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

Abstract

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.

1. 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

$$\begin{array}{c}{y}^{″}\left(x\right)+\mathbb{A}y\left(x-h\right)=\mathbb{B}u\left(x\right),x\in {\Omega }:=\left[0,x_1\right],\\ y\left(x\right)\equiv \psi \left(x\right),y^{\prime }\left(x\right)\equiv {\psi }^{\prime }\left(x\right),-h\le x\le 0,\end{array}$$

(1)

and the controllability and Hyers–Ulam stability of the corresponding nonlinear delay differential equations

$$\begin{array}{c}{y}^{″}\left(x\right)+\mathbb{A}y\left(x-h\right)=f\left(x,y\left(x\right)\right)+\mathbb{B}u\left(x\right),x\in {\Omega },\\ y\left(x\right)\equiv \psi \left(x\right),y^{\prime }\left(x\right)\equiv {\psi }^{\prime }\left(x\right),-h\le x\le 0,\end{array}$$

(2)

where $h>0$ is a delay; $x_1>\left(n-1\right)h$, $y\left(x\right)\in {\mathbb{R}}^n$, $\psi \in C\left(\left[-h,0\right],{\mathbb{R}}^n\right)$, $\mathbb{A}\in {\mathbb{R}}^{n\times n}$, and $\mathbb{B}\in {\mathbb{R}}^{n\times m}$ are matrices; $u\left(x\right)\in {\mathbb{R}}^m$ shows the control vector; and $f\in C\left({\Omega }\times {\mathbb{R}}^n,{\mathbb{R}}^n\right)$ is a given function.

Very recently, Elshenhab and Wang [11] gave a new representation of solutions of the linear differential equations with pure delay

$$\begin{array}{c}{y}^{″}\left(x\right)+\mathbb{A}y\left(x-h\right)=f\left(x\right),x\ge 0,\\ y\left(x\right)\equiv \psi \left(x\right),y^{\prime }\left(x\right)\equiv {\psi }^{\prime }\left(x\right),-h\le x\le 0,\end{array}$$

(3)

as follows:

$$\begin{array}{cc}\hfill y\left(x\right)& ={\mathcal{H}}_h\left(\mathbb{A}\left(x-h\right)\right)\psi \left(0\right)+{\mathcal{M}}_h\left(\mathbb{A}\left(x-h\right)\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)f\left(\vartheta \right)d\vartheta ,\hfill \end{array}$$

(4)

where ${\mathcal{H}}_h\left(\mathbb{A}\left(x\right)\right)$ and ${\mathcal{M}}_h\left(\mathbb{A}\left(x\right)\right)$ are called the delayed matrix functions formulated by

$${\mathcal{H}}_h\left(\mathbb{A}\left(x\right)\right):=\left\{\begin{array}{c}{\Theta },-\infty <x<-h,\\ \mathbb{I},-h\le x<0,\\ \mathbb{I}-\mathbb{A}\frac{x^2}{2!},0\le x<h,\\ ⋮⋮\\ \mathbb{I}-\mathbb{A}\frac{x^2}{2!}+{\mathbb{A}}^2\frac{{\left(x-h\right)}^4}{4!}\\ +\cdots +{(-1)}^r{\mathbb{A}}^r\frac{{\left(x-\left(r-1\right)h\right)}^{2r}}{\left(2r\right)!},\left(r-1\right)h\le x<rh,\end{array}\right.$$

(5)

and

$${\mathcal{M}}_h\left(\mathbb{A}\left(x\right)\right):=\left\{\begin{array}{c}{\Theta },-\infty <x<-h,\\ \mathbb{I}\left(x+h\right),-h\le x<0,\\ \mathbb{I}\left(x+h\right)-\mathbb{A}\frac{x^3}{3!},0\le x<h,\\ ⋮⋮\\ \mathbb{I}\left(x+h\right)-\mathbb{A}\frac{x^3}{3!}+{\mathbb{A}}^2\frac{{\left(x-h\right)}^5}{5!}\\ +\cdots +{(-1)}^r{\mathbb{A}}^r\frac{{\left(x-\left(r-1\right)h\right)}^{2r+1}}{\left(2r+1\right)!},\left(r-1\right)h\le x<rh,\end{array}\right.$$

(6)

respectively, where $r=0,1,2,\dots$, and the notations $\mathbb{I}$ is the $n\times n$ identity matrix and ${\Theta }$ is the $n\times n$ null matrix.

Applying Formula (4), the solution of (2) can be expressed as

$$\begin{array}{cc}\hfill y\left(x\right)& ={\mathcal{H}}_h\left(\mathbb{A}\left(x-h\right)\right)\psi \left(0\right)+{\mathcal{M}}_h\left(\mathbb{A}\left(x-h\right)\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)f\left(\vartheta ,y\left(\vartheta \right)\right)d\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)\mathbb{B}u\left(\vartheta \right)d\vartheta ,\hfill \end{array}$$

(7)

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 ${\Omega }=\left[0,x_1\right]$.

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 $\left[0,x_1\right]$. Finally, we give two examples to illustrate the main results.

2. Preliminaries

Throughout the paper, we refer to $C\left({\Omega },{\mathbb{R}}^n\right)$ as the Banach space of vector-valued continuous function from ${\Omega }\to {\mathbb{R}}^n$ endowed with the norm ${∥y∥}_{C\left({\Omega }\right)}={max}_{x\in {\Omega }}∥y\left(x\right)∥$ for a norm $∥·∥$ on ${\mathbb{R}}^n$, and the matrix norm as $∥\mathbb{A}∥={max}_{∥y∥=1}∥Ay∥$, where $\mathbb{A}:{\mathbb{R}}^n\to {\mathbb{R}}^n$. We define a space $C^1\left({\Omega },{\mathbb{R}}^n\right)=\left\{y\in C\left({\Omega },{\mathbb{R}}^n\right):y^{\prime }\in C\left({\Omega },{\mathbb{R}}^n\right)\right\}$. Let X, Y be two Banach spaces and $L_b\left(X,Y\right)$ be the space of bounded linear operators from X to Y. Now, $L^p\left({\Omega },Y\right)$ indicates the Banach space of functions $f:{\Omega }\to Y$ that are Bochner integrable normed by ${∥f∥}_{L^p\left({\Omega },Y\right)}$ for some $1<p<\infty$. Furthermore, we let ${∥\psi ∥}_C={max}_{s\in \left[-h,0\right]}∥\psi \left(s\right)∥$ and ${∥{\psi }^{\prime }∥}_C={max}_{s\in \left[-h,0\right]}∥{\psi }^{\prime }\left(s\right)∥$.

We recall some basic notations and fundamental definitions used throughout this paper.

