Abstract
There are lots of papers on the delay dependent stability criteria for differential delay equations (DDEs), stochastic differential delay equations (SDDEs) and hybrid SDDEs. A common feature of these existing criteria is that they can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions (namely, satisfy the linear growth condition). In other words, there is so far no delay-dependent stability criterion on nonlinear equations without the linear growth condition (we will refer to such equations as highly nonlinear ones). This paper is the first to establish delay dependent criteria for highly nonlinear hybrid SDDEs. It is therefore a breakthrough in the stability study of highly nonlinear hybrid SDDEs.
Highlights
Time-delay is encountered in many real-world systems in science and industry
It is known that Assumption 2.1 only guarantees that the stochastic differential delay equations (SDDEs) (2.1) with the initial data (2.2) has a unique maximal solution, which may explode to infinity at a finite time
In this paper we have established the new theory on the delaydependent stability criteria for highly nonlinear hybrid SDDEs
Summary
Time-delay is encountered in many real-world systems in science and industry. Differential delay equations (DDEs) (or more generally, functional differential equations) have been developed to model such time-delay systems. A common feature of the existing delaydependent stability criteria is that they can only be applied to the hybrid SDDEs where their coefficients are either linear or nonlinear but bounded by linear functions (namely, satisfy the linear growth condition). There is so far no delaydependent stability criterion on nonlinear hybrid SDDEs without the linear growth condition (we will refer to such equations as highly nonlinear ones). We should point out that there are already some papers on the asymptotic stability of highly nonlinear hybrid SDDEs (see, e.g., Hu, Mao, & Shen, 2013; Hu, Mao, & Zhang, 2013; Liu, 2012; Luo, Mao, & Shen, 2011) but these existing results are all delay independent.
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