Controllability and Hyers–Ulam Stability of Fractional Systems with Pure Delay

: Linear and nonlinear fractional-delay systems are studied. As an application, we derive the controllability and Hyers–Ulam stability results using the representation of solutions of these systems with the help of their delayed Mittag–Lefﬂer matrix functions. We provide some sufﬁcient and necessary conditions for the controllability of linear fractional-delay systems by introducing a fractional delay Gramian matrix. Furthermore, we establish some sufﬁcient conditions of controllability and Hyers–Ulam stability of nonlinear fractional-delay systems by applying Krasnoselskii’s ﬁxed-point theorem. Our results improve, extend, and complement some existing ones. Finally, numerical examples of linear and nonlinear fractional-delay systems are presented to demonstrate the theoretical results.

On the one hand, the controllability of systems is one of the most fundamental and significant concepts in modern control theory, which consists of determining the control parameters that steer the solutions of a control system from its initial state to its final state using a set of admissible controls, where initial and final states may vary over an entire space. In recent decades, there has been considerable interest in the controllability analysis of fractional-delay systems of order α ∈ (0, 1) and α ∈ (1, 2), and several methods for studying the controllability results have been developed, for example, the robust and universal methods [22]; the Laplace transform technique, the Mittag-Leffler function and fixed-point argument [23]; Martelli's fixed-point theorem, multivalued functions, and cosine and sine families [24]; the Mittag-Leffler matrix functions and the Schauder fixed-point theorem [20,25,26]; the Mittag-Leffler matrix function, the Gramian matrix, and the iterative technique [27]; the solution operator theory, fractional calculations, and fixed point techniques [28]; and the delayed fractional Gram matrix and the explicit solution formula [29].
On the other hand, the Hyers-Ulam stability of fractional delay systems has been studied recently by many authors; see, for example, [19,30,31] and the references therein.
However, to the best of our knowledge, no research has been conducted on the controllability of linear fractional-delay systems of the form and the controllability and Hyers-Ulam stability of the corresponding nonlinear fractionaldelay systems of the form where C D α 0 + is called the Caputo fractional derivative of order α ∈ (1, 2] with the lower index zero, h > 0 is a delay, x 1 > (n − 1)h, y(x) ∈ R n , ψ ∈ C([−h, 0], R n ), A ∈ R n×n and B ∈ R n×m are any matrices, f ∈ C(Ω × R n , R n ) is a given function, and u(x) ∈ R m shows control vector.
Elshenhab and Wang [11] have presented a novel formulation of solutions to the linear fractional-delay systems of the following form: where H h,α (Ax α ), M h,α (Ax α ), and S h,α (Ax α ) are known as the delayed Mittag-Lefflertype matrix functions formulated by and respectively, where the notation Θ and I are the n × n null and identity matrix, respectively, Γ is a gamma function, and r = 0, 1, 2, .... Applying Formula (4), the solution of (2) can be represented as Motivated by [11,16], the explicit solutions Formula (8) of (3) combined with the delayed Mittag-Leffler matrix functions are employed as an application to derive controllability results on Ω = [0, The rest of this paper is structured as follows: in Section 2, we present some preliminaries, some basic notation and definitions, and some useful lemmas. In Section 3, we derive sufficient and necessary conditions for the controllability of (1) by introducing a fractional delay Gramian matrix. In Section 4, we establish sufficient conditions of the controllability of (2) by applying Krasnoselskii's fixed-point theorem. In Section 5, we discuss the Hyers-Ulam stability of (2) on the finite-time interval [0, x 1 ]. Finally, we provide numerical examples of linear and nonlinear fractional-delay systems to demonstrate the theoretical results.

Preliminaries
Throughout the paper, we refer to C(Ω, R n ) as the Banach space of a vector-valued continuous function from Ω → R n endowed with the norm y C(Ω) = max x∈Ω y(x) for a norm · on R n , and the matrix norm as A = max y =1 Ay , where A : R n → R n . We define a space C 1 (Ω, R n ) = {y ∈ C(Ω, R n ) : y ∈ C(Ω, R n )}. Let X, Y be two Banach spaces and L b (X, Y) be the space of bounded linear operators from X to Y. Now, L p (Ω, Y) indicates the Banach space of functions f : Ω → Y that are Bochner integrable normed by f L p (Ω,Y) for some 1 < p < ∞. Furthermore, we let ψ C = max s∈[−h,0] ψ(s) and We mention some basic concepts and lemmas utilized throughout this paper.

Lemma 2.
Let α > 0 and ϕ ∈ C(Ω, R n ) be a solution of the inequality (9). Then there exists, for a given constant ε > 0, a solution ϕ * satisfying the inequality Proof. From Remark 1, the solution of the equation can be written as From Lemma 1, we obtain for all x ∈ Ω. This ends the proof.

Lemma 3.
(Krasnoselskii's fixed-point theorem, [34]). Let C be a closed, convex, and non-empty subset of a Banach space X. Suppose that the operators A and B are maps from C into X such that Ax + By ∈ C for every pair x, y ∈ C. If A is compact and continuous, B is a contraction mapping. Then, there exists z ∈ C such that z = Az + Bz.