Definition 1

([6]). The Mittag–Leffler function with two parameters is given by

$${\mathbb{E}}_{\alpha ,\gamma }\left(z\right)=\sum _{r=0}^{\infty }\frac{z^r}{{\Gamma }\left(\alpha r+\gamma \right)},\alpha ,\gamma >0,z\in \mathbb{C},$$

where Γ is a gamma function. Especially, if $\gamma =1$, then

$${\mathbb{E}}_{\alpha ,1}\left(z\right)={\mathbb{E}}_{\alpha }\left(z\right)=\sum _{r=0}^{\infty }\frac{z^r}{{\Gamma }\left(\alpha r+1\right)},\alpha >0.$$

Definition 2

([28]). The systems (1) or (2) are controllable on ${\Omega }=\left[0,x_1\right]$ if there exists a control function $u\in L^2\left({\Omega },{\mathbb{R}}^m\right)$ such that (1) or (2) have a solution $y:\left[-h,x_1\right]\to {\mathbb{R}}^n$ with $y\left(0\right)=y_0$, $y^{\prime }\left(0\right)=y_0^{\prime }$ satisfies $y\left(x_1\right)=y_1$ for all $y_0$, $y_0^{\prime }$, $y_1\in {\mathbb{R}}^n$.

Definition 3

([27]). The system (2) is Hyers—Ulam stable on $[0,x_1]$ if there exists, for a given constant $\epsilon >0$, a function $\phi \in C\left({\Omega },{\mathbb{R}}^n\right)$ satisfying the inequality

$$∥{\phi }^{″}\left(x\right)+\mathbb{A}\phi \left(x-h\right)-f\left(x,\phi \left(x\right)\right)-\mathbb{B}u\left(x\right)∥\le \epsilon ,x\in \left[0,x_1\right],$$

(8)

and there exists a solution $y\in C\left({\Omega },{\mathbb{R}}^n\right)$ of (2) and a constant $M>0$ such that

$$∥\phi \left(x\right)-y\left(x\right)∥\le M\epsilon ,forallx\in \left[0,x_1\right].$$

Remark 1

([27]). A function $\phi \in C\left({\Omega },{\mathbb{R}}^n\right)$ is a solution of the inequality (8) if and only if there exists a function $g\in C\left({\Omega },{\mathbb{R}}^n\right)$ such that

(i) 

$∥g\left(x\right)∥\le \epsilon$, $x\in {\Omega }$.

(ii) 

${\phi }^{″}\left(x\right)=-\mathbb{A}\phi \left(x-h\right)+f\left(x,\phi \left(x\right)\right)+\mathbb{B}u\left(x\right)+g\left(x\right)$, $x\in {\Omega }$.

Lemma 1.

For any $x\in \left[\left(m-1\right)h,mh\right]$, $m=1,2,\dots$, we have

$$∥{\mathcal{H}}_h\left(\mathbb{A}\left(x\right)\right)∥\le {\mathbb{E}}_2\left(∥\mathbb{A}∥x^2\right).$$

Proof. 

Using (5), we obtain the following

$$\begin{array}{cc}\hfill ∥{\mathcal{H}}_h\left(\mathbb{A}\left(x\right)\right)∥& \le 1+∥\mathbb{A}∥\frac{x^2}{2!}+{∥\mathbb{A}∥}^2\frac{{\left(x-h\right)}^4}{4!}\hfill \\ \hfill & +\cdots +{∥\mathbb{A}∥}^m\frac{{\left(x-\left(m-1\right)h\right)}^{2m}}{\left(2m\right)!}\hfill \\ \hfill & \le 1+∥\mathbb{A}∥\frac{x^2}{2!}+{∥\mathbb{A}∥}^2\frac{x^4}{4!}+\cdots +{∥\mathbb{A}∥}^m\frac{x^{2m}}{\left(2m\right)!}\hfill \\ \hfill & \le \sum _{k=0}^{\infty }\frac{{\left(∥\mathbb{A}∥x^2\right)}^k}{\left(2k\right)!}={\mathbb{E}}_2\left(∥\mathbb{A}∥x^2\right).\hfill \end{array}$$

This completes the proof. □

Lemma 2.

For any $x\in \left[\left(m-1\right)h,mh\right]$, $m=1,2,\dots$, we have

$$∥{\mathcal{M}}_h\left(\mathbb{A}\left(x\right)\right)∥\le \left(x+h\right){\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥{\left(x+h\right)}^2\right).$$

Proof. 

Using (6), we obtain the following

$$\begin{array}{cc}\hfill ∥{\mathcal{M}}_h\left(\mathbb{A}\left(x\right)\right)∥& \le \left(x+h\right)+∥\mathbb{A}∥\frac{x^3}{3!}+{∥\mathbb{A}∥}^2\frac{{\left(x-h\right)}^5}{5!}\hfill \\ \hfill & +\cdots +{∥\mathbb{A}∥}^m\frac{{\left(x-\left(m-1\right)h\right)}^{2m+1}}{\left(2m+1\right)!}\hfill \\ \hfill & \le \left(x+h\right)+∥\mathbb{A}∥\frac{{\left(x+h\right)}^3}{3!}+{∥\mathbb{A}∥}^2\frac{{\left(x+h\right)}^5}{5!}\hfill \\ \hfill & +\cdots +{∥\mathbb{A}∥}^m\frac{{\left(x+h\right)}^{2m+1}}{\left(2m+1\right)!}\hfill \\ \hfill & \le \sum _{k=0}^{\infty }\frac{{\left[∥\mathbb{A}∥{\left(x+h\right)}^2\right]}^k\left(x+h\right)}{\left(2k+1\right)!}=\left(x+h\right){\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥{\left(x+h\right)}^2\right).\hfill \end{array}$$

This completes the proof. □

Lemma 3.

Let $\phi \in C\left({\Omega },{\mathbb{R}}^n\right)$ be a solution of the inequality (8). Then, φ is a solution of the inequality

$$∥\phi \left(x\right)-{\phi }^{*}\left(x\right)∥\le \frac{x^2\epsilon }{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right),$$

where

$$\begin{array}{cc}\hfill {\phi }^{*}\left(x\right)& ={\mathcal{H}}_h\left(\mathbb{A}\left(x-h\right)\right)\psi \left(0\right)+{\mathcal{M}}_h\left(\mathbb{A}\left(x-h\right)\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)f\left(\vartheta ,\phi \left(\vartheta \right)\right)d\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)B{u}_{\phi }\left(\vartheta \right)d\vartheta .\hfill \end{array}$$

Proof. 

From Remark 1, the solution of the equation

$${\phi }^{″}\left(x\right)=-\mathbb{A}\phi \left(x-h\right)+f\left(x,\phi \left(x\right)\right)+\mathbb{B}u\left(x\right)+g\left(x\right),x\in {\Omega },$$

can be written as

$$\begin{array}{cc}\hfill \phi \left(x\right)& ={\mathcal{H}}_h\left(\mathbb{A}\left(x-h\right)\right)\psi \left(0\right)+{\mathcal{M}}_h\left(\mathbb{A}\left(x-h\right)\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)f\left(\vartheta ,\phi \left(\vartheta \right)\right)d\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)\mathbb{B}{u}_{\phi }\left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)g\left(\vartheta \right)d\vartheta .\hfill \end{array}$$

From Lemma 2, we obtain

$$\begin{array}{cc}\hfill ∥\phi \left(x\right)-{\phi }^{*}\left(x\right)∥& \le {\int }_0^x∥{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)∥∥g\left(\vartheta \right)∥d\vartheta \hfill \\ \hfill & \le \epsilon {\int }_0^x\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥{\left(x-\vartheta \right)}^2\right)d\vartheta \hfill \\ \hfill & \le \frac{x^2\epsilon }{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right),\hfill \end{array}$$

for all $x\in {\Omega }$. 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 $Ax+By\in C$ for every pair x, $y\in C$. If A is compact and continuous and B is a contraction mapping, then there exists $z\in C$ such that $z=Az+Bz$.

