The pth moment boundedness of stochastic functional differential equations with Markovian switching

Abstract

This paper gives some Razumikhin-type theorems on pth moment boundedness of stochastic functional differential equations with Markovian switching (SFDEwMS) by using Razumikhin technique and comparison principle. Some improved conditions on pth moment stability are also proposed. The main results of this paper allow the estimated upper bound of the diffusion operator associated with the underlying SFDEwMS of the Lyapunov function to have time-varying coefficients (the coefficients may even be sign-changing functions). Examples are provided to illustrate the effectiveness of the proposed results.

Introduction

Stochastic functional differential equations (SFDEs) which include stochastic delay differential equations (SDDEs) play a very important role in formulation and analysis in mechanical, electrical, control engineering and physical sciences, economic and social sciences [1]. Therefore, the theory of SFDEs has been developed very quickly. Recently, the investigation for SFDEs has attracted the considerable attention of researchers and many qualitative theories of SFDEs have been obtained. A large number of stability and boundedness criteria of SFDEs have been reported (see e.g., [1], [2], [3], [4], [5], [6], [7], [8] and the references cited therein). Some applications based on stability theory of SFDEs are also given (see e.g., [9], [10]). On the other hand, SFDEs or SDDEs may often experience abrupt changes in their structure and parameters and continuous-time Markov chains have been used to model these abrupt changes. Hence, SFDEs with Markovian switching (SFDEwMS), known also as hybrid SFDEs, have appeared frequently in practice [11]. One of the important issues in the study of SFDEwMS is the automatic control, with consequent emphasis being placed on the asymptotic analysis of stability. There is an intensive literature in the area of SFDEwMS and we mention, for example, [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. In particular, Mao and Yuan [11] is the first book in this area.

Boundedness is another important asymptotic property of dynamical systems, which plays an important role in investigating stability, invariant and attracting properties, chaotic behaviour, and so on. However, to the best of our knowledge, up till now, there are few reports on the boundedness of SFDEwMS [11], [21], [22], [27]. The work by Mao and Yuan [11] presents a theorem for checking boundedness of SFDEwMS based on an Lyapunov-type condition on the diffusion operator. More specifically, the diffusion operator $\mathcal{L}V$ of an Lyapunov function (to be defined in Section 2) is required to satisfy the following condition:

$$\mathcal{L}V(t,\phi ,i)\le \alpha -{\lambda }_1\left|\phi \right(0){|}^p+{\lambda }_2{\int }_{-\tau }^{0}w(u\left)\right|\phi \left(u\right){|}^p\mathrm{d}u$$

where $(t,\phi ,i)\in {\mathbb{R}}^{+}\times C\left(\right[-\tau ,0];{\mathbb{R}}^n)\times \mathbb{S}$, $w\in \mathcal{W}\left(\right[-\tau ,0];{\mathbb{R}}^{+})$, λ₁ and λ₂ are positive constants with ${\lambda }_1>{\lambda }_2$.

More recently, the authors [21] investigated stability and boundedness of nonlinear hybrid stochastic differential equations (HSDDEs) by introducing the auxiliary functions U₁ and U₂, which can be potentially used to cope with nonlinearities and delays encountered in a stochastic system with Markovian switching. The condition of the diffusion operator $\mathcal{L}V$ of an Lyapunov function is:

$$\mathcal{L}V(t,x,y,i)\le c_1-c_2{U}_2(t,x)+c_3{U}_2(t-\tau ,y)$$

for all $(t,x,y,i)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^n\times {\mathbb{R}}^n\times \mathbb{S}$, where c₁, c₂ and c₃ are three nonnegative constants with $c_2>c_3$, $U_1(t,x)\le V(t,x,i)\le U_2(t,x)$, $(t,x,i)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^n\times \mathbb{S}$, ${\lim }_{\left|x\right|\to \infty }\left({\inf }_{t\ge 0}{U}_1\right(t,x\left)\right)=\infty$.

While the authors [22] established some new results on asymptotic stability and boundedness for nonlinear HSDDEs without the linear growth condition. By using M-matrices and some inequality techniques, the bound of $\mathcal{L}V$ can be described as:

$$\mathcal{L}V(t,x,y,i)\le {\alpha }_1-{\alpha }_2|x{|}^p+{\alpha }_3|y{|}^p-{\alpha }_4|x{|}^q+{\alpha }_5|y{|}^q$$

where $q>p\ge 2$, ${\alpha }_2>{\alpha }_3$ and ${\alpha }_4>{\alpha }_5$.

We note that all the coefficients on the right-hand sides of the inequalities (1.1), (1.2), (1.3) are constants. From Eqs. (1.1), (1.2), (1.3), especially, when $\alpha =0$, $c_1=0$ and ${\alpha }_1=0$ respectively, the solutions of the corresponding systems tend to zero exponentially. However, there are many practical systems whose solutions only tend to zero asymptotically but not exponentially. If the estimated upper bound of the diffusion operator $\mathcal{L}V$ of a Lyapunov function has time-varying coefficients (even the coefficients are sign-changing sometimes), the results based on Eqs. (1.1), (1.2), (1.3) or other existing criteria for convergence and boundedness cannot be applied. Motivated by the discussions above, the main purpose of this paper is to give some new results, so that they can be applied to some more general SFDEwMS.

It is well known that the Razumikhin-type results on stability have been obtained for various kinds of SFDEs (see e.g., [3], [4]). The Razumikhin-type results allow the time delay to be a bounded variable only. For general SFDEwMS, Mao et al. proposed the Razumikhin-type theorem on pth moment exponential stability [12] and its applications to hybrid stochastic interval systems [13], while Huang et al. established Razumikhin-type criteria on pth moment asymptotic stability in [14], [15]. The Razumikhin-type criteria in [12], [13], [14], [15] require the diffusion operator of the Lyapunov function to be negative definite. While, the main results in [23] allow the diffusion operator of the Lyapunov function to be not always negative. In this paper, we employ comparison principle and Razumikhin technique to establish some Razumikhin-type results which can ascertain the solutions of Eq. (2.1) to be pth moment bounded, pth moment ultimately bounded. The main results of the present work allow the estimated upper bound of the diffusion operator to have time-varying coefficients, and the coefficients may be sign-changing functions. In particular, as a by-product, some improved conditions on pth moment asymptotic stability are also obtained.

The rest of this paper is organized as follows. In Section 2, some notations and definitions are given. In Section 3, some results on pth moment boundedness, pth ultimate boundedness, pth moment stability and pth moment asymptotic stability for SFDEwMS are established. In Section 4, several examples are given to illustrate our results. Finally, a conclusion is given in Section 5.