Stability of stochastic functional differential systems with semi-Markovian switching and Lévy noise by functional Itô’s formula and its applications

Abstract

This paper investigates the general decay stability on systems represented by stochastic functional differential equations with semi-Markovian switching and Lévy noise (SFDEs-SMS-LN). Based on functional Itô’s formula, multiple degenerate Lyapunov functionals and nonnegative semi-martingale convergence theorem, new pth moment and almost surely stability criteria with general decay rate for SFDEs-SMS-LN are established. Meanwhile, the diffusion operators are allowed to be controlled by multiple auxiliary functions with time-varying coefficients, which can be more adaptable to the non-autonomous SFDEs-SMS-LN with high-order nonlinear coefficients. Furthermore, as applications of the presented stability criteria, new delay-dependent sufficient conditions for general decay stability of the stochastic delayed neural network with semi-Markovian switching and Lévy noise (SDNN-SMS-LN) and the scalar non-autonomous SFDE-SMS-LN with non-global Lipschitz condition are respectively obtained in terms of binary diagonal matrices (BDMs) and linear matrix inequalities (LMIs). Finally, two numerical examples are given to demonstrate the effectiveness of the proposed results.

Introduction

In recent decades, stochastic functional differential equations (SFDEs), including stochastic differential equations (SDEs) and stochastic delayed differential equations (SDDEs) as their special cases, have aroused great concern because of their wide range of significant applications in economics, finance, physics, biology, engineering, etc. [1], [2]. As is well known, time delays commonly encountered in SFDEs may cause oscillation or instability, which would be harmful to the applications of SFDEs. Therefore, it is of significance to investigate the stability of SFDEs, and there have been a number of important results on this issue, see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] and references therein. For the present study, Razumikhin method and Lyapunov functionals are two primary techniques for stability analysis for SFDEs. The stochastic versions of Razumikhin method was developed (e.g., [1], [9], [10]) for studying moment asymptotic stability, however, different from its deterministic counterpart, the stochastic Razumikhin method has limited success [6], [11]. By specific Lyapunov functionals, [1], [7], [8] examined stability and related issues of SFDEs, but fundamental stability theory related to Lyapunov functionals for SFDEs has not yet been fully developed. Furthermore, the solution processes of SFDEs are no longer Markovian due to the existence of time delay which has defied bona fide operators and functional Itô’s formulas in the past [6]. Recently, by means of pathwise functional derivatives, Dupire generalized Itô’s formula to a functional circumstance in [12] which was subsequently further developed to the case of c$\stackrel{`}{\mathrm{a}}$dl$\stackrel{`}{\mathrm{a}}$g semi-martingales by [13]. Such functional Itô’s formulas ensure us to achieve bona fide operators for SFDEs and substantially ease the difficulties in facilitating the use of Lyapunov functionals to study stability and related issues for a much larger class of stochastic systems with time delays including SFDEs with semi-Markovian switching and Lévy noise. Most recently, based on functional Itô’s formula, [6] established the moment and almost sure exponential stability criteria for SFDEs by degenerate Lyapunov functionals, while [14] studied the almost sure and L^p stability of SFDEs with regime-switching by Lyapunov functionals. In the meanwhile, there are extensive references about polynomial and exponential stability for SFDEs, see, e.g. [1], [2], [15], [16]. The main difference between polynomial stability and exponential stability is the rate at which the solution decays to zero. Moreover, such stability concepts can be further generalized to the general decay stability, see [17], [18], [19], [20] and references therein.