3. 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

$$W_h^{\mathcal{M}}\left[0,x_1\right]={\int }_0^{x_1}{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}{\mathbb{B}}^T{\mathcal{M}}_h\left({\mathbb{A}}^T\left(x_1-h-\vartheta \right)\right)d\vartheta .$$

(9)

It follows from the definition of the matrix $W_h^{\mathcal{M}}\left[0,x_1\right]$ that it is always positive semidefinite for $x\ge 0$.

Theorem 1.

The linear system (1) is controllable if and only if $W_h^{\mathcal{M}}\left[0,x_1\right]$ is positive definite.

Proof. 

Sufficiency. Let $W_h^{\mathcal{M}}\left[0,x_1\right]$ be positive definite; then, it is non-singular and its inverse is well-defined. As a result, we can derive the associated control input $u\left(x\right)$, for any finite terminal conditions $y_1$, $y_1^{\prime }\in {\mathbb{R}}^n$, as

$$u\left(x\right)={\mathbb{B}}^T{\mathcal{M}}_h\left({\mathbb{A}}^T\left(x_1-h-x\right)\right){\left(W_h^{\mathcal{M}}\right)}^{-1}\left[0,x_1\right]\beta ,$$

(10)

where

$$\begin{array}{cc}\hfill \beta & =y_1-{\mathcal{H}}_h\left(\mathbb{A}\left(x_1-h\right)\right)\psi \left(0\right)-{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h\right)\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & +\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta .\hfill \end{array}$$

(11)

From (7), the solution $y\left(x_1\right)$ of (1) can be formulated as:

$$\begin{array}{cc}\hfill y\left(x_1\right)& ={\mathcal{H}}_h\left(\mathbb{A}\left(x_1-h\right)\right)\psi \left(0\right)+{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h\right)\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}u\left(\vartheta \right)d\vartheta .\hfill \end{array}$$

(12)

Substituting (10) into (12), we obtain the following:

$$\begin{array}{cc}\hfill & y\left(x_1\right)\hfill \\ \hfill & ={\mathcal{H}}_h\left(\mathbb{A}\left(x_1-h\right)\right)\psi \left(0\right)+{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h\right)\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}{\mathbb{B}}^T{\mathcal{M}}_h\left({\mathbb{A}}^T\left(x_1-h-\vartheta \right)\right)d\vartheta {\left(W_h^{\mathcal{M}}\right)}^{-1}\left[0,x_1\right]\beta .\hfill \end{array}$$

(13)

Using (9) and (11) in (13), we obtain

$$\begin{array}{cc}\hfill y\left(x_1\right)& ={\mathcal{H}}_h\left(\mathbb{A}\left(x_1-h\right)\right)\psi \left(0\right)+{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h\right)\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta +\beta \hfill \\ \hfill & =y_1.\hfill \end{array}$$

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 $W_h^{\mathcal{M}}\left[0,x_1\right]$ is not positive definite; there exists at least a nonzero vector $z\in {\mathbb{R}}^n$ such that $z^T{W}_h^{\mathcal{M}}\left[0,x_1\right]z=0$, which implies that

$$\begin{array}{cc}\hfill 0& =z^T{W}_h^{\mathcal{M}}\left[0,x_1\right]z\hfill \\ \hfill & ={\int }_0^{x_1}{z}^T{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}{\mathbb{B}}^T{\mathcal{M}}_h\left({\mathbb{A}}^T\left(x_1-h-\vartheta \right)\right)zd\vartheta \hfill \\ \hfill & ={\int }_0^{x_1}\left[z^T{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}\right]{\left[z^T{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}\right]}^{T}d\vartheta \hfill \\ \hfill & ={\int }_0^{x_1}∥z^T{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}∥d\vartheta .\hfill \end{array}$$

Hence,

$$z^T{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}=\left(0,\dots ,0\right):={\mathbf{0}}^T,\mathrm{for}\mathrm{all}\vartheta \in {\Omega },$$

(14)

where $\mathbf{0}$ denotes the n dimensional zero vector. Consider the initial points $y_0=y_0^{\prime }=\mathbf{0}$ and the final point $y_1=z$ at $x=x_1$. Since (1) is controllable, from Definition 2, there exists a control function $u_1\left(x\right)$ that steers the response from $\mathbf{0}$ to $y_1=z$ at $x=x_1$. Then,

$$\begin{array}{cc}\hfill y_1& =z=-\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}{u}_1\left(\vartheta \right)d\vartheta .\hfill \end{array}$$

(15)

Multiplying (15) by $z^T$ and using (14), we obtain $z^{T}z=0$. This is a contradiction to $z\ne \mathbf{0}$. Thus, $W_h\left[0,x_1\right]$ is positive definite. This ends the proof. □

Corollary 1.

Let $\mathbb{A}={\mathbb{A}}^2$ in (1). Then, Theorem 1 holds.

Corollary 2.

Let $\mathbb{A}={\mathbb{A}}^2$ in (1) such that $\mathbb{A}$ is a nonsingular $n\times n$ matrix. Then, the linear system (1) is controllable if and only if $W_h\left[0,x_1\right]$ is nonsingular, where $W_h\left[0,x_1\right]$ is defined as

$$W_h\left[0,x_1\right]={\mathbb{A}}^{-1}{\int }_0^{x_1}{sin}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)B{B}^T{sin}_h\left({\mathbb{A}}^T\left(x_1-h-\vartheta \right)\right),$$

and ${sin}_h\left(Ax\right)$ and ${cos}_h\left(Ax\right)$ are called the delayed matrix of sine and cosine type, respectively, defined in [9].

Proof. 

