Abstract
The aim of this paper is to study the controllability of fractional systems involving the Atangana–Baleanu fractional derivative using the Caputo approach. In the first step, the solution of a linear fractional system is obtained. Then, based on the obtained solution, some necessary and sufficient conditions for the controllability of such a system will be presented. Afterwards, the controllability of a nonlinear fractional system will be analyzed, based on these results. Our tool for the presentation of the sufficient conditions of controllability in this part is Schauder fixed point theorem. In the last step, the analytical results are illustrated by numerical examples.
1. Introduction
Fractional calculus, with its long history, has been used in a wide range of applications. The fractional-order systems, for instance, can describe some dynamic procedures in many physical systems such as heat conduction [1] and viscoelastic materials [2] and in biology [3], bioengineering [4], energy systems [5], and economics [6]. The long-memory property of fractional calculus makes it a potent instrument for describing and designing a specific set of nonlocal dynamic trends, associated with complex systems [7]. Formerly, there have been widespread discussions in the field of dynamic properties of fractional systems. The problem of stability, for example, has been studied in [8–11].
Another important issue of fractional systems is the concept of controllability. The controllability of linear systems is established by Klamka [12]. The sufficient conditions of controllability of semilinear systems with multiple variable delays in control are formulated in [13]. Buedo-Fernández and Nieto [14] presented the necessary and sufficient conditions for the controllability of a linear fractional system, with constant coefficients using the Caputo fractional derivative, based on the Kalman matrix. In [15], the controllability of linear fractional dynamical systems with different order is investigated. The constrained controllability of continuous-time fractional-order control systems with multiple delays in control is considered in [16]. A computational procedure for the controllability of linear and nonlinear fractional dynamical systems of order is provided by Balachandran and Govindaraj [17]. Balachandran et al. [18] established a set of sufficient conditions for the controllability of nonlinear fractional dynamical systems. In [19], the controllability of nonlinear fractional delay dynamical systems with prescribed controls is considered. A variational approach to study the finite approximate controllability for Sobolev-type fractional semilinear evolution equations with nonlocal conditions in Hilbert spaces is extended by Mahmudov [20].
So far, various types of fractional derivatives such as Riemann–Liouville and Caputo have been proposed [7, 21, 22]. Derivatives such as those mentioned above can not properly model some nonlocal phenomena due to having a singular kernel. To overcome the singularity problems in fractional derivative, Caputo and Fabrizio introduced a new fractional differential operator using the exponential function as a kernel [23]. In [24], a new formulation of time fractional optimal control problems governed by Caputo–Fabrizio fractional derivative is proposed. Also, Atangana and Baleanu proposed a new derivative based on the Mittag-Leffler function [25]. An open discussion is ongoing about the mathematical construction of the Atangana–Baleanu fractional operator in Caputo sense (ABC). Diethelm et al. [26] showed that it is possible to solve differential equations with nonsingular kernel derivatives only when a very restrictive and unnatural assumption is made on the initial condition. Giusti [27] indicated that this operator can be expressed as an infinite series involving Riemann–Liouville integrals. The authors in [28] showed that the ABC definition cannot be useful in modeling problems such as the fractional diffusion equation because the solutions obtained for these equations do not satisfy the initial condition. Ortigueira et al. [29] showed that the models involving the generalized fractional derivative with regular kernels poorly reflect the real-world data. In responses to these criticisms, Atangana and Gómez-Aguilar [30] emphasized the need to account for a fractional calculus approach without an imposed index law and with nonsingular kernels. Furthermore, Sabatier [31] showed that the papers [26, 27] are not correct and produce the wrong conclusion on the restriction imposed by nonsingular kernels. In a comment written by Baleanu [32], it has been shown that the opinions of Ortigueira et al. [29] are not consistent. Also, Atangana and Goufo [33] presented some interesting results to clarify the mistake and lack of understanding for those writing against derivatives with nonsingular kernels. Hristov [34] investigated the underlying physical meaning of the nonsingular kernel and also presented a collection of recent applications of fractional differentiation operators with nonsingular kernels.
Motivated by the references mentioned above, we study the controllability of a nonlinear fractional system, represented as follows:where is the Atangana–Baleanu fractional derivative in Caputo sense of order ; and are state and input vectors, respectively; and and are constant matrices. Also, and its Caputo derivative are continuous.
