Given an unstable hybrid stochastic differential equation (SDE), can we design a feedback control, based on the discrete-time observations of the state at times 0, τ, 2τ, • • • , so that the controlled hybrid SDE becomes asymptotically stable? It has been proved that this is possible if the drift and diffusion coefficients of the given hybrid SDE satisfy the linear growth condition. However, many hybrid SDEs in the real world do not satisfy this condition (namely, they are highly nonlinear) and there is no answer to the question yet if the given SDE is highly nonlinear. The aim of this paper is to tackle the stabilization problem for a class of highly nonlinear hybrid SDEs. Under some reasonable conditions on the drift and diffusion coefficients, we show how to design the feedback control function and give an explicit bound on τ (the time duration between two consecutive state observations), whence the new theory established in this paper is implementable.
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 SDDE
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. 1. Introduction. Systems in many branches of science and industry not only depend on the present state and the past ones but may also experience abrupt changes in their structures and parameters. Hybrid stochastic differential delay equations (SDDEs; also 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]). In asymptotic analysis, robust stability and boundedness have been two of most popular topics. For example, Ackermann [1] gave a nice motivation of robust stability. 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. In the aspect of robustness of stochastic stability, Haussmann [6] studied robust stability for a linear system and Ichikawa [11] for a semilinear system. Mao, 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. For more information on the stability and boundedness of hybrid SDDEs, please see, e.g., [12,22,23,25]. However, all of the papers, up to 2013, in this area only considered these robust
Stability criteria for neutral stochastic differential delay equations (NSDDEs) have been studied intensively for the past several decades. Most of these criteria can only be applied to NSDDEs where their coefficients are either linear or nonlinear but bounded by linear functions. This paper is concerned with the stability of hybrid NSDDEs without the linear growth condition, to which we will refer as highly nonlinear ones. The stability criteria established in this paper will be dependent on delays.
For the past few decades, the stability criteria for the stochastic differential delay equations (SD-DEs) have been studied intensively. Most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability criterion for highly nonlinear hybrid stochastic differential equations is investigated in [Fei, Hu, Mao and Shen, Automatica, 2017]. In this paper, we investigate a class of highly nonlinear hybrid stochastic integro-differential delay equations (SIDDEs). First, we establish the stability and boundedness of hybrid stochastic integro-differential delay equations. Then the delay-dependent criteria of the stability and boundedness of solutions to SIDDEs are studied. Finally, an illustrative example is provided.
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