Abstract
We study the Mittag-Leffler and class-K function stability of fractional differential equations with order . We also investigate the comparison between two systems with Caputo and Riemann-Liouville derivatives. Two examples related to fractional-order Hopfield neural networks with constant external inputs and a marine protected area model are introduced to illustrate the applicability of stability results.
1. Introduction
Recently, the topic of fractional differential equations has attracted many researchers which plays a main role in various applications. Many areas such as physics, biophysics, aerodynamics, control theory, viscoelasticity, capacitor theory, electrical circuit, description of memory, and hereditary properties (see [1–4]) used the fractional models instead of classical models. Recently, stability of fractional differential systems has attracted many authors for more investigation in this topic [5]. In 1996, Matignon [6] firstly studied the stability of linear fractional differential systems with the Caputo derivative. Since then, many researchers have done further studies on the stability of linear fractional differential systems (see [7, 8]). In nonlinear systems, Lyapunov’s direct method provides a way to analyze the stability as Mittag-Leffler stability of a system without explicitly solving the differential equations (see [9–11]). The method generalizes the idea which shows that the system is stable if there are some Lyapunov function candidates for the system. Lyapunov’s direct method is a sufficient condition to show the stability of the systems, which means that the systems may still be stable even if one cannot find Lyapunov’s function candidate to conclude the system stability property. For the nonlinear fractional differential systems, the stability analysis is much more difficult and only a few are available.
Some authors (see [12–14]) studied the following nonlinear fractional differential system:with initial values , where . Then, some literatures on the stability of linear fractional differential systems with have appeared (see [15, 16]). However, not all the fractional differential systems have fractional orders in . There exist fractional models which have fractional orders lying in , for example, super-diffusion. In [17], the authors study the stability of n-dimensional linear fractional differential systemsof order and initial conditionsThe corresponding perturbed system iswhich is investigated by authors in [17].
For fractional-order Hopfield neural networks in applications to parallel computation and signal processing, it is required that there be a well-defined computable solution for all possible initial states. From a mathematical viewpoint, the stability analysis of a unique equilibrium point is very necessary and valuable. Recently, several important and interesting results for the stability analysis of fractional-order Hopfield neural networks have been gained [4].
The marine protected area model [18] can describe the ecological linkage between the reserve and fishing ground by the autonomous linear systemHere, is the natural mortality (death) rate, is the dispersal rate, and is the harvesting rate. The corresponding fractional behavior of such system may exist in theoretical view as the fractional order .
Motivated by these articles, we investigate the stability of the nonautonomous perturbed fractional semilinear system of the formwhere , , , , and is a given continuous function such that . The fractional derivative is either Caputo or Riemann-Liouville.
2. Preliminaries and Notations
We introduce in this section several definitions and results that will be used later in the sequel.
Definition 1 ([19]). The Riemann-Liouville fractional integral of a function is defined by
Definition 2 ([19]). The Riemann-Liouville fractional derivative of order , is defined bywhere is the ordinary differential operator.
Theorem 3 ([19]). If and , then the equalityholds everywhere on , where .
Definition 4 ([20]). The Caputo derivative (left-sided) of a function , , is defined as
Lemma 5 ([21]). Let ; then
Definition 6. The Laplace transform of a real function is defined as The Laplace transform of the Caputo fractional derivative is given byThe Laplace transform of the Riemann-Liouville fractional derivative is given by [20]The Laplace transform of is given byThe convolution of the functions and is defined asThe inverse Laplace transform of the product of the functions and is given by
Lemma 7 ([20]). The Laplace transform of the Mittag-Leffler function is given byWe conclude thatThe Mittag-Leffler function [21] is very important in solving fractional differential equations and in many applications. The exponential function as special case of Mittag-Leffler which is frequently used in the solution of integer order differential equations. The Mittag-Leffler function of one parameter is defined asThe Mittag-Leffler function of two parameters is defined bywhich both satisfy the fact that , and
Lemma 8 ([22]). The system is asymptotically stable if where , and are the eigenvalues of matrix . The stable and unstable regions for and are shown in Figures 1 and 2, respectively.
Remark 9 ([22]). The perturbed system is asymptotically stable if , and satisfies the fact that , for some .
It may be observed that the behavior of Mittag-Leffler function is a relaxation for , it is exponential for , it becomes a damped oscillation for , and it oscillates for . The decay is very fast as and very slow as. See Figures 3, 4, and 5.
