Finite-time stability and optimal control of an impulsive stochastic reaction-diffusion vegetation-water system driven by L$ {\rm \acute{e}} $vy process with time-varying delay

Zixiao Xiong; Xining Li; Ming Ye; Qimin Zhang; Zixiao Xiong; Xining Li; Ming Ye; Qimin Zhang

doi:10.3934/mbe.2021419

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 6: 8462-8498. doi: 10.3934/mbe.2021419

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Finite-time stability and optimal control of an impulsive stochastic reaction-diffusion vegetation-water system driven by L$ {\rm \acute{e}} $vy process with time-varying delay

1.
School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China
2.
Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, USA
3.
Department of Earth, Ocean, and Atmospheric Science, Florida State University, Tallahassee, FL 32306, USA

Academic Editor: Glenn Ledder

Received: 02 August 2021 Accepted: 08 September 2021 Published: 27 September 2021

In this paper, a reaction-diffusion vegetation-water system with time-varying delay, impulse and L$ {\rm \acute{e}} $vy jump is proposed. The existence and uniqueness of the positive solution are proved. Meanwhile, mainly through the principle of comparison, we obtain the sufficient conditions for finite-time stability which reflect the effect of time delay, diffusion, impulse, and noise. Besides, considering the planting, irrigation and other measures, we introduce control variable into the vegetation-water system. In order to save the costs of strategies, the optimal control is analyzed by using the minimum principle. Finally, numerical simulations are shown to illustrate the effectiveness of our theoretical results.
- vegetation-water model,
- optimal control,
- finite-time stability,
- L$ {\rm \acute{e}} $vy process
Citation: Zixiao Xiong, Xining Li, Ming Ye, Qimin Zhang. Finite-time stability and optimal control of an impulsive stochastic reaction-diffusion vegetation-water system driven by L$ {\rm \acute{e}} $vy process with time-varying delay[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 8462-8498. doi: 10.3934/mbe.2021419

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Abstract

In this paper, a reaction-diffusion vegetation-water system with time-varying delay, impulse and L$ {\rm \acute{e}} $vy jump is proposed. The existence and uniqueness of the positive solution are proved. Meanwhile, mainly through the principle of comparison, we obtain the sufficient conditions for finite-time stability which reflect the effect of time delay, diffusion, impulse, and noise. Besides, considering the planting, irrigation and other measures, we introduce control variable into the vegetation-water system. In order to save the costs of strategies, the optimal control is analyzed by using the minimum principle. Finally, numerical simulations are shown to illustrate the effectiveness of our theoretical results.

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