Research Article | Open Access

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

**Academic Editor:**Irena Rachůnková

#### 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).

#### References

- K. Zhang and D. Q. Cao, “Further results on asymptotic stability of linear neutral systems with multiple delays,”
*Journal of the Franklin Institute. Engineering and Applied Mathematics*, vol. 344, no. 6, pp. 858–866, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. Q. Cao and P. He, “Stability criteria of linear neutral systems with a single delay,”
*Applied Mathematics and Computation*, vol. 148, no. 1, pp. 135–143, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. T. Nam and V. N. Phat, “An improved stability criterion for a class of neutral differential equations,”
*Applied Mathematics Letters*, vol. 22, no. 1, pp. 31–35, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Balasubramaniam, R. Krishnasamy, and R. Rakkiyappan, “Delay-dependent stability of neutral systems with time-varying delays using delay-decomposition approach,”
*Applied Mathematical Modelling*, vol. 36, no. 5, pp. 2253–2261, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Wang, “Exponential asymptotic stability for nonlinear neutral systems with multiple delays,”
*Nonlinear Analysis: Real World Applications*, vol. 8, no. 1, pp. 312–322, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons, New York, NY, USA, 1993. View at: MathSciNet - I. Podlubny,
*Fractional Differential Equations*, Academic Press, San Diego, Calif, USA, 1999. View at: MathSciNet - A. Anatoly Kilbas and H. M. Srivastava,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, The Netherlands, 2006. - T. Kaczorek,
*Selected Problems of Fractional Systems Theory*, Springer, Berlin, Germany, 2011. View at: Publisher Site | MathSciNet - Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for fractional neutral differential equations with infinite delay,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 7-8, pp. 3249–3256, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Liu, X. Li, W. Jiang, and X. Zhou, “Mittag-Leffler stability of nonlinear fractional neutral singular systems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 10, pp. 3961–3966, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X.-F. Zhou, J. Wei, and L.-G. Hu, “Controllability of a fractional linear time-invariant neutral dynamical system,”
*Applied Mathematics Letters*, vol. 26, no. 4, pp. 418–424, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Liu and W. Jiang, “Asymptotic stability of nonlinear descriptor systems with infinite delays,”
*Annals of Differential Equations*, vol. 26, no. 2, pp. 174–180, 2010. View at: Google Scholar | Zentralblatt MATH | MathSciNet - Z. Zhang and J. Wei, “Some results of the degenerate fractional differential system with delay,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1284–1291, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. P. Lazarević, “Finite time stability analysis of ${\mathrm{PD}}^{\alpha}$ fractional control of robotic time-delay systems,”
*Mechanics Research Communications*, vol. 33, no. 2, pp. 269–279, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Bonnet and J. R. Partington, “Analysis of fractional delay systems of retarded and neutral type,”
*Automatica*, vol. 38, no. 8, pp. 1133–1138, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Zhang, “Some results of linear fractional order time-delay system,”
*Applied Mathematics and Computation*, vol. 197, no. 1, pp. 407–411, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. P. Lazarević and A. M. Spasić, “Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach,”
*Mathematical and Computer Modelling*, vol. 49, no. 3-4, pp. 475–481, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,”
*Journal of Mathematical Analysis and Applications*, vol. 328, no. 2, pp. 1075–1081, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2014 Pang Denghao and Jiang Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.