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Stability in Mean of Partial Variables for Coupled Stochastic Reaction‐Diffusion Systems on Networks: A Graph Approach

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  • Yonggui Kao
  • Hamid Reza Karimi

Abstract

This paper is devoted to investigating stability in mean of partial variables for coupled stochastic reaction‐diffusion systems on networks (CSRDSNs). By transforming the integral of the trajectory with respect to spatial variables as the solution of the stochastic ordinary differential equations (SODE) and using Itô formula, we establish some novel stability principles for uniform stability in mean, asymptotic stability in mean, uniformly asymptotic stability in mean, and exponential stability in mean of partial variables for CSRDSNs. These stability principles have a close relation with the topology property of the network. We also provide a systematic method for constructing global Lyapunov function for these CSRDSNs by using graph theory. The new method can help to analyze the dynamics of complex networks. An example is presented to illustrate the effectiveness and efficiency of the obtained results.

Suggested Citation

  • Yonggui Kao & Hamid Reza Karimi, 2014. "Stability in Mean of Partial Variables for Coupled Stochastic Reaction‐Diffusion Systems on Networks: A Graph Approach," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:597502
    DOI: 10.1155/2014/597502
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    References listed on IDEAS

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    1. Xiao Liang & Linshan Wang, 2012. "Exponential Stability for a Class of Stochastic Reaction-Diffusion Hopfield Neural Networks with Delays," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-12, March.
    2. Jun Hu & Zidong Wang & Hongli Dong & Huijun Gao, 2013. "Recent Advances on Recursive Filtering and Sliding Mode Design for Networked Nonlinear Stochastic Systems: A Survey," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-12, December.
    3. Duncan J. Watts & Steven H. Strogatz, 1998. "Collective dynamics of ‘small-world’ networks," Nature, Nature, vol. 393(6684), pages 440-442, June.
    4. Ignatyev, Oleksiy, 2009. "Partial asymptotic stability in probability of stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 597-601, March.
    5. Xiao Liang & Linshan Wang, 2012. "Exponential Stability for a Class of Stochastic Reaction‐Diffusion Hopfield Neural Networks with Delays," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
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    1. Yanbo Li & Peng Zhang & Yonggui Kao & Hamid Reza Karimi, 2014. "Quantized State‐Feedback Stabilization for Delayed Markovian Jump Linear Systems with Generally Incomplete Transition Rates," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

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