Mittag–Leffler stability analysis of nonlinear fractional-order systems with impulses
Introduction
The development of the theory of the fractional calculus can be dated back to 17th century along with the classical inter-order calculus. Unfortunately, the application of fractional calculus did not gain adequate attention until specialists and scholars declare that fractional derivatives possess the superior merits of memory and hereditary properties [1], [2], [3]. In recent decades, fractional derivative has been applied to model mechanical and electrical properties of various real materials, meanwhile, fractional derivative has also been extensively incorporated into all kinds of dynamical systems, for example, see [4], [5], [6], [7], [8], [9] and reference therein.
In recent years, some scholars try their efforts to extend the theory of classical impulsive differential equation [10] to the case of fractional order, and the dynamical behaviors of impulsive evolution models of fractional order have become an active research topic. On the basis of Mittag–Leffler stability theory and the Lyapunov direct method proposed by Podlubny and his colleagues [11], [12], a considerable number of results of stability analysis for fractional impulsive dynamical systems have been reported, see [13], [14], [15], [16] and reference therein. In [13], [14], the authors proposed a series of conclusions on the asymptotically stable and Mittag–Leffler stable for impulsive fractional differential equations by applying Lyapunov direct method. In [15], the authors established a Lyaponov function with the term of Riemann–Liouville operator, and investigated the asymptotic stability for fractional network models. In [16], on account of the graph theory and Lyapunov method, the authors discussed the asymptotic stability and Mittag–Leffler stability for a class of feedback control systems of fractional differential equations on networks with impulsive effects. It should be recognized that in [17], the authors presented a general quadratic Lyapunov function which has been employed to establish some stability criteria for many fractional systems effectively, for example, see [18], [19]. As far as we know, there exists few result on the stability for nonlinear fractional-order systems with impulsive effects based on general quadratic Lyapunov function.
Motivated by the aforementioned discussions, the objective of this paper is to analyze the stability of a class of fractional-order nonlinear systems with impulses by utilizing the general quadratic Lyapunov function, and several sufficient conditions equipped with the terms of linear matrix inequalities will be presented.
Notations: and denote the n-dimensional Euclidean space and the set of all n × n real matrices, respectively. I stands for the identity matrix with appropriate dimension. QT means the transpose of matrix Q. stands for the Euclidean norm of a real vector u. For a real matrix A, λmax(A) and λmin(A) denote, respectively, the maximal and minimal eigenvalues of A, and A > 0( < 0) means the matrix A is symmetric and positive definite (or negative definite). In addition, let Ω be an open set in containing the origin, and .
Section snippets
Model description and preliminaries
In this paper, we consider the following fractional-order nonlinear systems with impulses: where denotes the Caputo fractional derivative of order α, (0 < α < 1), is the state vector, is a constant matrix, is the nonlinear term with Jk( · ) stands for the jump operator of impulsive, and the impulsive moments satisfy with .
Let u0 ∈ Ω. Denote by
Main results
Theorem 1 Assume that (H) holds, and Jk(u(tk)) satisfies. If there exist a symmetric and positive definite matrix Q > 0, and positive scalars ρ1 > 0 and 0 < μ1 ≤ 1, such that the matrix Q > 0 and the impulse matrix Bk satisfy the following inequalities
then system (1) is Mittag–Leffler stable. Proof Consider a Lyapunov function candidate: . When t ≠ tk, by calculating the α-order Caputo derivatives of V(t)
Numerical simulation examples
Example 1 Consider the following fractional-order nonlinear systems with impulses
where Apparently, g(t, u(t)) is continuous with Lipschitz constant . Let and . It is easy to verify that conditions (17) and (18) are satisfied. Therefore, according to Theorem 1, system (37) is Mittag–Leffler stable. Fig. 1 shows the time
Conclusions
In this paper, we have analyzed the Mittag–Leffler stability of nonlinear fractional-order systems with impulses. Several sufficient criteria of the form of linear matrix inequalities have been established by applying the theory of fractional differential equation and impulsive differential equation. Meanwhile, two simulation examples have been presented to illustrate the effectiveness and availability of the theory results. However, it should be pointed out that we have only obtained some
Acknowledgments
This work was supported by the Qatar National Research Fund, a member of the Qatar Foundation, through the National Priorities Research Program under Grant NPRP 4-1162-1-181, by the National Natural Science Foundation of China under Grants 61374078, 61273021 and 11501065, and in part by the Chongqing Research Program of Basic Research and Frontier Technology of cstc2015jcyjBX0052.
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