Controllability of Linear Fractional Delay System
In this section, we establish some sufficient and necessary conditions of controllability of (1) by introducing a fractional delay Gramian matrix defined by It follows from the definition of the matrix W h,α [0, x 1 ] that it is always positive semidefinite for x 1 ≥ 0.
Proof. Sufficiency. Let W h,α [0, x 1 ] be positive definite; then, it will be non-singular and its inverse will be well-defined. As a result, we can derive the associated control input u(x), for any finite terminal conditions y 1 , y 1 ∈ R n , as where From (8), the solution y(x 1 ) of (1) can be formulated as Substituting (11) into (13), we obtain From (10), (12), and (14), we obtain We can see from (3) and (4) that the boundary conditions hold. Thus, (1) is controllable. Necessity. Assume that (1) is controllable. For the sake of a contradiction, suppose that W h,α [0, x 1 ] is not positive definite, and there exists at least a nonzero vector z ∈ R n such that z T W h,α [0, x 1 ]z = 0, which implies that where 0 denotes the n dimensional zero vector. Consider the initial points y 0 = y 0 = 0 and the final point y 1 = z at x = x 1 . Since (1) is controllable, from Definition 3, there exists a control function u 1 (x) that steers the response from 0 to y 1 = z at x = x 1 . Then, Multiplying (16) by z T and using (15), we obtain z T z = 0. This is a contradiction to z = 0. Thus, W h,α [0, x 1 ] is positive definite. This ends the proof.

Remark 2.
We note in the case of α = 2 in (1) that Theorem 1 coincides with the conclusion of Corollary 1 in [16].

Remark 3.
Under condition A, a nonsingular n × n matrix, we note in the case of α = 2, A = A 2 in (1) that Theorem 1 coincides with the conclusion of Theorem 3.1 in [21] and Corollary 2 in [16].

Controllability of Nonlinear Fractional Delay System
In this section, we estabilish sufficient conditions of controllability of (2) using Krasnoselskii's fixed point theorem.
We impose the following assumptions: (G1) The function f : Ω × R n → R n is continuous, and there exists a constant L f ∈ L q (Ω, R + ) and q > 1 such that Suppose that Υ −1 exists and takes values in L 2 (Ω, R m )/ ker Υ, and there exists a constant M 1 > 0 such that Υ −1 ≤ M 1 .
To establish our result, we now employ Krasnoselskii's fixed point theorem.
Theorem 2. Let (G1) and (G2) hold. Then, the nonlinear system (2) is controllable if where Proof. Before we start to prove this theorem, we shall use the following assumptions and estimates: We consider the set . From (G1) and Hölder inequality, we obtain Furthermore, consider the following control function u y : for x ∈ Ω. From (18), (19), (G1), (G2), and Lemma 1, we obtain where We also define the operators L 1 , L 2 on B as follows: Now, we see that B is a closed, bounded, and convex set of C([−h, x 1 ], R n ). Therefore, our proof is divided into three main steps.
Step 1. We prove L 1 y + L 2 z ∈ B for all y, z ∈ B .
For each x ∈ Ω and y, z ∈ B , using (20), we obtain Thus, for some sufficiency large, and from (17), we have L 1 y + L 2 z ∈ B .
Step 3. We prove L 2 : Firstly, we show that L 2 is continuous. Let {y n } be a sequence such that y n → y as n → ∞ in B . Thus, for each x ∈ Ω, using (23) and Lebesgue's dominated convergence theorem, we obtain Next, we prove that L 2 is uniformly bounded on B . For each x ∈ Ω, y ∈ B , we have which implies that L 2 is uniformly bounded on B . It remains to be shown that L 2 is equicontinuous. For each x 2 , x 3 ∈ Ω, 0 < x 2 < x 3 ≤ x 1 and y ∈ B , using (23), we obtain and Now, we can check Ψ i → 0 as x 2 → x 3 , i = 1, 2. For Ψ 1 , we obtain For Ψ 2 , we obtain From (7), we know that the delayed Mittag-Leffler type matrix function S h,α (Ax α ) is uniformly continuous for x ∈ Ω. Thus, Therefore, we have Ψ i → 0 as x 2 → x 3 , i = 1, 2, which implies that, using (24), for all y ∈ B . Thus, the Arzelà-Ascoli theorem tells us that L 2 is compact on B .
Therefore, according to Krasnoselskii's fixed-point theorem (Lemma 3), L 1 + L 2 has a fixed point y on B . In addition, y is also a solution of (2) and (L 1 y + L 2 y)(x 1 ) = y 1 . This means that u y steers the system (2) from y 0 to y 1 in finite time x 1 , which implies that (2) is controllable on Ω. This completes the proof.

Remark 4.
We note in the case of α = 2 in (2) that Theorem 2 coincides with the conclusion of Corollary 3 in [16].

Remark 5.
Under condition A, there is a nonsingular n × n matrix; we note in the case of α = 2 and A = A 2 in (2) that Theorem 2 coincides with the conclusion of Theorem 4.1 in [21] and Corollary 4 in [16].

Hyers-Ulam Stability of Nonlinear Fractional Delay System
In this section, we discuss the Hyers-Ulam stability of (2) on the finite time interval [0, x 1 ]. Proof. With the help of Theorem 2, let ∈ C(Ω, R n ) be a solution of the inequality (9) and y be the unique solution of (2), that is, From Lemma 2, and by a similar way in the proof of Theorem 2 and by virtue of (21), we obtain This completes the proof.

Examples
In this section, we present applications of the results derived.

Conclusions
In this work, we established some sufficient and necessary conditions for the controllability of linear fractional-delay systems by using a fractional delay Gramian matrix and the representation of solutions of these systems with the help of their delayed Mittag-Leffler matrix functions. Furthermore, we established some sufficient conditions for the controllability and Hyers-Ulam stability of nonlinear fractional-delay systems by applying Krasnoselskii's fixed-point theorem and the representation of the solutions of these systems. Finally, the effectiveness of the obtained results was illustrated by numerical examples.
Our future work includes extending and complementing the results of this paper to derive the controllability and Hyers-Ulam stability results of fractional stochastic delay systems with compact analytic semigroups or using the delayed Mittag-Leffler matrix functions with various behaviors such as impulses and delays in multi-states.