- Open Access
Stability of fractional neutral systems
© Liu and Jiang; licensee Springer. 2014
- Received: 22 October 2013
- Accepted: 19 February 2014
- Published: 7 March 2014
In this paper, we investigate the stability of a class of nonlinear fractional neutral systems. We extend the Lyapunov-Krasovskii approach to nonlinear fractional neutral systems. Necessary and sufficient conditions for stability are obtained for the nonlinear fractional neutral systems.
MSC:34K20, 34K37, 34K40.
- fractional neutral systems
- Lyapunov-Krasovskii approach
In the past few decades, fractional calculus and fractional differential equations have attracted great attention. It has been proved that fractional-order calculus is more adequate to describe real world problems than the integer calculus. Therefore, not only mathematicians have currently a strong interest in the fractional calculus but also researchers in applied fields such as mechanics, physics, chemistry, biology, economics, control theory, and signal processing. For details and examples, see [1–7] and the references therein.
On the other hand, stability analysis is always one of the most important issues in the theory of differential equations and their applications. The analysis on stability of fractional-order differential equations is more complex than that of classical differential equations, since fractional derivatives are nonlocal and have weakly singular kernels. The earliest study on the stability of fractional systems started in ; the author has given a well-known stability criterion for a linear fractional differential system with constant coefficient matrix. Since then, the stability of fractional systems has attracted increasing interest. Many researchers have done further studies on the stability of fractional systems. In , by the frequency domain method, the BIBO-stability of fractional differential systems with delays was considered. Chen and Moore  considered the analytical stability bound for a class of fractional differential systems with time-delay. The authors derived a stability condition by applying the Laplace transformation and the Lambert function.  studied the linear system with multi-order Caputo derivative and derived a sufficient condition on Lyapunov global asymptotical stability. In  LMI stability conditions for linear fractional differential systems were given. The boundedness properties of system responses are very important from the engineering point of view. From this fact, finite-time stability for fractional differential systems with time-delay was introduced [13, 14]. In , the definitions of q-fractional calculus was presented and the stability of non-autonomous systems within the frame of the q-Caputo fractional derivative was studied. Recently, survey papers [16, 17] have provided more details about the stability results and the methods available to analyze the stability of fractional differential systems; the reader may refer to them and the references therein.
As is well known, Lyapunov’s second method provides a way to analyze the stability of a system without explicitly solving the differential equations. It is necessary to extend Lyapunov’s second method to fractional systems. In [18, 19], the fractional Lyapunov’s second method was proposed, and the authors extended the exponential stability of integer-order differential system to the Mittag-Leffler stability of fractional differential system. In , by using Bihari’s and Bellman-Gronwall’s inequality, an extension of Lyapunov’s second method for fractional-order systems was proposed. In [21–23], Baleanu et al. extended Lyapunov’s method to fractional functional differential systems and developed the Lyapunov-Krasovskii stability theorem, Lyapunov-Razumikhin stability theorem and Mittag-Leffler stability theorem for fractional functional differential systems. As far as we know, there are few papers with respect to the stability of fractional neutral systems. In this paper, we consider the stability of a class of nonlinear fractional neutral functional differential equations with the Caputo derivative. Motivated by Li et al.. [18, 19], Baleanu et al. , and Cruz and Hale , we aim in this paper to extend the Lyapunov-Krasovskii approach for the nonlinear fractional neutral systems.
The rest of the paper is organized as follows. In Section 2, we give some notations and recall some concepts and preparation results. In Sections 3, by using Lyapunov functionals, we extend the Lyapunov-Krasovskii approach for nonlinear fractional neutral systems, results of stability for nonlinear fractional neutral systems are presented. Finally, some concluding remarks end the paper.
In this section, we introduce notations, definitions, and preliminary facts needed here. Throughout this paper, let be a real n-dimensional linear vector space with the norm , denotes the induced norm of a matrix A, let be the space of continuous functions taking into with , defined by , , , are constants. If , and , then for any , we let be defined by , .
where is the gamma function.
For , we have .
- (ii)When , , we have
For , and , we have , .
In general, it is not true that is nondecreasing in t.
In , Cruz and Hale studied a class of functional difference operators which are very useful in stability theory and the asymptotic behavior of solutions of functional differential equations of neutral type. In a monograph , Hale presents the following definition and results of the difference operators.
is uniformly asymptotically stable.
- (2)There are constants and such that for any , any solution y of the nonhomogeneous equation
Remark 2.2 Let , let the matrix B be Schur stable, i.e., the spectrum of the matrix lies in the open unit disc of the complex plane; then is stable.
where , , the matrix B is Schur stable, is continuous, Lipschitz in . Here, we always assume that fractional neutral system (3.1) with initial condition (3.2) has a unique continuous solution which depends continuously upon , φ.
Definition 3.1 We say that the zero solution of (3.1) is stable if for any and any , there exists a such that any solution of (3.1) with initial value φ at , satisfies for .
For any given the functional is continuous in ϕ at the point 0, i.e., for any there exists such that the inequality implies .
Along the solutions of the system (3.1) the functional satisfies for .
Proof Sufficiency: Since the matrix B is Schur stable, there exist and such that the inequality holds for .
Since for a given functional is continuous in ϕ at the point 0, there exists such that for any , with . Here, we claim that . Suppose this is not the case; then there exists an initial function such that and . On the one hand, for this initial function we have . On the other hand, . The contradiction proves the desired inequality.
Therefore, the zero solution of system (3.1) is stable.
Necessity: Now, the zero solution of system (3.1) is stable, and we must prove that there exist a function and a functional that satisfy the conditions of the theorem.
that is, for a fixed the functional is continuous in φ at the point 0.
The proof is complete. □
Remark 3.1 The functional (3.9) has only academic value. Obviously, we cannot use such functionals in applications. The computation of practically useful Lyapunov functionals is a very difficult task.
- (3)along the solutions of the system (3.1) the functional is continuously differentiable and satisfies
then the zero solution of system (3.1) is stable.
Proof Note that the theorem’s condition imply that of Theorem 3.1, therefore, the zero solution of system (3.1) is stable. □
Theorem 3.3 Suppose that the assumptions in Theorem 3.2 are satisfied except for replacing by ; then one has the same result for stability.
Since , then . Then we obtain the same result for stability. □
In this paper, we have studied the stability of a class of nonlinear fractional neutral differential difference systems. We introduce the Lyapunov-Krasovskii approach for fractional neutral systems, which enrich the knowledge of both system theory and fractional calculus. By using Lyapunov functionals and the Lyapunov-Krasovskii technique, stability criteria are obtained for the nonlinear fractional neutral systems. Finally, we point out that since the computation of practically useful Lyapunov functionals is a very difficult task, the fractional Lyapunov method has its own limitations and should be generalized and verified for more complicated linear and nonlinear problems.
This work is supported by the National Natural Science Foundation of China (11371027), the Fundamental Research Funds for the Central Universities (2013HGXJ0226) and the Fund of Anhui University Graduate Academic Innovation Research (10117700004).
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