Stochastic stabilization of Markovian jump neutral systems with fractional Brownian motion and quantized controller

Abstract

This paper explores the delay dependent stochastic stabilization of Markovian jump neutral systems (MJNS) which are modeled by fractional Brownian motion(fBm) via a quantized controller. A function Round quantizer is introduced which solves the model uncertainties and the nonlinear part by a uniform operator. Then by structuring a Lyapunov–Krasovskii functional (LKF) and the aid of linear matrix inequalities (LMIs) method, a stochastic stability criterion is achieved. Last, different parameters are selected to simulate the effectiveness of our findings.

Introduction

Recently, the theory of neutral stochastic differential equation has been well generalized. Stochastic neutral system is taken into account the influence of time-delay and random disturbances, and it is considered that it is not only affected by the past and present state, but also related to the rate of change of the past state, which can better be in line with reality. Therefore, it has been widespread in finance, biology, machinery, control and so forth. It’s time and effort that quite a few scholars have been willing to invest in stochastic neutral models [4], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. Kao et al.[4] looked into the delay-dependent stability problem for unknown transition rates. Delay-dependent impulses were paid attention to in Fu and Zhu [15]. Especially linear matrix inequalities (LMIs) summarized by Boyd et al.[2] have also attracted much attention in the above work. Additionally, the exponential stability was solved by the aid of a comparison principle reductio ad absurdum in Ngoc [18], 19] that relatively few people adopted.

$W_t^H$ is a Gaussian process which stemmed from Kolmogorovs work [1] and is governed by the Hurst index $H\in (0,1)$. It is particularly notorious that fBm has two important properties of self-similarity and non-stationarity. Based on stochastic differential equation, fBm has achieved wider range of applications, we can refer to Khandani et al. [7], Chadha et al. [12], Zou et al. [20], Xu and Pei [21], Yu and Zhang [22], Duc et al. [23], Sathiyaraj and Balasubramaniam [24], Li et al. [25], Liu et al. [26], Lv and Yang [27]. Moreover, the results involving neutral stochastic functional differential equations are abundant, see, for instance, Boufoussi and Hajji [28], Dung [29], Cui and Yan [30], Duan and Ren [31], Ouahra et al. [32], Xu and Luo [33], Boufoussi and Mouchtabih [34], Boudaoui and Blouhi [35], Diop and Ezzinbi [36], Wang and Yan [37]. Among them, Cui [30] and Brahim Boufoussi et al. [34] paid close attention to the controllability. Relatively speaking, Duan and Ren [31], Ouahra et al.[32] and Diop and Ezzinbi [36] investigated the stability problem. Different $H$ will result in different models. For $H=\frac{1}{2}$, it corresponds to a standard Brownian motion which is favored by crowds of researchers. In the case of $0<H<\frac{1}{2}$, the time series exhibits anti-persistence, so it is obviously more volatile than pure random. Some surveys could be found in Xu and Luo [33], Boufoussi and Mouchtabih [34], Boudaoui and Blouhi [35]. When $\frac{1}{2}<H<1$, the time series is trend-increasing and follows the process of biased random walk. That means it produces a strong dependence on time. Hence, more literature is in favor by comparison of $\frac{1}{2}<H<1$. For more results, see, e.g., [28], [29], [30], [36], [37]. In [36], this work covers the stability for some impulsive integro-differential equations. Wang and Yan [37] probed the stochastic averaging method by exploiting the linear operator. However, up to now, few researchers have taken the extension of the stochastic neutral systems with the fBm, which is very novel and meaningful.

Inspired by the above comparatively rich discussions, we will concentrate on stochastic neutral model driven by fBm with $\frac{1}{2}<H<1$. It should be emphasized that on the basis of Xie and Kao [5] who addressed the problem of mean-square exponential stability, this paper is further concerned with control of the system with the hurst $\frac{1}{2}<H<1$ seriously. In the work, the expected LMIs depending on the established LKF after derivation, a series of lemmas and schur complement approach are requred. And then, The trajectories obtained by the Matlab software is used to verify the effectiveness of the previous theory.

In addition, main other contributions of the thesis are as follows: (1) As is known to all, Markovian jump model whose system is a kind of special switching system whose system mode is constrained by Markovian chain and describes various systems with sudden changes in structure and parameters. Therefore, it is also essential and necessary to look into it. Considerable attention has been paid to the stability and control of systems considering Markovian jump [43], [44], [45], [46]. Compared with the completed research of Wang and Yan [37], we also propose the case of Markovian jump process. (2) Futhermore, we also designed a quantitative controller in the system, which consists of the uncertainties section and the nonlinear one. Among them, the aim of the uncertainties portion is applied in Theorem 3.1 and the nonlinear portion is to avoid its interference of the quantized input. (3) Compared with the LKF structure of the existing and mature standard Brownian motion system which is widely used, the form of exponential function is employed to aid the proofs of stability. (4) In addition, it should be noting that the problem of variable time delay is explored in depth different from Xie and Kao [5].

Notation

In the work, let $(\Omega ,\mathbb{F},{\left\{{\mathbb{F}}_t\right\}}_{t\ge 0},\mathbb{P})$ signify a complete probability space accompanied by a filtration ${\left\{{\mathbb{F}}_t\right\}}_{t\ge 0}$. $A^T$ stands for the transpose of $A$. Given a square matrix $P$, $P>0(P<0)$ indicates a symmetric positive (negative) matrix. $\parallel ·\parallel$ refers to the Euclidean norm. $C({\mathbb{R}}_{+}\times S_h;{\mathbb{R}}_{+})$ demonstrates that functions defined on it are first-order derivable to $t$ and second-order derivable to $x$.