From the definition of ${\mathcal{H}}_h\left(\mathbb{A}\left(x\right)\right)$ and ${\mathcal{M}}_h\left(\mathbb{A}\left(x\right)\right)$ in the case of the matrix $\mathbb{A}={\mathbb{A}}^2$, we find that

$${\mathcal{H}}_h\left({\mathbb{A}}^2\left(x\right)\right)={cos}_h\left(\mathbb{A}\left(x\right)\right),{\mathcal{M}}_h\left({\mathbb{A}}^2\left(x\right)\right)={\mathbb{A}}^{-1}{sin}_h\left(\mathbb{A}\left(x\right)\right),$$

which implies that

$$\begin{array}{cc}\hfill W_h^{\mathcal{M}}\left[0,x_1\right]& ={\int }_0^{x_1}{\mathcal{M}}_h\left({\mathbb{A}}^2\left(x_1-h-\vartheta \right)\right)\mathbb{B}{\mathbb{B}}^T{\mathcal{M}}_h\left({\left({\mathbb{A}}^2\right)}^T\left(x_1-h-\vartheta \right)\right)d\vartheta \hfill \\ \hfill & ={\int }_0^{x_1}{\mathbb{A}}^{-1}{sin}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}{\mathbb{B}}^T{sin}_h\left({\mathbb{A}}^T\left(x_1-h-\vartheta \right)\right){\left({\mathbb{A}}^{-1}\right)}^{T}d\vartheta \hfill \\ \hfill & ={\mathbb{A}}^{-1}{\int }_0^{x_1}{sin}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}{\mathbb{B}}^T{sin}_h\left({\mathbb{A}}^T\left(x_1-h-\vartheta \right)\right)d\vartheta {\left({\mathbb{A}}^T\right)}^{-1}\hfill \\ \hfill & =W_h\left[0,x_1\right]{\left({\mathbb{A}}^T\right)}^{-1}.\hfill \end{array}$$

Hence,

$$W_h\left[0,x_1\right]=W_h^{\mathcal{M}}\left[0,x_1\right]{\mathbb{A}}^T.$$

(16)

From the conclusion of Theorem 1, we have that $W_h^{\mathcal{M}}\left[0,x_1\right]$ is nonsingular. Thus, from (16), we find that $W_h\left[0,x_1\right]$ is also nonsingular. This completes the proof. □

4. 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 $f:{\Omega }\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuous, and there exists a constant $L_f\in L^q\left({\Omega },{\mathbb{R}}^{+}\right)$ and $q>1$ such that

$$∥f\left(x,y_1\right)-f\left(x,y_2\right)∥\le L_f\left(x\right)∥y_1-y_2∥,\mathrm{for}\mathrm{all}x\in {\Omega },y_1,y_2\in {\mathbb{R}}^n.$$

Let ${sup}_{x\in {\Omega }}f\left(x,0\right)=M_f<\infty$.

(G2) 

The linear operator $Q:L^2\left({\Omega },{\mathbb{R}}^m\right)\to {\mathbb{R}}^n$ is defined by

$$Q={\int }_0^{x_1}{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)\mathbb{B}u\left(\vartheta \right)d\vartheta .$$

Suppose that $Q^{-1}$ exists and takes values in $L^2\left({\Omega },{\mathbb{R}}^m\right)/kerQ$, and there exists a constant $M_1>0$ such that $∥Q^{-1}∥\le M_1$.

To establish our result, we now employ Krasnoselskii’s fixed point theorem.

Theorem 2.

Let $\left(\mathbf{G}\mathbf{1}\right)$ and $\left(\mathbf{G}\mathbf{2}\right)$ hold. Then, the nonlinear system (2) is controllable if

$$M_2\left[1+\frac{M_1{x}_1^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)∥\mathbb{B}∥\right]<1,$$

(17)

where

$$M_2=\frac{x_1^{1+\frac{1}{p}}}{{\left(p+1\right)}^{\frac{1}{p}}}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right){∥L_f∥}_{L^q\left({\Omega },{\mathbb{R}}^{+}\right)}and\frac{1}{p}+\frac{1}{q}=1,p,q>1.$$

Proof. 

Before we start to prove this theorem, we shall use the following assumptions and estimates: we consider the set

$$B_{ϵ}=\left\{y\in C\left(\left[-h,x_1\right],{\mathbb{R}}^n\right):{∥y∥}_{C\left[-h,x_1\right]}=\underset{x\in \left[-h,x_1\right]}{sup}∥y\left(x\right)∥\le ϵ\right\}.$$

Let $x\in \left[0,x_1\right]$. From $\left(\mathbf{G}\mathbf{1}\right)$ and Hölder inequality, we obtain

$$\begin{array}{cc}\hfill & {\int }_0^x\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥{\left(x-\vartheta \right)}^2\right)L_f\left(\vartheta \right)d\vartheta \hfill \\ \hfill & \le {\left({\int }_0^x{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥{\left(x-\vartheta \right)}^2\right)\right)}^{p}d\vartheta \right)}^{\frac{1}{p}}{\left({\int }_0^x{L}_f^q\left(\vartheta \right)d\vartheta \right)}^{\frac{1}{q}}\hfill \\ \hfill & \le {\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right){\left({\int }_0^x{\left(x-\vartheta \right)}^{p}d\vartheta \right)}^{\frac{1}{p}}{\left({\int }_0^x{L}_f^q\left(\vartheta \right)d\vartheta \right)}^{\frac{1}{q}}\hfill \\ \hfill & =\frac{x^{2-\frac{1}{q}}}{{\left(p+1\right)}^{\frac{1}{p}}}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right){∥L_f∥}_{L^q\left({\Omega },{\mathbb{R}}^{+}\right)}.\hfill \end{array}$$

(18)

Furthermore, consider the following control function $u_y$:

$$\begin{array}{cc}\hfill u_y\left(x\right)& =Q^{-1}\left[y_1-{\mathcal{H}}_h\left(\mathbb{A}\left(x_1-h\right)\right)\psi \left(0\right)-{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h\right)\right){\psi }^{\prime }\left(0\right)\right.\hfill \\ \hfill & +\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta \hfill \\ \hfill & -{\int }_0^{x_1}\left.{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)f\left(\vartheta ,y\left(\vartheta \right)\right)d\vartheta \right]\left(x\right),\hfill \end{array}$$

(19)

for $x\in {\Omega }$. From (18), (19), $\left(\mathbf{G}\mathbf{1}\right)$, and $\left(\mathbf{G}\mathbf{2}\right)$ and Lemmas 1 and 2, we obtain

