Generalized Decay Synchronization of Directly Coupled Stochastic Dynamic Networks With Multiple Time Delays

Abstract

This article is dedicated to addressing the generalized decay synchronization problem of directly coupled stochastic dynamic networks with multiple time delays. First, by means of the presentations of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi $ </tex-math></inline-formula> -type function and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi $ </tex-math></inline-formula> -type stability, the definitions of decay synchronization <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\psi $ </tex-math></inline-formula> -type synchronization) are introduced. After that, by constructing appropriate feedback controllers and the Lyapunov functional method, we study the decay synchronization for multiple coupled dynamic networks. Moreover, for different coupling matrices, their zero eigenvalues are correspondingly denoted by different normalized left eigenvectors (NLEVec), so it is difficult to establish the functional based on these NLEVec. With the help of the weighted combination of NLEVec for multiple coupling matrices, under the assumption that the Chebyshev gap among NLEVec is less than an allowable deviation interval, some generalized decay (anti-) synchronization conditions are established. Furthermore, we discuss exponential synchronization and polynomial synchronization as special cases, and more relevant criteria are acquired. Examples are provided to verify the effectiveness of those obtained conditions.