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

: Dynamic systems of linear and nonlinear differential equations with pure delay are considered in this study. As an application, the representation of solutions of these systems with the help of their delayed Mittag–Lefﬂer matrix functions is used to obtain the controllability and Hyers–Ulam stability results. By introducing a delay Gramian matrix, we establish some sufﬁcient and necessary conditions for the controllability of linear delay differential systems. In addition, by applying Krasnoselskii’s ﬁxed point theorem, we establish some sufﬁcient conditions of controllability and Hyers–Ulam stability of nonlinear delay differential systems. Our results improve, extend, and complement some existing ones. Finally, two examples are given to illustrate the main results.


Introduction
Numerous processes in mechanical and technological systems were described using fractional delay differential equations. These systems are frequently utilized in the modelling of phenomena in technological and scientific problems. These models have applications in diffusion processes [1], viscoelastic systems [2,3], modeling disease [4], forced oscillations, signal analysis, control theory, biology, computer engineering, finance, and population dynamics; see for instance [5][6][7]. On the other hand, in 2003, Khusainov and Shuklin [8] constructed a novel notion of a delayed exponential matrix function to represent the solutions of linear delay differential equations. In 2008, Khusainov et al. [9] used this method to express the solutions of an oscillating system with pure delay by constructing a delayed matrix sine and a delayed matrix cosine. This pioneering research yielded plenty of novel results on the representation of solutions [10][11][12][13][14], which are applied in the stability analysis [15,16], and control problems [17,18] of time-delay systems. 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 the set of admissible controls, where initial and final states may vary over the entire space. In recent decades, the controllability of differential delay systems has been studied by many authors. There are a few recent studies in the literature on control theory [19][20][21][22][23][24] and Ulam stability [25][26][27][28] for delay differential equations.
However, to the best of our knowledge, no study exists dealing with the controllability of the linear delay differential equations y (x) + Ay(x − h) = Bu(x), x ∈ Ω := [0, and the controllability and Hyers-Ulam stability of the corresponding nonlinear delay differential equations where 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 matrices; u(x) ∈ R m shows the control vector; and f ∈ C(Ω × R n , R n ) is a given function. Very recently, Elshenhab and Wang [11] gave a new representation of solutions of the linear differential equations with pure delay as follows: where H h (A(x)) and M h (A(x)) are called the delayed matrix functions formulated by and respectively, where r = 0, 1, 2, . . . , and the notations I is the n × n identity matrix and Θ is the n × n null matrix. Applying Formula (4), the solution of (2) can be expressed as Motivated by [11,17], as an application, the explicit formula of solutions (7) of (3) and the delayed matrix functions are used to obtain controllability results on Ω = [0, The rest of this paper is arranged as follows: In Section 2, we give some preliminaries, basic notations and fundamental definitions, and some lemmas. Furthermore, we give two very important lemmas, which provide estimations of norms for the delayed matrix functions, which are used while discussing controllability and Hyers-Ulam stability. In Section 3, we give sufficient and necessary conditions of the controllability of (1) by introducing a 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 give two examples to illustrate the main results.

Preliminaries
Throughout the paper, we refer to C(Ω, R n ) as the Banach space of 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 recall some basic notations and fundamental definitions used throughout this paper.

Definition 3 ([27]). The system
and there exists a solution y ∈ C(Ω, R n ) of (2) and a constant M > 0 such that

Remark 1 ([27]).
A function ϕ ∈ C(Ω, R n ) is a solution of the inequality (8) if and only if there exists a function g ∈ C(Ω, R n ) such that Proof. Using (5), we obtain the following This completes the proof.

Lemma 3.
Let ϕ ∈ C(Ω, R n ) be a solution of the inequality (8). Then, ϕ is a solution of the inequality Proof. From Remark 1, the solution of the equation can be written as From Lemma 2, we obtain for all x ∈ Ω. This ends the proof.
Lemma 4 (Krasnoselskii's fixed point theorem, [29]). Let C be a closed, convex, and non-empty subset of a Banach space X. Suppose that the operators A and B be maps from C into X such that Ax + By ∈ C for every pair x, y ∈ C. If A is compact and continuous and B is a contraction mapping, then there exists z ∈ C such that z = Az + Bz.

Controllability of Linear Delay Differential System
In this section, we establish some sufficient and necessary conditions for controllability of (1) by introducing a delay Gramian matrix defined by It follows from the definition of the matrix W M h [0, x 1 ] that it is always positive semidefinite for x ≥ 0.

