Research Article | Open Access
D. Baleanu, S. J. Sadati, R. Ghaderi, A. Ranjbar, T. Abdeljawad (Maraaba), F. Jarad, "Razumikhin Stability Theorem for Fractional Systems with Delay", Abstract and Applied Analysis, vol. 2010, Article ID 124812, 9 pages, 2010. https://doi.org/10.1155/2010/124812
Razumikhin Stability Theorem for Fractional Systems with Delay
Fractional calculus techniques and methods started to be applied successfully during the last decades in several fields of science and engineering. In this paper we studied the stability of fractional-order nonlinear time-delay systems for Riemann-Liouville and Caputo derivatives and we extended Razumikhin theorem for the fractional nonlinear time-delay systems.
Fractional calculus is a good candidate to solve the dynamics of complex systems. During the last years fractional calculus was subjected to an intense debate. The fractional differential equations started to play an important role in modeling anomalous diffusion, processes having long-range dependence, and so on. Several open problems remain unsolved or there were partially solved with this type of calculus. Among those kinds of problems we mention the question of stability which is interest in nonlinear science and control theory. Also, the problem of time-delay system has been discussed over many years. For a survey the reader can check the study in . Time delay is very often encountered in different technical systems, for example, electric, pneumatic, and hydraulic networks, chemical processes, and long transmission lines. The existence of pure time delay, regardless of its presence in a control and/or state, may cause undesirable system transient response, or, generally, even an instability. Numerous reports have been published on this matter, with particular emphasis on the application of Lyapunov’s second method [8, 9].
In recent years, considerable attention has been paid to control systems whose processes and/or controllers are of fractional order. This is mainly due to the fact that many real-world physical systems are well characterized by fractional-order differential equations, that is, equations involving noninteger-order derivatives. In particular, it has been shown that viscoelastic materials having memory and hereditary effects  and dynamical processes such as semi-infinite lossy RC transmission , mass diffusion, and heat conduction , can be more adequately modeled by fractional-order models than integer-order models. Moreover, with the success in the synthesis of real noninteger differentiator and the emergence of new electrical circuit element called “fractance” , fractional-order controllers [14, 15] including fractional-order PID controllers  have been proposed to enhance the robustness and performance of control systems.
Some literatures published about stability of fractional-order linear time-delay systems [16–19]. In the base of Lyapunov’s second method, some work has been done in the field of stability of fractional-order nonlinear systems without delay [20, 21]. But it seems that a few attentions have been paid to the stability of fractional-order nonlinear time-delay systems.
When the system involves time delay, it should be regarded as a functional differential equation (FDE). In this case, analysis of the stability relies on the Lyapunov-Krasovskii functional [22, 23]. However, the Razumikhin stability theory is more widely used to prove the stability of time-delay systems [22, 23], since the construction of Lyapunov-Krasovskii functional is more difficult than that of Lyapunov-Razumikhin function.
The purpose of this paper is to develop the Razumikhin theorem for fractional-order nonlinear time-delay systems.
The manuscript is organized as follows. In Section 2 some basic definitions of fractional calculus are mentioned. Section 3 is devoted to fractional nonlinear time-delay systems. Section 4 presents the generalization of the fractional Razumikhin theorem when both fractional derivatives and delay are present.
2. Preliminaries and Definitions
In the fractional calculus the Riemann-Liouville and Caputo fractional derivatives are defined respectively [1–3]: where is an arbitrary differentiable function, , and are the Riemann-Liouville and Caputo fractional derivatives of order on , respectively, and denotes the Gamma function.
For we have
Property 1. When , we have In particular, if , we have
Property 2. For any , we have In particular, if , , and , then from Property 1, we have
Property 3. where and are arbitrary constants.
Property 4. From the definition of Caputo’s derivative when we have where
3. Fractional Nonlinear Time-Delay System
Let be the set of continuous functions mapping the interval to . In many situations, one may wish to identify a maximum time delay of a system. In this case, we are often interested in the set of continuous function mapping to , for which we simplify the notation to . For any and any continuous function of time , , let be a segment of function defined as , .