The following notations are used throughout this paper. For , denotes the matrix transpose of . We assume that is nonsingular and . We use the notations and , respectively, for the column space and the null space of the matrix . The symbols and are used for and , respectively, at .
This paper is organized as follows. Section 2 presents some preliminary concepts of fractional calculus. In Section 3, we obtain the solution of a linear fractional system and analyze the controllability of such a system. In Section 4, we establish the sufficient conditions for the controllability of a nonlinear fractional system. The obtained results are numerically confirmed in Section 5. Eventually, the conclusion is stated in Section 6.
2. Preliminaries
We start with a brief overview of some mathematical preliminaries.
Definition 1. Let , , , and be a suitable function. The Caputo fractional derivative of order is defined as [21, 35]Throughout this paper ; then,
Lemma 2. Assuming that the Laplace transform of the function exists, then the Laplace transform formula for the Caputo fractional derivative is [21]For , we have
Definition 2. The two-parameter Mittag-Leffler function, as an important function in fractional calculus, is defined as [36]In particular, if , then
Lemma 2. The Laplace transform of the function is [21]In particular, for , equation (3) becomesFor Mittag-Leffler function in matrix form, similar equations are established.
Definition 3. Let , , and . The Atangana–Baleanu fractional derivative in Caputo sense is given as [25]where is a normalization function that can be any function satisfying . Throughout this paper, we suppose .
Lemma 3. Assuming that the Laplace transform of the function exists, then the Laplace transform of the Atangana–Baleanu fractional derivative in Caputo sense is as follows [25]:Throughout this paper, for brevity, we use and instead of and , respectively.
3. Linear System
The continuous-time linear fractional system with the Atangana–Baleanu fractional derivative in Caputo sense is presented as follows:where and are state and input vectors, respectively, and and are constant matrices.
In the following theorem, we present the solution of system (12).
Theorem 1. The solution of system (12), starting from , , and assuming the existence of , is
Proof. Taking Laplace transform of system (12), we havewhich may be rewritten asMultiplying equation (15) by , we obtainTherefore, we can writeThen, adding and subtracting yieldsAccording to (5) and taking the inverse Laplace transform, we haveFinally, by applying the convolution theorem and equations (8), (9), and (19), we obtainSince , we have , and so the proof is complete.
First, we review the definition of controllability of the fractional dynamical system, in agreement with [37].
Definition 4. System (12) is controllable on , if for every , there exists a control defined on such that the corresponding solution of (12) satisfies .
Corresponding to system (12), we define the controllability Gramian matrix as follows:Then, we will present the controllability criteria of system (12).
Theorem 2. The continuous-time linear fractional system (12) is controllable on , if and only if the controllability Gramian matrix is nonsingular.
Proof. Sufficiency: suppose that is nonsingular. For every that satisfies , we choose the controlwith . It can be shown that exists and satisfies inCombining (13) at and (23), we obtainwhich shows linear system (12) is controllable.
Necessity: assume that linear system (12) is controllable. If is singular, then there exists a vector such that , which is equivalent toThe preceding equation implies thatLet . According to controllability assumption, there exists a control on such that , which meansMultiplying equation (27) by , we can writeEquations (26) and (28) yield , which is a contradiction to . Thus, is nonsingular.
Now, we present another criterion for the controllability of system (12).
Theorem 3. The continuous-time linear fractional system (12) is controllable on , if and only if the Kalman matrix is full rank.
Proof. Let . At first, we prove that . For this purpose, we consider as the set of reachable states based on system (12), with the zero initial condition. We show that for every , . We will accomplish the proof in three steps.
In the first step, we prove that . Let be every reachable state. Hence, there exists a control such thatwhich is equal towhereWriting (30) as a product, we haveBy Cayley–Hamilton theorem, any of these matrices is a linear combination of . Hence, . Therefore, .
In the second step, we prove that . We will show equivalently that , where and are the orthogonal complements of and , respectively. We need to show that if , then . Let . Thus, we have , which is equivalent toFor , . Setting , we haveDifferentiating times with respect to and taking the limit when implies that , for . It follows thatThis gives us that . Therefore, .
In the third step, we prove that . Let . Then, there exists such that . For every that satisfies , we define the control aswith . It can be shown that exists and satisfies inThen, the solution of system (12) with at isTherefore, . This proves that .
