On Controllability of Fractional Continuous-Time Systems

(e 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. (en, 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.


Introduction
Fractional calculus, with its long history, has been used in a wide range of applications. e 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]. e longmemory 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. e problem of stability, for example, has been studied in [8][9][10][11].
Another important issue of fractional systems is the concept of controllability. e controllability of linear systems is established by Klamka [12]. e 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. e 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 1 < α ≤ 2 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. e 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 ABC D α is the Atangana-Baleanu fractional derivative in Caputo sense of order 0 < α < 1; x(t) ∈ R n and u(t) ∈ R m are state and input vectors, respectively; and A ∈ R n×n and B ∈ R n×m are constant matrices. Also, f: [0, T] × R n × R m ⟶ R n and its Caputo derivative are continuous. e following notations are used throughout this paper. For A ∈ R n×n , A * denotes the matrix transpose of A. We assume that I − (1 − α)A is nonsingular and A � (I − (1− α)A) − 1 . We use the notations Im(A) and Ker(A), respectively, for the column space and the null space of the matrix A.
e symbols u 0 and f 0 are used for u(t) and f(t, x(t), u(t)), respectively, at t � 0. is 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. e obtained results are numerically confirmed in Section 5. Eventually, the conclusion is stated in Section 6.

Preliminaries
We start with a brief overview of some mathematical preliminaries.

Lemma 2. e Laplace transform of the function
In particular, for β � 1, equation (3) becomes For Mittag-Leffler function in matrix form, similar equations are established.

Lemma 3.
Assuming that the Laplace transform F(s) of the function f exists, then the Laplace transform of the Atangana-Baleanu fractional derivative in Caputo sense is as follows [25]: roughout this paper, for brevity, we use

Linear System
e continuous-time linear fractional system with the Atangana-Baleanu fractional derivative in Caputo sense is presented as follows: where x(t) ∈ R n and u(t) ∈ R m are state and input vectors, respectively, and A ∈ R n×n and B ∈ R n×m are constant matrices.
In the following theorem, we present the solution of system (12).

Theorem 1.
e solution of system (12), starting from Proof. Taking Laplace transform of system (12), we have which may be rewritten as Multiplying equation (15) by A, we obtain erefore, we can write en, adding and subtracting According to (5) and taking the inverse Laplace transform, we have Finally, by applying the convolution theorem and equations (8), (9), and (19), we obtain and so the proof is complete.
First, we review the definition of controllability of the fractional dynamical system, in agreement with [37].
such that the corresponding solution of (12) satisfies x(T) � x 1 .
Corresponding to system (12), we define the controllability Gramian matrix as follows: Mathematical Problems in Engineering en, we will present the controllability criteria of system (12).

Theorem 2. e continuous-time linear fractional system (12) is controllable on [0, T], if and only if the controllability Gramian matrix W T is nonsingular.
Proof. Sufficiency: suppose that W T is nonsingular. For every u 0 that satisfies Ax 0 + Bu 0 � 0, we choose the control with c � (α/1 − α). It can be shown that D α u(t) exists and u(t) satisfies in Combining (13) at t � T and (23), we obtain which shows linear system (12) is controllable. Necessity: assume that linear system (12) is controllable. If W T is singular, then there exists a vector z ≠ 0such that z * W T z � 0, which is equivalent to e preceding equation implies that Let x 0 � (E α,1 (αAAT α )) − 1 z. According to controllability assumption, there exists a control u(t) on [0, T] such that x(T) � 0, which means 0 � E α,1 αAAT α E α,1 αAAT α − 1 z

Mathematical Problems in Engineering
Multiplying equation (27) by z * , we can write Equations (26) and (28) yield z * z � 0, which is a contradiction to z ≠ 0. us, W T is nonsingular. Now, we present another criterion for the controllability of system (12). Proof. Let K ABC : � AB (AA)AB · · · (AA) n− 1 AB . At first, we prove that Im(K ABC ) � Im(W T ). For this purpose, we consider R t � ζ ∈ R n : there exists u such that x(t) � ζ { } as the set of reachable states based on system (12), with the zero initial condition. We show that for every t > 0, R t � Im(K ABC ) � Im(W T ). We will accomplish the proof in three steps.
In the first step, we prove that R t ⊂ Im(K ABC ). Let ζ ∈ R t be every reachable state. Hence, there exists a control u(t) such that which is equal to where Writing (30) as a product, we have By Cayley-Hamilton theorem, any of these matrices AB, (AA)AB, . . . , (AA) N AB is a linear combination of AB, (AA)AB, . . . , (AA) n− 1 AB. Hence, ζ ∈ Im(K ABC ). erefore, R t ⊂ Im(K ABC ).
In the third step, we prove that Im(W T ) ⊂ R t . Let x 1 ∈ Im(W T ). en, there exists y such that x 1 � W T y. For every u 0 that satisfies Ax 0 + Bu 0 � 0, we define the control u(t) as with c � (α/1 − α). It can be shown that D α u(t) exists and u(t) satisfies in en, the solution of system (12) erefore, x 1 ∈ R t . is proves that Im(W T ) ⊂ R t . Considering all three previous steps, we obtain Im(K ABC ) � Im(W T ). Since AA � AA, we have So, Im(K) � Im(K ABC ) � Im(W T ). erefore, the Kalman matrix K is full rank if and only if the Gramian matrix W T is nonsingular. Using eorem 2, the proof is completed.
Using eorem 2 and eorem 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 eorem 3 is similar to [14, eorem 2] and does not depend on α.

Nonlinear System
Consider the continuous-time nonlinear fractional system (1). Let Q be the Banach space of continuous R n × R m valued function, defined on [0, T] with the following norm: where Assume that the Caputo derivatives of u and f exist. en, for every (z, v) ∈ Q, a similar process of eorem 1 can be used to show that the solution of the linear fractional system is 6 Mathematical Problems in Engineering 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.
Hence, from eorem 2 or eorem 3, system (76) with control u(t) defined in (22) is controllable. We consider x 1 (t) and x 2 (t) as the solutions of system (76) for initial conditions x 1 (0) and x 2 (0), respectively. Here e control u(t) steers system (76) from x 1 (0) and x 2 (0) to x 1 (2) and x 2 (2), respectively, where x 1 (2) � 5 1 , Now, consider the following system: where A, B, α, and t 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.

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. e 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.