$$\begin{array}{cc}\hfill ∥u_y\left(x\right)∥& \le ∥Q^{-1}∥\left(∥y_1∥+∥{\mathcal{H}}_h\left(\mathbb{A}\left(x_1-h\right)\right)∥∥\psi \left(0\right)∥\right.\hfill \\ \hfill & +∥{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h\right)\right)∥∥{\psi }^{\prime }\left(0\right)∥\hfill \\ \hfill & +∥\mathbb{A}∥{\int }_{-h}^0∥{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-2h-\vartheta \right)\right)∥∥\psi \left(\vartheta \right)∥d\vartheta \hfill \\ \hfill & +{\int }_0^{x_1}∥{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)∥∥f\left(\vartheta ,y\left(\vartheta \right)\right)∥d\vartheta )\hfill \\ \hfill & \le M_1∥y_1∥+M_1{\mathbb{E}}_2\left(∥\mathbb{A}∥{\left(x_1-h\right)}^2\right){∥\psi ∥}_C\hfill \\ \hfill & +M_1{x}_1{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right){∥{\psi }^{\prime }∥}_C\hfill \\ \hfill & +M_1∥\mathbb{A}∥{∥\psi ∥}_C{\int }_{-h}^0\left(x_1-h-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥{\left(x_1-h-\vartheta \right)}^2\right)d\vartheta \hfill \\ \hfill & +M_1{\int }_0^{x_1}\left(x_1-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥{\left(x_1-\vartheta \right)}^2\right)L_f\left(\vartheta \right)∥y\left(\vartheta \right)∥d\vartheta \hfill \\ \hfill & +M_1{\int }_0^{x_1}\left(x_1-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥{\left(x_1-\vartheta \right)}^2\right)∥f\left(\vartheta ,0\right)∥d\vartheta \hfill \\ \hfill & \le M_1∥y_1∥+M_1{\mathbb{E}}_2\left(∥\mathbb{A}∥{\left(x_1-h\right)}^2\right){∥\psi ∥}_C\hfill \\ \hfill & +M_1{x}_1{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right){∥{\psi }^{\prime }∥}_C+\frac{M_1∥\mathbb{A}∥{∥\psi ∥}_C{x}_1^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)\hfill \\ \hfill & +\frac{M_1{x}^{2-\frac{1}{q}}}{{\left(p+1\right)}^{\frac{1}{p}}}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right){∥L_f∥}_{L^q\left({\Omega },{\mathbb{R}}^{+}\right)}{∥y∥}_{C\left({\Omega }\right)}\hfill \\ \hfill & +\frac{M_1{M}_f{x}_1^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)\hfill \\ \hfill & \le M_1∥y_1∥+M_1{M}_2ϵ+M_1\theta \left(x_1\right),\hfill \end{array}$$

(20)

where

$$\begin{array}{cc}\hfill \theta \left(x\right)& ={\mathbb{E}}_2\left(∥\mathbb{A}∥{\left(x-h\right)}^2\right){∥\psi ∥}_C+x{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right){∥{\psi }^{\prime }∥}_C\hfill \\ \hfill & +\frac{x^2\left(∥\mathbb{A}∥{∥\psi ∥}_C+M_f\right)}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right).\hfill \end{array}$$

Furthermore,

$$\begin{array}{cc}\hfill & ∥u_y\left(x\right)-u_z\left(x\right)∥\hfill \\ \hfill & \le M_1{\int }_0^{x_1}∥{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)∥∥f\left(\vartheta ,y\left(\vartheta \right)\right)-f\left(\vartheta ,z\left(\vartheta \right)\right)∥d\vartheta \hfill \\ \hfill & \le M_1{\int }_0^{x_1}∥{\mathcal{M}}_h\left(\mathbb{A}\left(x_1-h-\vartheta \right)\right)∥L_f\left(\vartheta \right)∥y\left(\vartheta \right)-z\left(\vartheta \right)∥d\vartheta \hfill \\ \hfill & \le M_1{M}_2{∥y-z∥}_{C\left({\Omega }\right)}.\hfill \end{array}$$

(21)

We also define the operators ${\mathcal{L}}_1$, ${\mathcal{L}}_2$ on ${\mathbb{B}}_{ϵ}$ as follows:

$$\begin{array}{cc}\hfill \left({\mathcal{L}}_{1}y\right)\left(x\right)& ={\mathcal{H}}_h\left(\mathbb{A}\left(x-h\right)\right)\psi \left(0\right)+{\mathcal{M}}_h\left(\mathbb{A}\left(x-h\right)\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)\mathbb{B}{u}_y\left(\vartheta \right)d\vartheta ,\hfill \end{array}$$

(22)

$$\left({\mathcal{L}}_{2}y\right)\left(x\right)={\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)f\left(\vartheta ,y\left(\vartheta \right)\right)d\vartheta .$$

(23)

Now, we see that $B_{ϵ}$ is a closed, bounded, and convex set of $C\left(\left[-h,x_1\right],{\mathbb{R}}^n\right)$. Therefore, our proof is divided into three main steps.

Step 1. We prove ${\mathcal{L}}_{1}y+{\mathcal{L}}_{2}z\in B_{ϵ}$ for all y, $z\in B_{ϵ}$.

For each $x\in {\Omega }$ and y, $z\in B_{ϵ}$, using (20), we obtain

$$\begin{array}{cc}\hfill & {∥{\mathcal{L}}_{1}y+{\mathcal{L}}_{2}z∥}_{C\left[-h,x_1\right]}\hfill \\ \hfill & =\underset{x\in \left[-h,x_1\right]}{sup}∥\left({\mathcal{L}}_{1}y+{\mathcal{L}}_{2}z\right)\left(x\right)∥\hfill \\ \hfill & \le \underset{x\in \left[-h,x_1\right]}{sup}\left\{∥{\mathcal{H}}_h\left(\mathbb{A}\left(x-h\right)\right)∥∥\psi \left(0\right)∥+∥{\mathcal{M}}_h\left(\mathbb{A}\left(x-h\right)\right)∥∥{\psi }^{\prime }\left(0\right)∥\right.\hfill \\ \hfill & +∥\mathbb{A}∥{\int }_{-h}^0∥{\mathcal{M}}_h\left(\mathbb{A}\left(x-2h-\vartheta \right)\right)∥∥\psi \left(\vartheta \right)∥d\vartheta \hfill \\ \hfill & +{\int }_0^x∥{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)∥∥\mathbb{B}∥∥u_y\left(\vartheta \right)∥d\vartheta \hfill \\ \hfill & +{\int }_0^x\left.∥{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)∥∥f\left(\vartheta ,z\left(\vartheta \right)\right)∥d\vartheta \right\}\hfill \\ \hfill & \le {\mathbb{E}}_2\left(∥\mathbb{A}∥{\left(x-h\right)}^2\right){∥\psi ∥}_C+x{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right){∥{\psi }^{\prime }∥}_C\hfill \\ \hfill & +\frac{x^2∥\mathbb{A}∥{∥\psi ∥}_C}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right)+\frac{M_f{x}^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right)\hfill \\ \hfill & +\frac{x^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right)∥\mathbb{B}∥\left(M_1∥y_1∥+M_1{M}_2ϵ+M_1\theta \left(x_1\right)\right)d\vartheta \hfill \\ \hfill & +\frac{x^{2-\frac{1}{q}}}{{\left(p+1\right)}^{\frac{1}{p}}}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right){∥L_f∥}_{L^q\left({\Omega },{\mathbb{R}}^{+}\right)}{∥z∥}_{C\left({\Omega }\right)}\hfill \\ \hfill & \le \theta \left(x_1\right)+M_2ϵ+\frac{M_1{x}^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)∥\mathbb{B}∥∥y_1∥\hfill \\ \hfill & +\frac{M_1{M}_2ϵx^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)∥\mathbb{B}∥+\frac{M_1\theta \left(x_1\right)x^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)∥\mathbb{B}∥\hfill \\ \hfill & \le \theta \left(x_1\right)\left[1+\frac{M_1{x}_1^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)∥\mathbb{B}∥\right]+\frac{M_1{x}_1^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)∥\mathbb{B}∥∥y_1∥\hfill \\ \hfill & +M_2\left[1+\frac{M_1{x}_1^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)∥\mathbb{B}∥\right]ϵ.\hfill \end{array}$$

Thus, for some $ϵ$ sufficiently large and from (17), we have ${\mathcal{L}}_{1}y+{\mathcal{L}}_{2}z\in B_{ϵ}$.

