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Theory and application of stability for stochastic reaction diffusion systems

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Abstract

So far, the Lyapunov direct method is still the most effective technique in the study of stability for ordinary differential equations and stochastic differential equations. Due to the shortage of the corresponding Itô formula, this useful method has not been popularized in stochastic partial differential equations. The aim of this work is to try to extend the Lyapunov direct method to the Itô stochastic reaction diffusion systems and to establish the corresponding Lyapunov stability theory, including stability in probablity, asymptotic stability in probability, and exponential stability in mean square. As the application of the obtained theorems, this paper addresses the stability of the Hopfield neural network and points out that the main results obtained by Holden Helge and Liao Xiaoxin et al. can be all regarded as the corollaries of the theorems presented in this paper.

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Authors and Affiliations

  1. College of Information and Control, Nanjing University of Information Science & Technology, Nanjing, 210044, China

    Luo Qi

  2. College of Automation, South China University of Technology, Guangzhou, 410640, China

    Deng FeiQi

  3. Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH, Scotland, UK

    Mao XueRong

  4. College of Mathematics Science, Inner Mongolia Normal University, Huhhot, 010000, China

    Bao JunDong

  5. College of Mathematics and Physics, Nanjing University of Information Science & Technology, Nanjing, 210044, China

    Zhang YuTian

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  1. Luo Qi
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  2. Deng FeiQi
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  3. Mao XueRong
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  4. Bao JunDong
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  5. Zhang YuTian
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Correspondence to Luo Qi.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 60574042)

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Luo, Q., Deng, F., Mao, X. et al. Theory and application of stability for stochastic reaction diffusion systems. Sci. China Ser. F-Inf. Sci. 51, 158–170 (2008). https://doi.org/10.1007/s11432-008-0020-6

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  • DOI: https://doi.org/10.1007/s11432-008-0020-6

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