Delay-dependent robust stability of uncertain neutral-type It$\hat{o}$ stochastic systems with Markovian jumping parameters

Abstract

This paper investigates the problem of mean-square exponential stability for uncertain neutral stochastic systems with time-delays and Markovian jumping parameters. Based on the new results on expectations of stochastic cross terms containing the It$\hat{o}$ integral by Song et al. (2013), a new Lyapunov–Krasovskii function is established, and then an improved mean-square exponential stability criterion is derived. The derived results extend the conclusions recently presented in Song et al. (2013). In fact, the system discussed in Song et al. (2013) is a special case of ours. Finally, two examples are provided to demonstrate the effectiveness of the proposed results.

Introduction

Neutral-type systems or neutral systems, described by neutral function differential equations [1], involve the delays in both states and the derivatives of the states. It has been shown that a lot of practical systems can be modeled as neutral systems, such as HIV infection with drug therapy, distributed networks containing lossless transmission lines, robots in contact with rigid environments, inferred grinding model, nuclear reactor, slip stabilization, population dynamic mode and space navigation systems [2], [3], [4], [5], [6]. Due to modeling and measurements errors, linear approximations, etc., dynamic system is often disturbed by parameter uncertainties, causing poor performance or undesirable dynamic behaviors. Many results on stability analysis, robust control and filtering for neutral systems have been published in recent years [3], [7], [8], [9], [10], [11], [12], [13].

In practice, since systems in the real world are always perturbed by “noisy”, a degree of randomness must be incorporated into the model. Kolmanovskii and Nosov [17], [18] introduced the neutral stochastic differential functional equations. Mao initiated the study of exponential stability of neutral-type stochastic functional equations [14], [15] and asymptotic properties of neutral-type stochastic delay differential equations [16]. And neutral stochastic systems with time-delays or uncertainties have been widely studied over recent years. Stability analysis and control problems for neutral stochastic systems were considered in [19], [20], [21], [22], [23] and the references therein. As a result, some industrial systems, which can not be appropriately described by the famous state-space representation, are adequately described by the class of stochastic switching systems called piecewise deterministic systems. Such systems have two components in the state vector. The first component takes values in ${\mathcal{R}}^n$ evolving continuously in time and represents the classical state vector used in modern control theory. The second takes values in a finite set and switches in a random manner between the finite number of states, which is always described by a continuous-time Markov process. This class of stochastic systems with Markovian jumping parameters has been successfully used in model different practical systems, such as communication systems, economics systems, manufacturing systems and so on [17]. Systems with Markovian jumping parameters, which were introduced by Krasovskii and Lidskii [24] in 1961, are a set of hybrid systems with transitions among the models governed by a Makrov chain taking values in a finite set and receive great attention [25], [26], [27], [28], [29], [30], [31]. Fortunately, the stability analysis, filtering, optimization and control issues for neutral stochastic systems with Makrovian jumping parameters have been widely studied. W. Mao and X. Mao [32] investigate the approximations of solutions to neutral stochastic systems with Makrovian jumping parameters under non-Lipschitz conditions. The stability analysis for neutral systems with Markovian jumping parameters are discussed in [33], [34]. S. He and F. Liu have discussed the exponential stability for uncertain neutral systems with Makrovian jumps in [35]. The fuzzy $H_{\infty }$ filtering and gain scheduled $L_2$-$L_{\infty }$ filtering for neutral systems with Makrovian jumping have been considered in [36], [37], respectively. The almost sure asymptotic stability theorems for neutral stochastic delay differential equations with Makrovian switching have been established in [38]. Although these above results are effective, there are still some problems to be pointed out for neutral stochastic systems with Makrovian jumps.

Is the Newton–Leibniz formula still valid in stochastic systems with Makrovian jumps? From the preface of [39] and discussions in [20], we know that the Newton–Leibniz formula is not valid in stochastic differential equations. Therefore, the Newton–Leibniz formula is not valid in stochastic systems driven by Makrovian jumps too. According to [2], consider the following neutral stochastic differential equation with Markovian jumping parameters

$$\mathrm{d}[x(t)-D({\gamma }_t)(x_t)]=f(t\text{,}{x}_t\text{,}{\gamma }_t)\mathrm{d}t+g(t\text{,}{x}_t\text{,}{\gamma }_t)\mathrm{d}w(t)\text{,}t⩾0\text{,}$$

with the initial data $x_0=x(\theta ):-h⩽\theta ⩽0=\xi (\theta )\in {\mathcal{C}}_{{\mathcal{F}}_0}^b([-h\text{,}0]\text{;}{\mathcal{R}}^n)$, it is known in [2] that Eq. (1) is interpreted as meaning the stochastic integral equation

$$x(t)-D({\gamma }_t)(x_t)=x(0)-D({\gamma }_t)(x_0)+{\int }_0^{t}f(s\text{,}{x}_s\text{,}{\gamma }_s)\mathrm{d}s+{\int }_0^{t}g(s\text{,}{x}_s\text{,}{\gamma }_s)\mathrm{d}w(s)\text{.}$$