Step 2. We prove that ${\mathcal{L}}_1:B_{ϵ}\to C\left(\left[-h,x_1\right],{\mathbb{R}}^n\right)$ is a contraction.

For each $x\in {\Omega }$ and y, $z\in B_{ϵ}$, using (21), we obtain

$$\begin{array}{cc}\hfill ∥\left({\mathcal{L}}_{1}y\right)\left(x\right)-\left({\mathcal{L}}_{1}z\right)\left(x\right)∥& \le {\int }_0^x∥{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)∥∥\mathbb{B}∥∥u_y\left(\vartheta \right)-u_z\left(\vartheta \right)∥d\vartheta \hfill \\ \hfill & \le ∥\mathbb{B}∥M_1{M}_2{∥y-z∥}_{C\left({\Omega }\right)}{\int }_0^x∥{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)∥d\vartheta \hfill \\ \hfill & \le \frac{x_1^2∥\mathbb{B}∥M_1{M}_2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right){∥y-z∥}_{C\left({\Omega }\right)}\hfill \\ \hfill & \le \mu {∥y-z∥}_{C\left({\Omega }\right)},\hfill \end{array}$$

where $\mu :=\frac{x_1^2∥\mathbb{B}∥M_1{M}_2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)$. From (17), note $\mu <1$, we conclude that ${\mathcal{L}}_1$ is a contraction mapping.

Step 3. We prove ${\mathcal{L}}_2:B_{ϵ}\to C\left(\left[-h,x_1\right],{\mathbb{R}}^n\right)$ is a continuous compact operator.

Firstly, we show that ${\mathcal{L}}_2$ is continuous. Let $\left\{y_n\right\}$ be a sequence such that $y_n\to y$ as $n\to \infty$ in $B_{ϵ}$. Thus, for each $x\in {\Omega }$, using (23) and Lebesgue’s dominated convergence theorem, we obtain

$$\begin{array}{cc}\hfill & ∥\left({\mathcal{L}}_2{y}_n\right)\left(x\right)-\left({\mathcal{L}}_{2}y\right)\left(x\right)∥\hfill \\ \hfill & \le {\int }_0^x∥{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)∥∥f\left(\vartheta ,y_n\left(\vartheta \right)\right)-f\left(\vartheta ,y\left(\vartheta \right)\right)∥d\vartheta \hfill \\ \hfill & \le {\int }_0^x\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥{\left(x-\vartheta \right)}^2\right)L_f\left(\vartheta \right)∥y_n\left(\vartheta \right)-y\left(\vartheta \right)∥d\vartheta \to 0,\mathrm{as}n\to \infty .\hfill \end{array}$$

Hence, ${\mathcal{L}}_2:B_{ϵ}\to C\left(\left[-h,x_1\right],{\mathbb{R}}^n\right)$ is continuous.

Next, we prove that ${\mathcal{L}}_2$ is uniformly bounded on $B_{ϵ}$. For each $x\in {\Omega }$, $y\in B_{ϵ}$, we have

$$\begin{array}{cc}\hfill ∥{\mathcal{L}}_{2}y∥& =\underset{x\in {\Omega }}{sup}∥\left({\mathcal{L}}_{2}y\right)\left(x\right)∥\hfill \\ \hfill & \le \underset{x\in {\Omega }}{sup}\left\{{\int }_0^x∥{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)∥∥f\left(\vartheta ,y\left(\vartheta \right)\right)∥d\vartheta \right\}\hfill \\ \hfill & \le \frac{x^{2-\frac{1}{q}}}{{\left(p+1\right)}^{\frac{1}{p}}}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right){∥L_f∥}_{L^q\left({\Omega },{\mathbb{R}}^{+}\right)}{∥y∥}_{C\left({\Omega }\right)}\hfill \\ \hfill & +\frac{M_f{x}^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right)\hfill \\ \hfill & \le M_2ϵ+\frac{M_f{x}_1^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right),\hfill \end{array}$$

which implies that ${\mathcal{L}}_2$ is uniformly bounded on $B_{ϵ}$.

It remains to show that ${\mathcal{L}}_2$ is equicontinuous. For each $x_2$, $x_3\in {\Omega }$, $0<x_2<x_3\le x_1$ and $y\in B_{ϵ}$, using (23), we obtain

$$\begin{array}{cc}\hfill & \left({\mathcal{L}}_{2}y\right)\left(x_3\right)-\left({\mathcal{L}}_{2}y\right)\left(x_2\right)\hfill \\ \hfill & \le {\int }_0^{x_3}{\mathcal{M}}_h\left(\mathbb{A}\left(x_3-h-\vartheta \right)\right)f\left(\vartheta ,y\left(\vartheta \right)\right)d\vartheta \hfill \\ \hfill & -{\int }_0^{x_2}{\mathcal{M}}_h\left(\mathbb{A}\left(x_2-h-\vartheta \right)\right)f\left(\vartheta ,y\left(\vartheta \right)\right)d\vartheta \hfill \\ \hfill & =K_1+K_2,\hfill \end{array}$$

where

$${{\Psi }}_1={\int }_{x_2}^{x_3}{\mathcal{M}}_h\left(\mathbb{A}\left(x_3-h-\vartheta \right)\right)f\left(\vartheta ,y\left(\vartheta \right)\right)d\vartheta ,$$

and

$${{\Psi }}_2={\int }_0^{x_2}\left[{\mathcal{M}}_h\left(\mathbb{A}\left(x_3-h-\vartheta \right)\right)-{\mathcal{M}}_h\left(\mathbb{A}\left(x_2-h-\vartheta \right)\right)\right]f\left(\vartheta ,y\left(\vartheta \right)\right)d\vartheta .$$

Thus,

$$∥\left({\mathcal{L}}_{2}y\right)\left(x_3\right)-\left({\mathcal{L}}_{2}y\right)\left(x_2\right)∥\le ∥{{\Psi }}_1∥+∥{{\Psi }}_2∥.$$

(24)

Now, we can check $∥{{\Psi }}_i∥\to 0$ as $x_2\to x_3$, $i=1$, 2. For ${{\Psi }}_1$, we obtain

$$\begin{array}{cc}\hfill {{\Psi }}_1& \le {\int }_{x_2}^{x_3}\left(x_3-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥{\left(x_3-\vartheta \right)}^2\right)L_f\left(\vartheta \right)∥y\left(\vartheta \right)∥d\vartheta \hfill \\ \hfill & +{\int }_{x_2}^{x_3}\left(x_3-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥{\left(x_3-\vartheta \right)}^2\right)∥f\left(\vartheta ,0\right)∥d\vartheta \hfill \\ \hfill & \le \frac{{\left(x_3-x_2\right)}^{2-\frac{1}{q}}}{{\left(p+1\right)}^{\frac{1}{p}}}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_3^2\right){∥L_f∥}_{L^q\left({\Omega },{\mathbb{R}}^{+}\right)}{∥y∥}_{C\left({\Omega }\right)}\hfill \\ \hfill & +\frac{M_f{\left(x_3-x_2\right)}^2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_3^2\right)\to 0,\mathrm{as}{x}_2\to x_3.\hfill \end{array}$$