Considering all three previous steps, we obtain . Since , we haveSo, . Therefore, the Kalman matrix is full rank if and only if the Gramian matrix is nonsingular. Using Theorem 2, the proof is completed.
Using Theorem 2 and Theorem 4, we obtain the necessary and sufficient conditions for the controllability of classical linear system [38]. It should be noted that the presented criterion in Theorem 3 is similar to [14, Theorem 2] and does not depend on .
4. Nonlinear System
Consider the continuous-time nonlinear fractional system (1). Let be the Banach space of continuous valued function, defined on with the following norm:where and .
Assume that the Caputo derivatives of and exist. Then, for every , a similar process of Theorem 1 can be used to show that the solution of the linear fractional systemis
In order to analyze the controllability of nonlinear system (1), we propose the following theorem that presents sufficient conditions for the controllability of such a system.
Theorem 4. Let the continuous function with its continuous Caputo derivative satisfy the conditionuniformly for . If linear system (12) is controllable, then nonlinear system (1) is controllable on .
Proof. Consider the operator with . For every that satisfies , we definewith , andWe are going to provide the conditions for using the Schauder fixed point theorem to show that has a fixed point. According to the continuity of the Mittag-Leffler function and from (44), there exist positive numbers and , such that for , we havewhereIt should be noted that exists and satisfies inSimilarly, from (45) and (48), there exist positive numbers and such that for ,Next, defining and , we can write (46) and (49) asFrom [39], since continuous function satisfies assumption (43), for every pair of positive constants and , there exists a positive constant such that, if , thenNext, we introduce the set . We show that maps into itself. For every , we have and . Therefore, , for all . It follows that . Then, from (50) and (51), we get and . Hence, and . Since is continuous, the operator is continuous. It should be noted that is closed, bounded, and convex, so by the Schauder fixed point theorem [40, Theorem 4.14], has a fixed point , such that . Therefore, we rewrite (17) and (18) asandwhich shows that is the solution of nonlinear system (1). Now, setting in (54), we get , so nonlinear system (1) is controllable.
Proposition 1. The values of positive numbers , , , and used in Theorem 4 are calculated bywhere
Proof. According to (53) and (56), we can writeDefiningyieldsSimilarly, it follows from (54) thatUsing (48) yieldsCombining (60) and (61) results inThen, definingwe deriveIt should be noted that due to the existence of or or both of them in the calculation of the Caputo derivative of , we present the following proposition, which is an appropriate tool for examining condition (43).
Proposition 2. Assume that the notations , , and are described in Proposition 1. Then, the derivatives of and exist and are bounded.
Proof. According to (53), we can writeSettingwe haveDefiningyieldsSimilarly, it follows from (54) thatSettingwe obtainThen, using (61) yieldsDefiningwe derive
5. Numerical Examples
In this section, we utilize some examples to illustrate the validity of our results.
Example 1. We consider the following linear system:where , , andFrom (21), the controllability Gramian matrix iswhereThen, defining and yieldsTherefore, (78) becomeswhich is nonsingular.
Also, the Kalman matrix is full rank. Hence, from Theorem 2 or Theorem 3, system (76) with control defined in (22) is controllable. We consider and as the solutions of system (76) for initial conditions and , respectively. HereThe control steers system (76) from and to and , respectively, whereNow, consider the following system:where , , , and are as above. According to the above description, this system is also controllable. Using Matlab, the trajectories of solutions are depicted in Figures 1 and 2.
Example 2. We consider the nonlinear system as follows:where and , , , and are the same as in the Example 1. The Caputo derivative of isFrom Proposition 2, satisfies condition (43). Also, according to Example 1, the linear system is controllable. Therefore, nonlinear system (85) is controllable.
6. Conclusion
In this paper, we obtained the solution of the linear fractional system involving the Atangana–Baleanu derivative with the Caputo approach. According to the definition of controllability, we presented some necessary and sufficient conditions for the controllability of such a system based on the controllability Gramian matrix and the Kalman matrix. Using the Schauder fixed point theorem and the obtained results for the linear system, we also established sufficient conditions for the controllability of the nonlinear system. The reliability of the analytical process used in our work has been checked by numerical examples.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.