Finite-Time Stability of Neutral Fractional Time-Delay Systems via Generalized Gronwalls Inequality

Abstract

This paper studies the finite-time stability of neutral fractional time-delay systems. With the generalized Gronwall inequality, sufficient conditions of the finite-time stability are obtained for the particular class of neutral fractional time-delay systems.

1. Introduction

In this paper, we consider a neutral fractional time-delay system: where denotes the Caputo fractional derivative of order , the vector function , , , are constant system matrices of appropriate dimensions, the constant parameter represents the delay argument, and is a given continuously differentiable function on .

The neutral time-delay systems have received increasing attention (see [1–5]) due to their successful applications in population ecology, distributed networks containing lossless transmission lines, heat exchangers, robots in contact with rigid environments, partial element equivalent circuit (PEEC), the control of constrained manipulators with time-delay measurements, the systems which need the information of the past state variables, and so on.

Recently, with the development of theories of fractional differential equations (see [6–9]), there has been a surge in the study of neutral fractional time-delay systems (see [10–12]). In particular, the problem of stability analysis of such systems has been one of the most interesting topics in control theory because stability analysis is one of the most important issues for control systems (see [13–16]). But stability of these systems proves to be a more complex issue because the systems involve the derivative of the time-delayed state and the existence of time-delay is frequently the source of instability although this problem has been investigated for time-delay systems over many years. In the previous literatures, many scholars have utilized the Lyapunov technique, characteristic equation method, state solution approach, or Gronwall’s approach to derive sufficient conditions for stability of the systems. In this paper, motivated by the papers [17, 18], we discuss the stability of the neutral fractional system with delay via generalized Gronwall’s approach.

The organization of this paper is as follows. In Section 2, we summarize some notations and give preliminary results which will be used in this paper. In Section 3, we present our main results.

2. Preliminaries and Lemmas

Let us start with some definitions and lemmas which are used throughout this paper.

Definition 1 (see [7]). Euler's gamma function is defined as where denotes the complex plane.

Remark 2 (see [7]). (i) ; and for ;
(ii) , ;
(ii) .

Definition 3 (see [7]). The fractional integral of order with the lower limit zero for any function , , is defined as where , and is the gamma function.

Definition 4 (see [7]). The Riemann-Liouville derivative of order with the lower limit zero for any function , , is defined as

Definition 5 (see [7]). The Caputo derivative of order for any function , , is defined as

Remark 6 (see [7]). (i) If a function , , then ;
(ii) , , and is a constant.

Definition 7 (see [9]). The Mittag-Leffler function in two parameters is defined as where , and .

Remark 8 (see [9]). (i) For , and , ;
(ii) For , the matrix extension of the aforementioned Mittag-Leffler function has the following representation: , , and .

Lemma 9 (see [19] generalized Gronwall’s inequality). Suppose , are nonnegative and local integrable on , some , and is a nonnegative, nondecreasing continuous function defined on ; constant with on this interval. Then

Lemma 10 (see [19]). Under the hypothesis of Theorem 13, let be a nondecreasing function on . Then where is the Mittag-Leffler function.

3. Main Results

In this section, we discuss some problems of the neutral fractional time-delay system (1).

Let us denote by the space of all continuous real functions defined on and by the Banach space of continuous functions mapping the interval into with the topology of uniform convergence. Let , , and designate the norm of an element in by

Let and , be equipped with the norm where obviously .

Let be the largest singular value of matrix , namely, For convenience, we denote by , by , by , by , and by , respectively.

Definition 11 (see [18]). The system given by (1) and satisfying initial condition , for , is finite stable with respect to , , if and only if implies

Theorem 12. If is a solution of the systems (1), then there exists a positive constant such that

Proof. According to the properties of the fractional order , one can obtain a solution in the form of the equivalent Volterra integral equation [12]:
Using appropriate property of the norm in (16) and applying (10), it follows that
For , , (17) can be rewritten as
From Definition 3, we can see is an increasing function of if . So and are both increasing functions with regard to . Taking into account (18) and (11), it yields that
Let us introduce a function such as where the function is nondecreasing apparently.
Now, with the corollary of the generalized Gronwall inequality (9), we can obtain
Similarly, the same argument implies the following estimate:
From Definition 7, we know that the Mittag-Leffler function is an increasing function with regard to . Therefore, there exists such that and .
Relationships (21) and (22) suggest the following general expression:
To prove formula (23) by induction we have to show that it holds for because of formula (21) and if it holds for , then it holds also for . Indeed, for , ; on the one hand using formula (22),we have
On the other hand, using formula (23) we obtain
Taking into account (24) and (25) we conclude that
That is, The proof is completed.

Theorem 13. The neutral fractional time-delay systems given by (1) are finite-time stable with respect to , , if the following condition is satisfied:

Proof. From Theorem 12 we obtain
Hence, using Definition 11 and the basic condition of Theorem 13, it follows that The proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research was jointly supported by National Natural Science Foundation of China (no. 11371027 and no. 11071001), Doctoral Fund of Ministry of Education of China (no. 20093401110001) and Major Program of Educational Commission of Anhui Province of China (no. KJ2011A020).