Robust exponential stability of uncertain impulsive delays differential systems

doi:10.1016/j.neucom.2016.01.011

Neurocomputing

Volume 191, 26 May 2016, Pages 12-18

https://doi.org/10.1016/j.neucom.2016.01.011 Get rights and content

Abstract

This paper deals with the robust stability of a class of uncertain impulsive control systems with infinite delays. By employing the formula for the variation of parameters and estimating the Cauchy matrix, several criteria on robust exponential stability of the systems are derived, these criteria are less restrictive than those in the earlier publications. Moreover, the criteria can be applied to stabilize the unstable continuous systems with infinite delays and uncertainties by utilizing impulsive control. Finally, two numerical examples are given to illustrate the effectiveness and advantages of the proposed method.

Introduction

Recently, impulsive control has attracted great interest of many researchers [1], [2], [3], [4], [5], [6]. Such control systems arise naturally in a wide variety of applications, such as dosage supply in pharmacokinetics [1], orbital transfer of satellite [2], [3], ecosystems management [4], [5] and control of saving rates in a financial market [6]. Moreover, time delays and uncertainties [7], [8], [9], [10], [11] occur frequently in engineering, biological and economical systems, and sometimes they depend on the histories heavily and result in oscillation and instability of systems [8], [12]. [14], [15], [16], [17], [18], [19], [20], [21], [22] are the cases of finite delays. Yang and Xu presented several interesting criteria on robust stability for uncertain impulsive control systems with time-varying delays [16]. Liu established several criteria on asymptotic stability for impulsive control systems with time delays [18]. [23], [24], [25] are the cases of infinite delays. However, the corresponding theory for impulsive control systems with infinite delays has been relatively less developed. In fact, an infinite delays deserve study intensively because they are not only an extension of finite delays but also describing the adequate mathematical models in many fields [17]. Therefore, it is necessary to further investigate the stability of uncertain impulsive control systems with infinite delays. Meanwhile, it is challenging to address the issue since we must utilize impulsive effects to handle the instability which may be caused by the infinite delays and uncertainties. Hence, techniques and methods for uncertain impulsive control systems with infinite delays should be further developed and explored.

This paper is inspired by [16]. In this paper, we present some criteria for the robust exponential stability of uncertain impulsive control systems with infinite delays by using the formula for the variation of parameters and estimating the Cauchy matrix. More importantly, the robust stability criteria do not require the stability of the corresponding continuous systems and so it can be more widely applied to stabilize the unstable continuous systems with infinite delays and uncertainties by using impulsive control. Finally, two examples are given to show the effectiveness and advantages of the obtained results.

Section snippets

Preliminaries

Let $N = 1, 2, \dots$ , $I$ be the identity matrix, $λ_{\min} (\cdot)$ and $λ_{\max} (\cdot)$ be the smallest and the largest eigenvalues of a symmetrical matrix, respectively. For $ϕ : R \to R^{n}$ , denote $ϕ (t^{+}) = \lim_{s \to 0^{+}} ϕ (t + s), ϕ (t^{-}) = \lim_{s \to 0^{-}} ϕ (t + s)$ . For $x \in R^{n}$ and $A \in R^{n \times n}$ , let $∥ x ∥$ be any vector norm, $∥ ϕ ∥_{α} = \sup_{s \leq 0} ∥ ϕ (s) ∥$ and denote the induced matrix norm and the matrix measure, respectively, by $∥ A ∥ = \sup_{x \neq 0} \frac{∥ Ax ∥}{∥ x ∥}, μ (A) = \lim_{h \to 0^{+}} \frac{∥ I + hA ∥ - 1}{h} .$ The usual norms and measures of vectors and matrices are: $∥ x ∥_{1} = \sum_{j = 1}^{n} | x_{j} |, ∥ A ∥_{1} = \max_{1 \leq j \leq n} \sum_{i = 1}^{n} | a_{ij} |, μ_{1} (A) = \max_{1 \leq j \leq n} {a_{jj} +$

Main results

Theorem 3.1

Let $ρ = \sup_{k \in N} {t_{k} - t_{k - 1}} < \infty$ . Suppose that there exists a constant $0 < γ < 1$ satisfy $∥ I + C_{k} ∥ \leq γ$ and $\frac{a + b + ∥ B ∥ + (c + ∥ C ∥) M}{γ} + μ (A) + \frac{\ln γ}{ρ} < 0,$ where $M = \int_{0}^{+ \infty} | h (s) | e^{η s} ds$ . Then the zero solution of the system (3) is robustly exponentially stable.