Since the Newton–Leibniz formula is not valid in stochastic systems with Makrovian jumps, when using the stochastic integral Eq. (2) to derive delay-dependent conditions, there will occur the following stochastic cross terms $x^{\mathrm{T}}(t)M{\int }_{t-h}^{t}g(s\text{,}{x}_s\text{,}{\gamma }_s)\mathrm{d}w(s)\text{,}x{(t-h)}^{\mathrm{T}}N{\int }_{t-h}^{t}g(s\text{,}{x}_s\text{,}{\gamma }_s)\mathrm{d}w(s)\text{,}{({\int }_{t-h}^{t}f(s\text{,}{x}_s\text{,}{\gamma }_s)\mathrm{d}s)}^{\mathrm{T}}L{\int }_{t-h}^{t}g(s\text{,}{x}_s\text{,}{\gamma }_s)\mathrm{d}w(s)$. For these stochastic cross terms of stochastic delay systems, it is not enough reasonable to considered that the expectations of these stochastic terms are all equal to zero just as [20], [40].

Based on the above discussions and motivated by the new results on expectations of stochastic cross terms containing the It$\hat{o}$ integral in [20], this paper is concerned with the exponentially stability in mean square for uncertain neutral stochastic systems with time-delays and Markovian jumping parameters. And we design a new Lypapunov–Krasovskii function with Markovian jumping parameters, which is different from the one in [20] and less conservatism to avoid bounding stochastic cross terms. More specially, each transition rate of the Makrov chain is not a precise value, but its upper and lower bounds could estimated and completely known. By now, there have been few results about the mean-square exponential stability for the underlying systems. Section 2 is Preliminaries. In Section 3, based on Lemma 1, a new exponentially stability in mean square criterion for uncertain delayed Markovian jumping stochastic systems is presented. By establishing a Lyapunov–Krasovskii functional, the mathematical development avoids bounding stochastic cross terms, and neither the model transformation method nor free-weighting-matrix method is used. Thus the method leads to a simple criterion and shows less conservatism. In Section 4, two numerical examples are provided to show the effectiveness of the proposed results. Section 5 is conclusion.

$\mathbf{Notation}\text{.}$ Throughout the paper, let $(\mathrm{\Omega }\text{,}\mathcal{F}\text{,}{\{{\mathcal{F}}_t\}}_{t⩾0}\text{,}\mathcal{P})$ be a complete probability space with a natural filtration ${\{{\mathcal{F}}_t\}}_{t⩾0}$ and $\mathcal{E}\{·\}$ be the expectation operator with respect to the probability measure $\mathcal{P}$. Let $h>0$ and $\mathcal{C}([-h\text{,}0]\text{;}{\mathcal{R}}^n)$ denote the family of all continuous ${\mathcal{R}}^n-\mathit{valued}$ functions $\xi$ on $[-h\text{,}0]$ with the norm $\Vert \xi \Vert =\sup \{|\xi (\theta )|:-h⩽\theta ⩽0\}$. Let ${\mathcal{C}}_{{\mathcal{F}}_0}^b([-h\text{,}0]\text{;}{\mathcal{R}}^n)$ be the family of all ${\mathcal{F}}_0$-measurable bounded $\mathcal{C}([-h\text{,}0]\text{;}{\mathcal{R}}^n)-\mathit{valued}$ random variables. If A is a vector or matrix, its transpose is denoted by $A^{\mathrm{T}}$. If P is a square matrix, $P>0(P<0)$ means that is a symmetric positive (negative) definite matrix of appropriate dimensions while $P⩾0(P⩽0)$ is a symmetric positive (negative) semi-definite matrix. I stands for the identity matrix of appropriate dimensions. Denote by ${\lambda }_{\max }(·)$ or ${\lambda }_{\min }(·)$ the maximum or minimum eigenvalue of a given matrix. Let $|·|$ denote the Euclidean norm of a vector and its induced norm of a matrix. The symbol $\ast$ within a matrix represents the symmetric term of the matrix. Unless explicitly specified, matrices are assumed to have real entries and compatible dimensions. Let $\{{\gamma }_t\text{,}t⩾0\}$ be a right-continuous Markov process on the probability space which takes values in the finite space $\mathbb{S}=\{1\text{,}2\text{,}\dots \text{,}N\}$ with the generator $\mathrm{\Lambda }=\{{\pi }_{\mathit{ij}}\}(i\text{,}j\in \mathbb{S})$ (also called the transition rate matrix) given by

$$\mathcal{P}\{{\gamma }_{t+▵}=j|{\gamma }_t=i\}=\left\{\begin{array}{ccc}{\pi }_{\mathit{ij}}▵+o(▵)\text{,}\hfill & \mathrm{if}\hfill & j\ne i\text{;}\hfill \\ 1+{\pi }_{\mathit{ii}}▵+o(▵)\text{,}\hfill & \mathrm{if}\hfill & j=i\text{;}\hfill \end{array}\right.$$

$▵>0$ and ${\lim }_{▵\to 0}\frac{o(▵)}{▵}=0$. Here, ${\pi }_{\mathit{ij}}⩾0$ is the transition rate from i to j if $j\ne i$ and ${\pi }_{\mathit{ii}}=-{\sum }_{j\ne i}{\pi }_{\mathit{ij}}$.