It should be noted that the above-mentioned works concentrate only on the SFDEs driven by Brownian motion. It is well known that, as a continuous stochastic process, the Brownian motion is incapable of describing many real systems whose structures containing stochastic abrupt changes such as earthquakes, hurricanes, epidemics, stochastic failures, fluctuations in financial markets, large disasters in life sciences, sudden changes in the environment [21]. It is recognized that the SFDEs with Markovian switching and Lévy noise are more appropriate to describe such discontinuous stochastic systems, but it definitely increases the difficulty of analysis due to the discontinuity of their c$\stackrel{`}{\mathrm{a}}$dl$\stackrel{`}{\mathrm{a}}$g sample paths (i.e., paths that are right continuous with left limits). Moreover, it is acknowledged that the stability analysis of SFDEs with jump cannot be directly derived from SFDEs without jumps [22], in fact, the stochastic integral relating to Brownian motion (called Itô integral) and the one corresponding to the Poisson random measure (called Poisson integral [21]) is of significantly essential difference. In recent years, there have been a large number of representative works on the stability analysis for SFDEs with Markovian switching or Lévy noise [20], [23], [24], [25], [26], [27]. Using Lyapunov function and the Razumikhin method, sufficient conditions of almost sure stability and stability in distribution for hybrid SDDEs with Lévy jumps have been established in [23]. By Kunita’s inequality and Burkholder-Davis-Gundy inequality, [24] discussed the existence and asymptotic estimations of solutions to pantograph stochastic equations with diffusion and Lévy jumps. [25] established the Razumikhin-type stability theorem for SFDEs with Markovian switching and Lévy noise while [27] studied the stability of SFDEs with Lévy noise, where the time-varying delay is needless to be differentiable. In [20], [26], the authors were devoted to studying almost sure stability with general decay rate on neutral SDDEs with Lévy noise, and the input-to-state stability on switched SDDEs with Lévy noise, respectively.

Furthermore, for Markovian jump systems (MJSs), it is generally assumed that the sojourn time (the time duration between two successive jumps) of each mode follows exponential distribution which is of memorylessness, thus the transition rate is constant. However, it is difficult to ensure MJSs satisfy this assumption in practice such as the bunch-train cavity interaction [28], the fault-tolerant control system [29], DNA analysis [30] and risk minimizing option pricing [31]. To make this assumption less restrictive, the semi-Markovian jump systems (SMJSs) whose sojourn-time is allowed to obey the non-exponential distribution are proposed [28], [29], [30], [31], [32], [33]. It is clear that SMJSs can be specialized as MJSs, which briefly illustrates that SMJSs have much wider application domain than MJSs due to the relaxation on probability distributions of sojourn-time. Recent attentions and interests have been paid to both theoretical and practical cases on SMJSs [28], [29], [30], [31], [33]. The control problem of singularly perturbed SMJSs applied on the interaction of bunch-train cavity has been discussed in [28]. The application of SMJSs to the fault-tolerant control system has been investigated in [29], while the application of SMJSs and hidden SMJSs toward reliability and DNA analysis was addressed in [30]. For SMJSs with mode transition dependent sojourn-time distributions, [33] studied the stochastic stability and stabilization problem. Recently, [34] discussed the asymptotic stability of stochastic SMJSs without the constraint of bounded transition rates.

However, to the best of our knowledge, it lacks related results on stability analysis of stochastic functional differential equations with semi-Markovian switching and Lévy noise (SFDEs-SMS-LN) owing to their complexity and mathematical difficulty. Motivated by the work of [6], [13], by using functional Itô’s formula, multiple degenerate Lyapunov functionals, binary diagonal matrices (BDMs) and LMIs, this paper is aiming to close this gap by establishing new stability criteria with general decay rate for SFDEs-SMS-LN. Compared with the previous results, the main contributions of this article are as follows.

(i) A new functional Itô’s formula for SFDEs-SMS-LN is established by applying the well-known functional Itô’s formula for c$\stackrel{`}{\mathrm{a}}$dl$\stackrel{`}{\mathrm{a}}$g semi-martingale.

(ii) By the new functional Itô’s formula, nonnegative semi-martingale convergence theorem and multiple degenerate Lyapunov functionals, new pth moment and almost sure general stability criteria for SFDEs-SMS-LN are proposed.

(iii) The diffusion operators are improved to be controlled by the multiple auxiliary functions with time-varying coefficients, which can be more adaptable to the non-autonomous SFDEs-SMS-LN with high-order nonlinear coefficients.

(iv) As applications of the presented results, new delay-dependent sufficient conditions in terms of LMIs for general decay stability of SDNN-SMS-LN (stochastic delayed neural network with semi-Markovian switching and Lévy noise) and the scalar non-autonomous SFDE-SMS-LN with non-global Lipschitz condition are provided, where a vertex method (Lemma 2) has been employed to reduce the conservatism.

The remainder of this paper is organized as follows. Necessary notations and main problems are stated in Section 2, and several new general decay stability criteria for SFDEs-SMS-LN and their applications are provided in Section 3. Section 4 shows two numerical examples. A concluding remark is given in Section 5.