Stability in mean of partial variables for coupled stochastic reaction-diffusion systems on networks: A graph approach

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 Ito 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.

[1]  J. Kurths,et al.  Network synchronization, diffusion, and the paradox of heterogeneity. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  T E Browder,et al.  Observation of the D(sJ)(2317) and D(sJ)(2457) in B decays. , 2003, Physical review letters.

[3]  Changhong Wang,et al.  Global stability analysis for stochastic coupled reaction–diffusion systems on networks☆ , 2013 .

[4]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[5]  Mohammad Javad Yazdanpanah,et al.  Partial stabilization of uncertain nonlinear systems. , 2012, ISA transactions.

[6]  Michael Y. Li,et al.  Global-stability problem for coupled systems of differential equations on networks , 2010 .

[7]  Wu-Hua Chen,et al.  Impulsive synchronization of chaotic Lur'e systems via partial states , 2008 .

[8]  Hong-ke Wang,et al.  On the Exponential Stability of Stochastic Differential Equations , 2009, ICFIE.

[9]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[10]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[11]  Yisheng Zhong,et al.  Output consensus analysis and design for high-order linear swarm systems: Partial stability method , 2012, Autom..

[12]  Yisheng Zhong,et al.  Output consensus for high-order linear time-invariant swarm systems , 2012, Int. J. Control.

[13]  Jun Hu,et al.  Quantised recursive filtering for a class of nonlinear systems with multiplicative noises and missing measurements , 2013, Int. J. Control.

[14]  Changsong Deng,et al.  Statistics and Probability Letters , 2011 .

[15]  Jun Hu,et al.  State estimation for a class of discrete nonlinear systems with randomly occurring uncertainties and distributed sensor delays , 2014, Int. J. Gen. Syst..

[16]  Yonggui Kao,et al.  Delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen-Grossberg neural networks with mixed delays , 2012, J. Frankl. Inst..

[17]  Oleksiy A. Ignatyev,et al.  Partial asymptotic stability in probability of stochastic differential equations , 2009 .

[18]  Jinde Cao,et al.  A unified synchronization criterion for impulsive dynamical networks , 2010, Autom..

[19]  Huijun Gao,et al.  Recent Advances on Recursive Filtering and Sliding Mode Design for Networked Nonlinear Stochastic Systems: A Survey , 2013 .

[20]  Magdi S. Mahmoud,et al.  Finite-time analysis and H∞ control for switched stochastic systems , 2012, J. Frankl. Inst..

[21]  Yu. G. Martyshenko,et al.  On partial detectability of the nonlinear dynamic systems , 2009 .

[22]  Magdi S. Mahmoud,et al.  Stability and H∞ Performance Analysis of Switched Stochastic Neutral Systems , 2013, Circuits Syst. Signal Process..

[23]  Jinde Cao,et al.  Synchronization control of switched linearly coupled neural networks with delay , 2010, Neurocomputing.

[24]  Xuerong Mao,et al.  Stochastic differential equations and their applications , 1997 .

[25]  Yonggui Kao,et al.  Delay-Dependent Robust Exponential Stability of Impulsive Markovian Jumping Reaction-Diffusion Cohen-Grossberg Neural Networks , 2012, Neural Processing Letters.

[26]  M H Shafiei,et al.  Partial stabilization-based guidance. , 2012, ISA transactions.

[27]  C. Wu.  On the relationship between pinning control effectiveness and graph topology in complex networks of dynamical systems. , 2008, Chaos.

[28]  Yonggui Kao,et al.  Stability in mean of partial variables for stochastic reaction-diffusion systems with Markovian switching , 2014, J. Frankl. Inst..

[29]  V. Vorotnikov.  On stability and stabilization of motion with respect to a part of the variables , 1982 .

[30]  D. West.  Introduction to Graph Theory , 1995 .

[31]  Qi Luo,et al.  Stabilization of stochastic Hopfield neural network with distributed parameters , 2004, Science in China Series F: Information Sciences.

[32]  Stefano Longhi.  Spiral waves in optical parametric oscillators , 2001 .

[33]  Stability in mean of partial variables for stochastic reaction diffusion systems , 2009 .

[34]  Guowei Yang,et al.  Exponential stability of impulsive stochastic fuzzy reaction-diffusion Cohen-Grossberg neural networks with mixed delays , 2012, Neurocomputing.

[35]  Magdi S. Mahmoud,et al.  Robust H∞ filtering for switched stochastic systems under asynchronous switching , 2012, J. Frankl. Inst..

[36]  Yonggui Kao,et al.  Global stability of coupled Markovian switching reaction–diffusion systems on networks☆ , 2014 .

[37]  Jinde Cao,et al.  Exponential Synchronization of Linearly Coupled Neural Networks With Impulsive Disturbances , 2011, IEEE Transactions on Neural Networks.

[38]  S. Ibrir.  Static output feedback and guaranteed cost control of a class of discrete-time nonlinear systems with partial state measurements , 2008 .

[39]  Jun Hu,et al.  Recursive filtering with random parameter matrices, multiple fading measurements and correlated noises , 2013, Autom..

[40]  Xiongping Dai.  Weakly Birkhoff recurrent switching signals, almost sure and partial stability of linear switched dynamical systems , 2011 .

[41]  Adilson E Motter,et al.  Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? , 2003, Physical review letters.

[42]  Alessandro Vespignani,et al.  Epidemic dynamics and endemic states in complex networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Qi Luo,et al.  Almost sure exponential stability of stochastic reaction diffusion systems , 2009 .