Finite-time stability analysis of fractional singular time-delay systems
© Pang and Jiang; licensee Springer. 2014
Received: 11 May 2014
Accepted: 26 September 2014
Published: 13 October 2014
This paper studies the finite-time stability of fractional singular time-delay systems. First, by the method of the steps, we discuss the existence and uniqueness of the solutions for the equivalent systems to the fractional singular time-delay systems. Furthermore, we give the Mittag-Leffler estimation of the solutions for the equivalent systems and obtain the sufficient conditions of the finite-time stability for the original systems.
In the past 30 years or so, fractional calculus has attracted many physicists, mathematicians, and engineers, and notable contributions have been made to both the theory and the applications of fractional differential equations (see [1–9]). Moreover, the different techniques have been applied to investigate the stability of various fractional dynamical systems, such as the principle of contraction mappings , the Lyapunov direct method , linear matrix inequalities , Gronwall inequalities [13–16] and fixed-point theorems .
At the same time, we notice that large numbers of practical systems, such as economic systems, power systems and so on, are singular differential systems which are also named differential-algebraic systems or descriptor systems. Such systems have some particular properties including regularity and impulse behavior which does not need to be considered in normal systems. In [18–22], the authors discuss singular systems with or without delay and obtain some important results. However, in the previous literature, there are few results on the stability of fractional singular systems, especially the fractional singular systems with time delay. In this regard, it is necessary and important to study the stability problems for fractional singular dynamical systems. Motivated by this consideration, in this paper, we investigate the stability of fractional singular dynamical systems with state delay via the generalized Gronwall approach.
where denotes the Caputo fractional derivative of order ; the vector function is a state vector; are constant matrices; is a singular matrix i.e. ; the constant parameter represents the delay argument and is a given sufficiently often differentiable function on .
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
Definition 2.1 (see )
where ℂ denotes the complex plane.
Definition 2.2 (see )
where , is the gamma function.
Definition 2.3 (see )
Definition 2.4 (see )
Remark 2.1 (see )
- (i)The Laplace transform of the Caputo derivative is(2.5)
- (ii)The Caputo fractional derivative is a linear operator satisfying the relation(2.6)
where λ and μ are scalars.
Lemma 2.1 (see )
Definition 2.5 (see )
where , , and .
Remark 2.2 (see )
For , , and , ;
for , the matrix extension of the aforementioned Mittag-Leffler function has the following representation: , and ;
- (iii)we have the Laplace transform of the Mittag-Leffler function in two parameters(2.9)
where denotes the real parts of s.
Next, we introduce some fundamental definitions and lemmas about singular systems.
Definition 2.6 (see )
For any given two matrices , the pencil is called regular if there exists a constant scalar such that , or the polynomial .
Lemma 2.2 (see )
where ; ; is nilpotent; , are identity matrices.
Remark 2.3 (see )
- (i)is nilpotent (the nilpotent index is denoted by h), and we have(2.11)
the system (1.1) will be termed regular if the pencil is regular.
In the following, we present the first equivalent form (FE1) of system (1.1) by the coordinate transformation, which is also called the standard decomposition of a singular system.
where ; ; ; ; ; ; N is nilpotent.
The following definitions and lemmas will play important roles in our next analysis.
Definition 2.7 (see )
for all , in X and scalars α and β.
Remark 2.4 (see )
We say the linear operator is a bounded linear operator from X to Y if there is a finite constant such that for all x in X.
Lemma 2.3 (see )
is continuous at 0.
Lemma 2.4 (see ; Generalized Gronwall Inequality)
Lemma 2.5 (see )
where is the Mittag-Leffler function.
3 Main results
In this section, we discuss some problems of the singular fractional time-delay system (1.1).
where is an identity matrix.
Proof Obviously, the fractional differential operator is linear. According to Lemma 2.3, we are only necessary to show that is continuous at 0. Let any sequences , , and , all we finally need to do is to show that .
Therefore, the operator is bounded. □
To give the solution of systems (2.12), let us define a new function.
Definition 3.1 (see )
is called an function, where is the Dirac delta function.
Remark 3.1 (see )
The Laplace transformation of the function is .
Theorem 3.2 If the system (1.1) is regular, the solution for the system (2.12) exists uniquely.
Obviously, by the method of steps, once the solution of the system (2.12) on is known, continuing the above process, we can easily obtain the solution of the system (2.12) on . Thus the solution of the system (2.12) on exists uniquely. □
Furthermore, we give the following theorems as regards the Mittag-Leffler estimation of the solution and finite-time stability for this singular system.
Definition 3.2 (see )
obviously, it is nondecreasing.
From Definition 2.5, we know that the Mittag-Leffler function is an increasing function with regard to t. Therefore, there exists such that and .
The proof is completed. □
The proof is completed. □
The authors are sincerely thankful to the reviewers for their valuable suggestions and insightful comments. This research was jointly supported by National Natural Science Foundation of China (no. 11371027 and no. 11471015), Program of Natural Science of Colleges of Anhui Province (KJ2013A032, KJ2011A020) and the Special Research Fund for the Doctoral Program of the Ministry of Education of China (20093401110001).
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