Theorem 1. The linear system (1) is controllable if and only if W
then, it is non-singular and its inverse is 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 (7), the solution y(x 1 ) of (1) can be formulated as: Substituting (10) into (12), we obtain the following: Using (9) and (11) in (13), we obtain We can see from (3) and (4) Hence, 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 2, there exists a control function u 1 (x) that steers the response from 0 to y 1 = z at x = x 1 . Then, Multiplying (15) by z T and using (14), we obtain z T z = 0. This is a contradiction to z = 0.
Thus, W h [0, x 1 ] is positive definite. This ends the proof. (1). Then, Theorem 1 holds. (1) such that A is a nonsingular n × n matrix. Then, the linear system (1) is controllable if and only if W h [0, and sin h (Ax) and cos h (Ax) are called the delayed matrix of sine and cosine type, respectively, defined in [9].
Proof. From the definition of H h (A(x)) and M h (A(x)) in the case of the matrix A = A 2 , we find that From the conclusion of Theorem 1, we have that W M h [0, x 1 ] is nonsingular. Thus, from (16), we find that W h [0, x 1 ] is also nonsingular. This completes the proof.

Controllability of Nonlinear Delay Differential System
In this section, we establish the 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 Let Suppose that Q −1 exists and takes values in L 2 (Ω, R m )/ ker Q, and there exists a constant M 1 > 0 such that Q −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), and (G2) and Lemmas 1 and 2, we obtain where Furthermore, 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 sufficiently large and from (17), we have L 1 y + L 2 z ∈ B .
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 show 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 Thus, 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 (6), we know that M h (Ax) is uniformly continuous for x ∈ Ω. Hence, 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 4), 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.   (2) such that A is a nonsingular n × n matrix. Then, Theorem 2 coincides with Theorem 4.1 in [17].
Proof. Since M h A 2 (x) = A −1 sin h (Ax). From (G1) and Hölder inequality, we obtain By a similar way in the proof of Theorem 2 at A = A 2 and by virtue of (25) and (26), we obtain the same conclusion in Theorem 4.1 in [17]. This ends the proof.

Remark 2.
We note that Corollary 1 extends Theorems 3.1 and 4.1 in [17] by choosing the matrix A as an arbitrary, not necessarily squared matrix, and Corollaries 2 and 4 coincide with Theorems 3.1 and 4.1 in [17]. Therefore, our results in Corollaries 1-4 extend and improve Theorems 3.1 and 4.1 in [17] by removing the condition that A is a nonsingular matrix.

Hyers-Ulam Stability of Nonlinear Delay Differential 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 z ∈ C(Ω, R n ) be a solution of the inequality (8) and y be the unique solution of (2), that is, From Lemma 3, in a similar way to the proof of Theorem 2, and by virtue of (21), we obtain Therefore, Thus, This completes the proof.

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

Example 1.
Consider the following linear delay differential controlled system: where We note that B ∈ R 2×1 and u(x) ∈ R shows the control vector. Constructing the corresponding delay Gramian matrix of (27) via (9), we obtain Therefore, we see that W M 0.5 [0, 1] is positive definite. Furthermore, for any finite terminal conditions y 1 , y 1 ∈ R 2 such that y(x 1 ) = y 1 = (y 11 , y 12 ) T , y (x 1 ) = y 1 = y 11 , y 12 T ; as a result, we can establish the corresponding control as follows: Hence, the system (27) is controllable on [0, 1] by Theorem 1.

Example 2.
Consider the following nonlinear delay differential controlled system: where for all x ∈ Ω 1 , and y(x), z(x) ∈ R 2 . We set L f (x) = |0.5(x − 0.6)| such that L f ∈ L q (Ω 1 , R + ) in (G1). By choosing p = q = 2, we have Then, we obtain which implies that all the conditions of Theorems 2 and 3 are satisfied. Therefore, the system (28) is controllable and Hyers-Ulam stable.

Remark 3.
It is worth noting that Theorems 3.1 and 4.1 in [17] are not applicable to ascertaining the controllability of the systems (27) and (28) because the square of matrix A is used in [17] rather than A, and the systems (27) and (28) are considered with matrix A rather than A 2 . That is, the term A 2 y(x − h) is replaced by Ay(x − h); then, the definition of sin h Ax and cos h Ax must be modified by using the square root √ A instead of A. However, √ A, in the general case, does not exist as in Example 1 or may not be unique (including the possibility of infinitely many different square roots as in Example 2). Therefore, these two examples demonstrate the effectiveness of the obtained results.

Conclusions
In this work, we established some sufficient and necessary conditions for the controllability of linear delay differential systems by using a delay Gramian matrix and the representation of solutions of these systems with the help of their delayed matrix functions. Furthermore, we established some sufficient conditions of controllability and Hyers-Ulam stability of nonlinear delay differential systems by applying Krasnoselskii's fixed point theorem and the representation of solutions of these systems. Finally, we gave two exam-ples to demonstrate the effectiveness of the obtained results. The results are applicable to all singular, non-singular and arbitrary matrices, not necessarily squared. As a result, our results improve, extend, and complement the existing ones in [17].
One possible direction in which to extend the results of this paper is toward fractional differential and conformable fractional differential systems of order α ∈ (1, 2]. Another challenge is to find out if similar results can be derived in the case of variable delays in (1) and (2).
Author Contributions: All authors contributed equally in this research paper. All authors have read and agreed to the published version of the manuscript.