Stability and boundedness of nonlinear hybrid stochastic differential delay equations

Abstract

One of the important issues in the study of hybrid SDDEs is the automatic control, with consequent emphasis being placed on the asymptotic analysis of stability and boundedness (see e.g. [5], [10], [11], [13], [14], [15], [17], [19] and [21]). The method of Lyapunov functions is one of the most powerful techniques in the study of stability and boundedness. So far, most of the results in this area do not only require the Lyapunov functions in different modes have the same feature (e.g. polynomials with the same degree) but also that the diffusion operator in different modes be bounded by the same type of functions. These requirements are restrictive and often cannot be met by those hybrid SDDEs that have different nonlinear structures in different modes. To study the stability and boundedness of such hybrid SDDEs, we will in this paper use different types of Lyapunov functions (e.g. polynomials with different degrees) for different modes. Moreover, the condition on the diffusion operator is relaxed significantly

Structured Robust Stability and Boundedness of Nonlinear Hybrid Delay Systems

“…In terms of mathematics, conditions (4.2) and (4.3) describe the difference in structure. More understandably, condition (4.2) means that the hybrid SDDE in S 1 -modes satisfies the classical Khasminskii-type condition (see, e.g., [14,23]) while condition (4.2) means that the hybrid SDDE in S 2 -modes satisfies the generalized Khasminskii-type condition (see, e.g., [10]). In layman's terms, the coefficients of the SDDE in S 1 -modes may grow linearly in the delay component x(t -\tau ) while in S 2 -modes it may grow polynomially.…”

“…Noting that in Assumption 4.2, we only require \alpha i2 \in R for all i \in S. According to the Khasminskii-type theorems (see, e.g., [14,10,23]), the solution of the hybrid SDDE may grow exponentially. But our aim in this paper is to study the asymptotic boundedness and stability.…”

Stability of a Class of Hybrid Neutral Stochastic Differential Equations with Unbounded Delay

“…For example, in the neural network model, stochastic unbounded variable 2 Discrete Dynamics in Nature and Society delay differential system must be considered to model transmission and transformation of the signal in a better fashion (see [19]). Some related works on unbounded delay can be found in [20][21][22][23][24]. References [22,24] investigated the existence and uniqueness, as well as the pathwise stability of the global solutions to SDEs and NSDEs with unbounded delay, respectively.…”

“…Reference [23] studied the stability of SFDEs with unbounded delay. In Section 4.2 of [21], the stability and boundedness of nonlinear hybrid SDEs were discussed when the delay function ( ) was given as (0 < < 1). Particularly, [20] presented the existence and uniqueness of the exact solution for a class of NSDEwMs with unbounded delay and also showed the th moment exponential stability and almost sure exponential stability results under the bounded delay condition and established the Euler-Maruyama method under both cases.…”

“…The contributions of this paper are twofold. The first one is that we improve the delay function ( ) = (0 < < 1) in [21] to the general unbounded delay and add the neutral term. The other one is that we greatly generalize and advance the th moment exponential stability and almost sure exponential stability results in [20] where the delay term was bounded.…”