Proof

Since $\frac{a + b + ∥ B ∥ + (c + ∥ C ∥) M}{γ} + μ (A) + \frac{\ln γ}{ρ} < 0,$ then we choose small enough $λ \in (0, η)$ such that $\frac{a + (b + ∥ B ∥) e^{λ τ} + (c + ∥ C ∥) M}{γ} + μ (A) + \frac{\ln γ}{ρ} + λ < 0 .$ Furthermore, for any $ε \in (0, λ)$ , we have $0 \leq \frac{a + (b + ∥ B ∥) e^{(λ - ε) τ} + (c + ∥ C ∥) M}{γ} \leq - μ (A) - \frac{\ln γ}{ρ} - (λ - ε) .$ By the formula for the variation of parameters, the solution of (3) can be

Applications

In this section, we present two numerical examples to illustrate that our results can be applied to stabilize the unstable continuous system by using impulsive control.

Example 4.1

Consider the following uncertain impulsive control system ${\begin{matrix} \dot{x} (t) = [A + Δ A] x (t) + [B + Δ B] x (t - r (t)) + [C + Δ C] \int_{0}^{+ \infty} h (s) x (t - s) ds, & t \neq t_{k}, \\ Δ x (t_{k}) = x (t_{k}^{+}) - x (t_{k}^{-}) = C_{k} x (t_{k}^{-}), & k \in N, \end{matrix}$ where $r (t) \in [0, τ]$ , $τ$ is any given positive constant, $h (s) = 0.1 e^{- 1.2 s}, s > 0,$ $∥ Δ A ∥ \leq 0.1, ∥ Δ B ∥ \leq 0.2, ∥ Δ C ∥ \leq 0.3$ and $A = [\begin{matrix} 1.2 & - 1.1 \\ 0.7 & 0.8 \end{matrix}], B = [\begin{matrix} - 1.3 & 0.7 \\ - 0.9 & 0.5 \end{matrix}], C = [\begin{matrix} - 0.1 & - 0.22 \\ 0.43 & 0.65 \end{matrix}], C_{k} = [\begin{matrix} - 0.4 & 0.1 \\ - 0.1 & - \end{matrix}$

Conclusion

In this paper, the robust stability problem of uncertain impulsive control systems with infinite delays is concerned. By employing the formula for the variation of parameters and estimating the Cauchy matrix, several criteria on robust exponential stability are derived, which are less restrictive than those in the earlier literature. Two examples are given to illustrate the effectiveness of the theoretical results. In addition, we should point out that the method used in this paper can be

Acknowledgments

This work was jointly supported by National Natural Science Foundation of China (No. 11301308), China Postdoctoral Science Foundation founded project (2014M561956, 2015T80737) and Research Fund for International Cooperation Training Programme of Excellent Young Teachers of Shandong Normal University, National Natural Science Foundation of China (No. 61471226, No. 61201441), research funding from Shandong Province (JQ201516), and research funding from Jinan City (No. 201401221, No. 20120109).

Dengwang Li was born in Shanxi Province, China. He received his B.S. degree from Shandong University, in China. He earned his Ph.D. degree also from Shandong University, and won a joint Ph.D. program with the University of Sydney sponsored by CSC, in Australia. His research focused on impulsive systems, signal processing and biomedical engineering.

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Xiaodi Li was born in Shandong province, China. He received the B.S. and M.S. degrees from Shandong Normal University, Jinan, China, in 2005 and 2008, respectively, and the Ph.D. degree from Xiamen University, Xiamen, China, in 2011, all in applied mathematics. He is currently a Professor with the Department of Mathematics, Shandong Normal University. From 2014 to now, he is a Visiting Research Fellow at Laboratory for Industrial and Applied Mathematics in York University, Canada. He has authored or coauthored more than 50 research papers. His current research interests include stability theory, delay differential equations, impulsive control theory, artificial neural networks, and applied mathematics.

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Robust exponential stability of uncertain impulsive delays differential systems

Abstract

Introduction

Section snippets

Preliminaries

Main results

Applications

Conclusion

Acknowledgments

Syst. Control Lett.

Automatica

J. Math. Anal. Appl.

Appl. Math. Comput.

Math. Comput. Model.

J. Frankl. Instit.