Abstract
Taking different structures in different modes into account, the paper has developed a new theory on the structured robust stability and boundedness for nonlinear hybrid stochastic differential delay equations (SDDEs) without the linear growth condition. A new Lyapunov function is designed in order to deal with the effects of different structures as well as those of different parameters within the same modes. Moreover, a lot of effort is put into showing the almost sure asymptotic stability in the absence of the linear growth condition.
Highlights
Systems in many branches of science and industry depend on the present state and the past ones but may experience abrupt changes in their structures and parameters
One of the important issues in the study of hybrid stochastic differential delay equations (SDDEs) is the asymptotic analysis of stability and boundedness
To make our theory more general, we consider the case where the space of modes, S, of a given hybrid system can be divided into two proper subspaces, S1 and S2, such that the system is described by the same type of SDDEs for modes in S1 but by a different type of SDDEs for modes in S2
Summary
Systems in many branches of science and industry depend on the present state and the past ones but may experience abrupt changes in their structures and parameters. Hybrid stochastic differential delay equations (SDDEs; known as SDDEs with Markovian switching) have been widely used to model these systems (see, e.g., the books [23, 24] and the references therein). One of the important issues in the study of hybrid SDDEs is the asymptotic analysis of stability and boundedness (see, e.g., [3, 5, 13, 19]). Robust stability and boundedness have been two of most popular topics. Hinrichsen and Pritchard [7, 8] presented an excellent discussion of the stability radii of linear systems with structured perturbations. Su [26] and Tseng, Fong, and Su [27] discussed robust stability for linear delay equations. Koroleva, and Rodkina [21] discussed the robust stability of uncertain linear or semilinear stochastic delay systems. Mao [20] investigated the stability of the stochastic delay interval system with Markovian switching. All of the papers, up to 2013, in this area only considered these robust
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