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Finite-time stability analysis of fractional singular time-delay systems
Advances in Difference Equations volume 2014, Article number: 259 (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.
In this paper, we consider the following fractional singular time-delay system:
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 )
The Euler gamma function is defined as
where ℂ denotes the complex plane.
Definition 2.2 (see )
The fractional integral of order α with the lower limit zero for any function , is defined as
where , is the gamma function.
Definition 2.3 (see )
The Riemann-Liouville derivative of order α with the lower limit zero for any function , is defined as
Definition 2.4 (see )
The Caputo derivative of order α for any function , , is defined as
Remark 2.1 (see )
The Laplace transform of the Caputo derivative is(2.5)
The Caputo fractional derivative is a linear operator satisfying the relation(2.6)
where λ and μ are scalars.
Lemma 2.1 (see )
Let , then we have
Definition 2.5 (see )
The Mittag-Leffler function in two parameters is defined as
where , , and .
Remark 2.2 (see )
For , , and , ;
for , the matrix extension of the aforementioned Mittag-Leffler function has the following representation: , and ;
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 )
The pencil is regular if and only if two nonsingular matrices Q, P may be chosen such that
where ; ; is nilpotent; , are identity matrices.
Remark 2.3 (see )
is nilpotent (the nilpotent index is denoted by h), and we have(2.11)
where h is also called the index of the matrix pair ;
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.
For convenience, we denote by , from Lemma 2.1 and Remark 2.4, we deduce the following statement. Assume that the system (1.1) is regular throughout this paper, there exist two nonsingular matrices Q and P such that the system (1.1) is a restricted system equivalent to
with the coordinate transformation
where ; ; ; ; ; ; N is nilpotent.
The following definitions and lemmas will play important roles in our next analysis.
Definition 2.7 (see )
If X and Y are normed linear spaces, an operator is linear if
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 )
If is a linear operator from a normed linear space X to a normed linear space Y, the following are equivalent:
is continuous at 0.
Lemma 2.4 (see ; Generalized Gronwall Inequality)
Suppose , are nonnegative and local integrable on ; some , and is a nonnegative, nondecreasing continuous function defined on ; , where is a constant, with
on this interval. Then
Lemma 2.5 (see )
Under the hypothesis of Theorem 3.2, 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 singular fractional time-delay system (1.1).
Let be the Caputo fractional differential operator of order , and . It is not difficult to verify the following:
where is an identity matrix.
Theorem 3.1 The fractional differential operator is bounded i.e. there exists a positive constant M such that for we have
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 .
According to , we have
and for ,
Combining Remark 2.3 and (3.4) yields
Therefore, the operator is bounded. □
To give the solution of systems (2.12), let us define a new function.
Definition 3.1 (see )
Let α obey (), the function
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.
Proof From Remark 2.3, we know that the pencil is regular if the system (1.1) is regular. By the coordinate transformation, the system (2.12) is equivalent to the system (1.1). For , then , the system (2.12)(a) may be written as
Let . Obviously, if is the known function, then (3.7) may be written as
Applying the Laplace transformation on both sides of (3.8) and using (2.5) yield
Applying the Laplace inverse transformation on both sides of (3.9) and using (2.9) yield
As for the system (2.12)(b), it may be rewritten as
Similarly, let , and is the known sufficiently often differentiable function, then (3.11) may be written as
Taking the Laplace transformation on both sides of (3.12), we have
According to Remark 3.1, the inverse Laplace transformation of yields
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.
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 φ in C by
Let and , be equipped with the norm
Definition 3.2 (see )
The system given by (1.1) satisfying the initial condition , for is finite-time stable w.r.t. , , if and only if
Theorem 3.3 If is a solution of the system (2.12), then there exist positive constants a and b such that
Proof According to Lemma 2.1, the system (2.12)(a) may be rewritten in the form of the equivalent Volterra integral equation
Using the appropriate property of the norm on (3.19), it follows that
As for the system (2.12)(b), we have
Applying the appropriate property of the norm and Theorem 3.1, we have
Combining (3.20) and (3.22) yields
For , , (3.23) can be written as
From Definition 2.2, we know that is an increasing function of t, if . So and are both increasing functions with regard to t. Taking into account (3.24) and (3.16) yields
Let and , we can see the function in Lemma 2.5 to be
obviously, it is nondecreasing.
An application of the corollary of the generalized Gronwall inequality (2.18) yields
Similarly, the same argument implies the following estimate:
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 .
Equations (3.27) and (3.28) suggest the following general expression:
To prove (3.29) by induction we have to show that it holds for because of (3.27) and if it holds for , then it also holds for . Indeed, for , so that , on the one hand, using (3.28), we have
On the other hand, using (3.29) we obtain
Taking into account (3.30) and (3.31) we conclude that
The proof is completed. □
Theorem 3.4 The fractional singular time-delay system given by (1.1) is finite-time stable w.r.t. , , if the following condition is satisfied:
Proof From the coordinate transformation (2.13), we have
From Theorem 3.3 we obtain
Hence, using Definition 3.2 and the basic condition of Theorem 3.4, it follows that
The proof is completed. □
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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).
The authors declare that they have no competing interests.
The authors contributed equally to the manuscript. Both of them read and approved the final manuscript.
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Pang, D., Jiang, W. Finite-time stability analysis of fractional singular time-delay systems. Adv Differ Equ 2014, 259 (2014). https://doi.org/10.1186/1687-1847-2014-259
- finite-time stability
- singular systems
- time delay
- fractional calculus
- Mittag-Leffler estimation
- generalized Gronwall inequality