For ${{\Psi }}_2$, we obtain

$$\begin{array}{cc}\hfill ∥{{\Psi }}_2∥& \le ϵ{\int }_0^{x_2}∥{\mathcal{M}}_h\left(\mathbb{A}\left(x_3-h-\vartheta \right)\right)-{\mathcal{M}}_h\left(\mathbb{A}\left(x_2-h-\vartheta \right)\right)∥L_f\left(\vartheta \right)d\vartheta \hfill \\ \hfill & +M_f{\int }_0^{x_2}∥{\mathcal{M}}_h\left(\mathbb{A}\left(x_3-h-\vartheta \right)\right)-{\mathcal{M}}_h\left(\mathbb{A}\left(x_2-h-\vartheta \right)\right)∥d\vartheta .\hfill \end{array}$$

From (6), we know that ${\mathcal{M}}_h\left(Ax\right)$ is uniformly continuous for $x\in {\Omega }$. Hence,

$$∥{\mathcal{M}}_h\left(\mathbb{A}\left(x_3-h-\vartheta \right)\right)-{\mathcal{M}}_h\left(\mathbb{A}\left(x_2-h-\vartheta \right)\right)∥\to 0,\mathrm{as}{x}_2\to x_3.$$

Therefore, we have $∥{{\Psi }}_i∥\to 0$ as $x_2\to x_3$, $i=1$, 2, which implies that, using (24),

$$∥\left({\mathcal{L}}_{2}y\right)\left(x_3\right)-\left({\mathcal{L}}_{2}y\right)\left(x_2\right)∥\to 0,\mathrm{as}{x}_2\to x_3,$$

for all $y\in B_{ϵ}$. Thus, the Arzelà-Ascoli theorem tells us that ${\mathcal{L}}_2$ is compact on $B_{ϵ}$.

Therefore, according to Krasnoselskii’s fixed point theorem (Lemma 4), ${\mathcal{L}}_1+{\mathcal{L}}_2$ has a fixed point y on $B_{ϵ}$. In addition, y is also a solution of (2) and $\left({\mathcal{L}}_{1}y+{\mathcal{L}}_{2}y\right)\left(x_1\right)=y_1$. 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 ${\Omega }$. This completes the proof. □

Corollary 3.

Let $\mathbb{A}={\mathbb{A}}^2$ in (2). Then, Theorem 2 holds.

Corollary 4.

Let $\mathbb{A}={\mathbb{A}}^2$ in (2) such that $\mathbb{A}$ is a nonsingular $n\times n$ matrix. Then, Theorem 2 coincides with Theorem 4.1 in [17].

Proof. 

Since ${\mathcal{M}}_h\left({\mathbb{A}}^2\left(x\right)\right)={\mathbb{A}}^{-1}{sin}_h\left(\mathbb{A}x\right)$. From $\left(\mathbf{G}\mathbf{1}\right)$ and Hölder inequality, we obtain

$$\begin{array}{cc}\hfill & {\int }_0^x∥{\mathcal{M}}_h\left({\mathbb{A}}^2\left(x-h-\vartheta \right)\right)∥L_f\left(\vartheta \right)d\vartheta \hfill \\ \hfill & =∥{\mathbb{A}}^{-1}∥{\int }_0^x∥\mathbb{A}{\mathcal{M}}_h\left({\mathbb{A}}^2\left(x-h-\vartheta \right)\right)∥L_f\left(\vartheta \right)d\vartheta \hfill \\ \hfill & \le ∥{\mathbb{A}}^{-1}∥{\int }_0^x∥{sin}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)∥L_f\left(\vartheta \right)d\vartheta \hfill \\ \hfill & \le ∥{\mathbb{A}}^{-1}∥{\int }_0^x∥sinh\left(∥\mathbb{A}∥\left(x-\vartheta \right)\right)∥L_f\left(\vartheta \right)d\vartheta \hfill \\ \hfill & \le ∥{\mathbb{A}}^{-1}∥{\left({\int }_0^x{\left(sinh\left(∥\mathbb{A}∥\left(x-\vartheta \right)\right)\right)}^{p}d\vartheta \right)}^{\frac{1}{p}}{\left({\int }_0^x{L}_f^q\left(\vartheta \right)d\vartheta \right)}^{\frac{1}{q}}\hfill \\ \hfill & =∥{\mathbb{A}}^{-1}∥{\left({\int }_0^x\frac{exp\left(∥\mathbb{A}∥p\left(x-\vartheta \right)\right)}{2^p}d\vartheta \right)}^{\frac{1}{p}}{\left({\int }_0^x{L}_f^q\left(\vartheta \right)d\vartheta \right)}^{\frac{1}{q}}\hfill \\ \hfill & =∥{\mathbb{A}}^{-1}∥{\left(\frac{1}{2^p∥\mathbb{A}∥p}\left(exp\left(∥\mathbb{A}∥px-1\right)\right)\right)}^{\frac{1}{p}}{∥L_f∥}_{L^q\left({\Omega },{\mathbb{R}}^{+}\right)}.\hfill \end{array}$$

(25)

and

$$\begin{array}{cc}\hfill & {\int }_0^x∥{\mathcal{M}}_h\left({\mathbb{A}}^2\left(x-h-\vartheta \right)\right)∥∥f\left(\vartheta ,0\right)∥d\vartheta \hfill \\ \hfill & =∥{\mathbb{A}}^{-1}∥{\int }_0^x∥{sin}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)∥∥f\left(\vartheta ,0\right)∥d\vartheta \hfill \\ \hfill & \le M_f∥{\mathbb{A}}^{-1}∥{\int }_0^x∥sinh\left(∥\mathbb{A}∥\left(x-\vartheta \right)\right)∥d\vartheta \hfill \\ \hfill & =\frac{M_f∥{\mathbb{A}}^{-1}∥}{∥\mathbb{A}∥}∥cosh\left(∥\mathbb{A}∥x\right)-1∥.\hfill \end{array}$$

(26)

By a similar way in the proof of Theorem 2 at $\mathbb{A}={\mathbb{A}}^2$ 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 $\mathbb{A}$ 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 $\mathbb{A}$ is a nonsingular matrix.

5. Hyers–Ulam Stability of Nonlinear Delay Differential System

In this section, we discuss the Hyers–Ulam stability of (2) on the finite time interval $\left[0,x_1\right]$.

Theorem 3.

