Finite‐time cluster synchronisation of Markovian switching complex networks with stochastic perturbations

Abstract

In this study, the authors study the finite-time cluster synchronisation problem for a class of Markovian switching complex networks with stochastic noise perturbations. By constructing the suitable stochastic Lyapunov-Krasovskii functional, using finite-time stability theorem, inequality techniques and the properties of Weiner process, sufficient conditions are obtained to ensure finite-time cluster synchronisation for the complex networks with or without time delays. The effects of control parameters on cluster synchronisation speed and time delays are also analysed. Since finite-time cluster synchronisation means the optimality in convergence time and has better robustness and disturbance rejection properties, this study has important theory significance and practical application value. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results. NomenclatureThroughout this paper, R n and R n×m denote, respectively, the n-dimensional Euclidean space and the set of all n × m real matrices. The superscript 'T' denotes the transpose and the notation X ≥ Y (respectively, X > Y ) where X and Y are symmetric matrices, means that X − Y is positive semi-definite (respectively, positive definite); I N is the identity matrix with compatible dimension. · refers to the Euclidean vector norm; the notation A ⊗ B stands for the Kronecker product of matrices A and B. If A is a matrix, λ min (·) denotes the minimum eigenvalue. diag {· · · } stands for a block-diagonal matrix. E[x] means the expectation of the random variable x. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

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