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

and Applied Analysis 3 Theorem 12. If x(t) = x(t, φ) is a solution of the systems (1), then there exists a positive constant ] such that ‖x (t)‖ ≤ (1 + 2] 2 ) 󵄩 󵄩 󵄩 󵄩 φ 󵄩 󵄩 󵄩 󵄩 E α (]tα) ∀t ∈ J = [0, T] . (15) Proof. According to the properties of the fractional order 0 < α < 1, one can obtain a solution in the form of the equivalent Volterra integral equation [12]: x (t) = x (0) + C (x (t − τ) − x (−τ))


Introduction
In this paper, we consider a neutral fractional time-delay system: where  D  denotes the Caputo fractional derivative of order 0 <  ≤ 1, the vector function () ∈   , , ,  are constant system matrices of appropriate dimensions, the constant parameter  > 0 represents the delay argument, and () is a given continuously differentiable function on [−, 0].The neutral time-delay systems have received increasing attention (see [1][2][3][4][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 timedelay 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][7][8][9]), there has been a surge in the study of neutral fractional time-delay systems (see [10][11][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][14][15][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.

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 C denotes the complex plane.