Consider fractional nonlinear time-delay system where , , and . As such, to determine the future evolution of the state, it is necessary to specify the initial state variables in a time interval of length , say from to , that is, where is given. In other words .
Definition 3.1. Suppose that for all . The solution of (3.1) is said to be stable if for any , , there is a such that implies that for . The solution of (3.1) is said to be asymptotically stable if it is stable and there is a such that implies that as . The solution is said to be uniformly stable if the number in the definition is independent of . The solution of (3.1) is uniformly asymptotically stable if it is uniformly stable and there is a such that, for every , there is a such that implies that for for every .
4. Fractional Razumikhin Theorem
As in the study of systems without delay, an effective method for determining the stability of a time-delay system is Lyapunov method. Since in a time-delay system the “state” at time required the value of in the interval , that is, , it is natural to expect that, for a time-delay system, corresponding Lyapunov function be a functional depending on , which also should measure the deviation of from the trivial solution .
Let be differentiable, and let be the solution of (3.1) at time with initial condition . Then we calculate the Caputo derivative of with respect to and evaluate it at as follow, respectively: where .
Theorem 4.1. Suppose that in (3.1) maps into bounded sets in , and are continuous nondecreasing functions, are positive for , and , strictly increasing. If there exists a continuously differentiable function such that
and the Caputo fractional derivative of along the solution of (3.1) satisfies
for and , then system (3.1) is uniformly stable.
If, in addition, for and there exists a continuous nondecreasing function for such that condition (4.4) is strengthened to for and , then system (3.1) is uniformly asymptotically stable.If in addition , then system (3.1) is globally uniformly asymptotically stable.
Proof. To prove uniform stability, for any given , let . Then for any given and , with , we have for . Let be solution of (3.1) with initial condition . According to (4.4), as increases, whenever and for , , therefore, by Property 4 and (4.3), for all . Due to the continuity of , it is therefore impossible for to exceed . In other words, we have for , but this implies that for .
To complete the proof of the theorem, suppose that , are such that . Such numbers always exist by our hypotheses on and . In fact, since and for , one can preassign and then determine a such that the desired relation is satisfied. If as , then one can fix arbitrarily and determine such that . This remark and reasoning that follows will prove the uniform asymptotic stability of as well as the fact that is a globally uniformly asymptotically stable.
If , the same argument as in the proof of uniform stability shows that implies that , for . Suppose that is arbitrary. We need to show that there is a number such that for any and the solution of (3.1) satisfies , . This will be true if we show that , for .
From the properties of function , there is a number such that for . Let be the first positive integer such that , and let and .
We now show that for all . First, we show that for . If , for , then, since for all , it follows that for .
Hypothesis (4.5) implies that for . Consequently, and hence by Property 4, for all we have on the same interval. The positive property (4.2) of implies that at . But this implies that for all , since is negative by condition (4.5), and therefore is negative when .
Now, let , , , and assume that, for some integer , in the interval , we have By the same type reasoning as above, we have for , and we have and we have if . Consequently, and, finally, for . This completes the induction and we have for all . This proves the theorem.
Lemma 4.2. Let and , then
Proof. By using Property 1 we have . Because and , we obtained that .
Theorem 4.3. Suppose that the assumptions in Theorem 4.1 are satisfied except replacing by , then one has the same result for uniform stability, uniform asymptotic stability, and global uniform asymptotic stability.
Proof. It follows from Lemma 4.2 and that which implies that for all . Following the same proof as in Theorem 4.1 yields uniform stability, uniform asymptotic stability, and global uniform asymptotic stability.
The combination of the fractional calculus and delay techniques seems to describe better the dynamics of the complex systems, namely, because both theories take into account the memory effects. Having in mind these aspects, in this paper we generalized the fractional Razumikhin theorem in presence of Caputo fractional derivative and delay. By using the Caputo and the Riemann-Liouville we have proved two corresponding theorems. The obtained theorems contain as particular case the fractional calculus version as well as the time-delay one.
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