Definition 10 ([23]). The constant is an equilibrium point of fractional differential systemif and only if, for all , where means either Caputo or Riemann-Liouville fractional operator.
For convenience, we state all definitions and theorems for stability in Caputo sense when the equilibrium point is the origin of , i.e., . There is no loss of generality in doing so because any equilibrium point can be shifted to the origin via a change of variables. Indeed, suppose that the equilibrium point is , then consider the change of variable . Therefore, the Caputo derivative of iswhere , and in the new variable , the system has an equilibrium point at the origin.
Definition 11 ([23]). The zero solution of system (22) is said to be stable if there exists such that , for all .
Definition 12 ([23]). The zero solution of system (22) is said to be asymptotically stable if , as .
A function is Lipschitz function if there exists a positive constant such that
Definition 13 ([23]). Let be a domain containing the origin. The zero solution of system (22) is said to satisfy the Mittag-Leffler criteria ifwhere , , , and is Lipschitz function on .
Definition 14 ([23]). Let be a domain containing the origin. The zero solution of system (22) is said to satisfy the generalized Mittag-Leffler criteria ifwhere , , , , , , and is Lipschitz function on .
Remark 15. If , the generalized Mittag-Leffler is reduced to Mittag-Leffler criteria. Moreover, the Mittag-Leffler stability implies the asymptotic stability. According to investigations in [17], the above definitions can be applied to the case of for some systems.
Definition 16 ([24]). A function is said to belong to a class-K function if is a continuous function satisfying , and it is strictly increasing.
Definition 17 ([9]). Let be an equilibrium point for system (22), and let defined on an open set containing and is continuously differentiable on . Suppose further thatThen is said to be stable if , on .
A function satisfying the above two conditions is called a Lyapunov function.
Lemma 18. Let be Lipschitz function; then the unique solution of (6) is
Proof. The existence and uniqueness of solution for (6) can be obtained by applying Banach fixed point Theorem (see [17]). By applying Laplace transform for (6), we haveand, then,Set ; then we getwhich impliesNow, taking Laplace inverse to both sides and using Lemma 7, we obtainBy convolution theorem of Laplace transform on the third term of (33), we haveThus, we get the result.
Lemma 19. Let be continuously differentiable function. The solution of the fractional differential equation iswhere , and
Proof. By taking Laplace transformation to both sides of (35), we get ThenSo,By applying the inverse Laplace transform and using Lemma 7, we obtain the solution.
Lemma 20. Let be continuously differentiable function. The solution of the fractional differential equation is where , .
Proof. By taking Laplace transform to both sides of (40), we get which implies thatBy taking for both sides, we obtain This finishes the proof.
Remark 21. In Lemma 20, if we replace Riemann-Liouville derivative by Caputo derivative, then the solution iswhere , and
Lemma 22. Let be continuously differentiable function. The solution of the fractional differential equation iswhere
Proof. By taking Laplace transform for both sides of (46), we get and thenSo,By taking and using Lemma 7, we obtain the results.
3. Stability Results
In this section, we study the stability of nonlinear fractional differential systems with Caputo derivative by utilizing a Lyapunov-like function, taking into account the relation between asymptotical stability and generalized Mittag-Leffler criteria.
The Lyapunov direct method is a sufficient condition to show the stability of systems, which means the system may still be stable even if one cannot find a Lyapunov function candidate to conclude the system stability property.
Hereafter, we assume that the Lyapunov function is continuously differentiable with respect to , Lipschitz with respect to , and .
Theorem 23. Let be an equilibrium point of system (6) and let satisfy andwhere , , and are positive real numbers. Then, the zero solution is asymptotically stable whenever , and .
Proof. Inequality (52) implies the existence of a nonnegative function such thatUsing Lemma 20, we getwhere The negativity of the last term of (54) will imply thatIn accordance with (51), we obtain Now, set ; then , as Therefore, the zero solution of system (6) is asymptotically stable.
Theorem 24. Let be an equilibrium point of system (6) and let satisfywhere , , , and are arbitrary positive constants. Then, the zero solution satisfies the Mittag-Leffler criteria whenever , and .
Proof. Using (58), there exists a nonnegative function such thatBy Lemma 18 and noting that is nonnegative, we deduce that where , and . Then,In accordance with (57), we obtain , for Then, the zero solution of system (6) satisfies the Mittag-Leffler criteria.
Theorem 25. Let be an equilibrium point of system (6) and let satisfy and where , , and are arbitrary positive constant. Then, the zero solution satisfies the generalized Mittag-Leffler criteria whenever , and .
