Abstract
Robust stability has been one of the most popular topics in the study of stochastic functional differential equations with Markovian switching (SFDEwMSs), including stochastic delay differential equations with Markovian switching (hybrid SDDEs). Most of the existing results on the robust stability require that the drift coefficient f and diffusion coefficient g of the stochastic system are either linear or nonlinear with linear growth condition. Recently Hu, Mao and Zhang (IEEE Trans. Autom. Control 58:2319-2332, 2013) obtained some new results for the robust stability of nonlinear hybrid SDDEs, requiring $\boldsymbol {x}^{T}\boldsymbol {f}$ and $\vert \mathbf{g}\vert ^{2}$ to be bounded by polynomials with the same orders. However, there are many SFDEwMSs which do not satisfy the above requirement. Hence the existing criteria on the robust stability are not applicable and we see the necessity to develop some new criteria. Our aim in this paper is to establish some new criteria for the robust stability of a class of SFDEwMSs, where $\boldsymbol {x}^{T}\boldsymbol {f}$ and $\vert \mathbf{g}\vert ^{2}$ are controlled by polynomials with different orders.
Highlights
1 Introduction In general, time delays and system uncertainty are commonly encountered problems in deterministic or stochastic dynamical systems, which usually result in instability
Continuous-time Markovian chains have been introduced to cope with such situations. These systems can be described as stochastic functional differential equations with Markovian switching (SFDEwMSs) including stochastic delay differential equations with Markovian switching (SDDEwMSs or hybrid SDDEs), and many researchers have studied the stability theory of SFDEwMSs
It is necessary to develop some new theory of the robust stability for SFDEwMSs with some new conditions similar to ( )
Summary
Time delays and system uncertainty are commonly encountered problems in deterministic or stochastic dynamical systems, which usually result in instability (see [ ]). We would like to mention the work of Hu et al [ ] They studied the robust stability and robust boundedness for a stochastically perturbed system of the determined nonlinear delay differential equation having the form dx(t) = f x(t), x(t – τ ), t, r(t) dt + g x(t), x(t – τ ), t, r(t) dB(t), on t ≥ , where τ > , f : Rn ×Rn ×R+ ×S → Rn, g : Rn ×Rn ×R+ ×S → Rn×m, r(t) and B(t) have the same definitions as with system ( ). Ρ(i) x(t + θ) dη(θ ) + c(i) a(i) – b(i) x(t + θ ) dη(θ )
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