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China

Received: 26 March 2021 Accepted: 14 June 2021 Published: 22 June 2021

In this paper, we introduce a class of stochastic harvesting population system with Fractional Brownian Motion (FBM), which is still unclear when the stochastic noise has the character of memorability. Stochastic optimal control problems with FBM can not be studied using classical methods, because FBM is neither a Markov pocess nor a semi-martingale. When the external environment impact on the system of FBM, the necessary and sufficient conditions for the optimization are offered through the stochastic maximum principle, Hamilton function and $ It\hat{o} $ formula in our work. To illustrate our study, we provide an example to demonstrate the obtained theoretical results, which is the expansion of certainty population system.
- optimal harvesting control,
- $ It\hat{o} $ Formula,
- Fractional Brownian Motion (FBM),
- maximum principle
Citation: Chaofeng Zhao, Zhibo Zhai, Qinghui Du. Optimal control of stochastic system with Fractional Brownian Motion[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 5625-5634. doi: 10.3934/mbe.2021284

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Abstract

In this paper, we introduce a class of stochastic harvesting population system with Fractional Brownian Motion (FBM), which is still unclear when the stochastic noise has the character of memorability. Stochastic optimal control problems with FBM can not be studied using classical methods, because FBM is neither a Markov pocess nor a semi-martingale. When the external environment impact on the system of FBM, the necessary and sufficient conditions for the optimization are offered through the stochastic maximum principle, Hamilton function and $ It\hat{o} $ formula in our work. To illustrate our study, we provide an example to demonstrate the obtained theoretical results, which is the expansion of certainty population system.

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