Proof. There exists a nonnegative function such thatBy using Lemma 20, we can deduce as in the proof of proceeding results thatwhere , , Therefore,Now, set ; thenNow, set , and, therefore, the zero solution of system (6) satisfies the generalized Mittag-Leffler criteria.
Theorem 26. Let be an equilibrium point of system (6). Assume there exists a class-K function satisfyingfor some nonnegative constant . Then, the zero solution is stable.
Proof. Using (69), there exists a nonnegative function such that . Thenwhere , and . Now, (68) and (71) imply thatTherefore,Then, by (70), we have Therefore, the zero solution of system (6) is stable.
Theorem 27. Let be an equilibrium point of system (6) and assume that there exists a class-K function satisfyingwhere Then, the zero solution is asymptotically stable whenever .
Proof. As in Theorem 23, we havewhere Now, inequalities (74) and (76) imply thatTherefore , for . Hence, the continuity of implies that as . Therefore, the zero solution of system (6) is asymptotically stable.
4. The Comparison Results on Stability
In this section we discuss the comparison of the stability for the two systems with Caputo and Riemann-Liouville derivatives. It is well known that the comparison method is an effective way in judging the stability of ordinary differential systems. We will discuss similar results on the stability of fractional differential systems by using the comparison method.
Consider the semilinear system where , , , and is a continuous function such that . Moreover, satisfies the Lipschitz condition with respect to to ensure the existence and uniqueness of solution (78).
Applying the Laplace technique as in Lemma 18 but using (14), we deduce the following result.
Lemma 28. For , the solution of system (78) is
In the next result, we recall that the set of vectors can be partially ordered if their components are so.
Theorem 29. Let be an equilibrium of system (78). Assume that is a class-K function and there exists a function such that , and for each , the following conditions hold:
(i) , for any solution of system (6).
(ii) .
(iii) has nonnegative finite components.
(iv)
Moreover, assume that is nondecreasing. Then
(1) if the zero solution of system (78) is stable, then the zero solution of system (6) is stable;
(2) if the zero solution of system (78) is asymptotically stable, then the zero solution of the system (6) is asymptotically stable.
Proof. (1) Let be vector-valued function defined on whose components are nonnegative and by (ii) it satisfies the following:As in the proof of Theorem 24, we havewhere , and . Using condition (iv), we have ; hence condition (iii) implies thatWe deduce thatNow, condition (i) implies thatSince, the zero solution of system (78) is stable, then there exists such thatwhich implies that Therefore, the zero solution of system (6) is stable.
(2) Let the zero solution of system (78) be asymptotically stable, then . Again, using , we deduce that . With the continuity of and the condition , we have ; then the zero solution of system (6) is asymptotically stable.
5. Applications
The Hopfield neural networks with constant external inputs are described by [4]where and is the number of units in a neural network, is the state of the th unit at time , denotes the activation function of the th neuron, denotes the constant connection weight of the th neuron on the th neuron, represents the rate with which the th neuron resets its potential to the resting state when disconnected from the network, denotes the constant external inputs, and means the approach of the fractional order to classical first-order derivative from right side. System (6) can be easily obtained by letting , , where , for and , for , and , where .
Example 30. Consider the scalar Hopfield neural networks system for . Assume that the Lyapunov function is and that , ; then,Thus, the conditions of Theorem 24 hold with Therefore, the zero solution of system (87) satisfies the Mittag-Leffler criteria.
To illustrate the biological applications [18], we consider two habitat areas, with a fish population dispersing between the two areas, whilst fishing takes place only in region 2, with region 1 established as a no-fishing zone. To describe the ecological linkage between the reserve and fishing ground we propose the fractional alternative model of the autonomous linear systemHere, is the natural mortality (death) rate, is the dispersal rate, and is the harvesting rate. If we assume that , and , we obtain the autonomous systemThe corresponding nonautonomous system can be modeled as (6).
Example 31. Consider the systemfor , and , . Let , and , ; then system (91) can be transformed as , where . The function is Lipschitz function which ensures the existence of a solution given by Lemma 28, and ensures the existence of zero solution for (91). The argument of eigenvalues of is ; then the solution of (91) is stable (see [17], Theorem 5.1). Therefore, we can always find a constant , such that , for any . Choose the Lyapunov candidate as ; then , and . If , then . It is obvious now that Hence all hypotheses of Theorem 27 are satisfied; then system (91) is asymptotically stable.
Data Availability
All data generated or analyzed during this study are included in this published article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.