Let $\left(\mathbf{G}\mathbf{1}\right)$, $\left(\mathbf{G}\mathbf{2}\right)$ and (17) be satisfied. Then, the system (2) is Hyers–Ulam stable.

Proof. 

With the help of Theorem 2, let $z\in C\left({\Omega },{\mathbb{R}}^n\right)$ be a solution of the inequality (8) and y be the unique solution of (2), that is,

$$\begin{array}{cc}\hfill y\left(x\right)& ={\mathcal{H}}_h\left(\mathbb{A}\left(x-h\right)\right)\psi \left(0\right)+{\mathcal{M}}_h\left(\mathbb{A}\left(x-h\right)\right){\psi }^{\prime }\left(0\right)\hfill \\ \hfill & -\mathbb{A}{\int }_{-h}^0{\mathcal{M}}_h\left(\mathbb{A}\left(x-2h-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)f\left(\vartheta ,y\left(\vartheta \right)\right)d\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)\mathbb{B}{u}_y\left(\vartheta \right)d\vartheta .\hfill \end{array}$$

From Lemma 3, in a similar way to the proof of Theorem 2, and by virtue of (21), we obtain

$$\begin{array}{cc}\hfill ∥z\left(x\right)-y\left(x\right)∥& \le ∥z\left(x\right)-z^{*}\left(x\right)∥+∥z^{*}\left(x\right)-y\left(x\right)∥\hfill \\ \hfill & \le \frac{x^2\epsilon }{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x^2\right)\hfill \\ \hfill & +{\int }_0^x∥{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)∥∥\mathbb{B}∥∥u_z\left(\vartheta \right)-u_y\left(\vartheta \right)∥d\vartheta \hfill \\ \hfill & +{\int }_0^x∥{\mathcal{M}}_h\left(\mathbb{A}\left(x-h-\vartheta \right)\right)∥∥f\left(\vartheta ,z\left(\vartheta \right)\right)-f\left(\vartheta ,y\left(\vartheta \right)\right)∥d\vartheta \hfill \\ \hfill & \le \frac{x_1^2\epsilon }{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)\hfill \\ \hfill & +\frac{x_1^2∥\mathbb{B}∥M_1{M}_2}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right){∥z-y∥}_{C\left({\Omega }\right)}\hfill \\ \hfill & +M_2{∥z-y∥}_{C\left({\Omega }\right)}\hfill \\ \hfill & =\frac{x_1^2\epsilon }{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)\hfill \\ \hfill & +M_2\left(1+\frac{x_1^2∥\mathbb{B}∥M_1}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)\right){∥z-y∥}_{C\left({\Omega }\right)}.\hfill \end{array}$$

Therefore,

$${∥z-y∥}_{C\left({\Omega }\right)}\le \frac{x_1^2\epsilon }{2\left(1-\rho \right)}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right),$$

where

$$\rho :=M_2\left(1+\frac{x_1^2∥\mathbb{B}∥M_1}{2}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right)\right).$$

Thus,

$$∥z\left(x\right)-y\left(x\right)∥\le N\epsilon ,N=\frac{x_1^2}{2\left(1-\rho \right)}{\mathbb{E}}_{2,2}\left(∥\mathbb{A}∥x_1^2\right).$$

This completes the proof. □

6. Examples

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

Example 1.

Consider the following linear delay differential controlled system:

$$\begin{array}{c}{y}^{″}\left(x\right)+\mathbb{A}y\left(x-0.5\right)=\mathbb{B}u\left(x\right),\mathrm{for}x\in {\Omega }:=\left[0,1\right],\\ y\left(x\right)\equiv \psi \left(x\right),y^{\prime }\left(x\right)\equiv {\psi }^{\prime }\left(x\right)\mathrm{for}-0.5\le x\le 0,\end{array}$$

(27)

where

$$\mathbb{A}=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),\mathbb{B}=\left(\begin{array}{c}1\\ 2\end{array}\right),\psi \left(x\right)=\left(\begin{array}{c}2x\\ x\end{array}\right),{\psi }^{\prime }\left(x\right)=\left(\begin{array}{c}2\\ 1\end{array}\right).$$

We note that $\mathbb{B}\in {\mathbb{R}}^{2\times 1}$ and $u\left(x\right)\in \mathbb{R}$ shows the control vector. Constructing the corresponding delay Gramian matrix of (27) via (9), we obtain

$$\begin{array}{cc}\hfill W_{0.5}^{\mathcal{M}}\left[0,1\right]& ={\int }_0^1{\mathcal{M}}_{0.5}\left(\mathbb{A}\left(0.5-\vartheta \right)\right)\mathbb{B}{\mathbb{B}}^T{\mathcal{M}}_{0.5}\left({\mathbb{A}}^T\left(0.5-\vartheta \right)\right)d\vartheta \hfill \\ \hfill & =:O_1+O_2,\hfill \end{array}$$

where

$$O_1={\int }_0^{0.5}{\mathcal{M}}_{0.5}\left(\mathbb{A}\left(0.5-\vartheta \right)\right)\mathbb{B}{\mathbb{B}}^T{\mathcal{M}}_{0.5}\left({\mathbb{A}}^T\left(0.5-\vartheta \right)\right)d\vartheta ,$$

for $\left(0.5-\vartheta \right)\in \left(0,0.5\right)$,

$$O_2={\int }_{0.5}^1{\mathcal{M}}_{0.5}\left(\mathbb{A}\left(0.5-\vartheta \right)\right)\mathbb{B}{\mathbb{B}}^T{\mathcal{M}}_{0.5}\left({\mathbb{A}}^T\left(0.5-\vartheta \right)\right)d\vartheta ,$$

for $\left(0.5-\vartheta \right)\in \left(-0.5,0\right)$, where

$${\mathcal{H}}_{0.5}\left(\mathbb{A}\left(x\right)\right):=\left\{\begin{array}{c}{\Theta },-\infty <x<-0.5,\\ \mathbb{I},-0.5\le x<0,\\ \mathbb{I}-\mathbb{A}\frac{x^2}{2},0\le x<0.5,\\ \mathbb{I}-\mathbb{A}\frac{x^2}{2}+{\mathbb{A}}^2\frac{{\left(x-0.5\right)}^4}{4!},0.5\le x<1,\end{array}\right.$$

and

$${\mathcal{M}}_{0.5}\left(\mathbb{A}\left(x\right)\right):=\left\{\begin{array}{c}{\Theta },-\infty <x<-0.5,\\ \mathbb{I}\left(x+0.5\right),-0.5\le x<0,\\ \mathbb{I}\left(x+0.5\right)-\mathbb{A}\frac{x^3}{3!},0\le x<0.5,\\ \mathbb{I}\left(x+0.5\right)-\mathbb{A}\frac{x^3}{3!}+{\mathbb{A}}^2\frac{{\left(x-0.5\right)}^5}{5!},0.5\le x<1.\end{array}\right.$$

Next, we can calculate that

$$O_1=\left(\begin{array}{cc}0.28242& 0.57396\\ 0.57396& 1.1667\end{array}\right),O_2=\left(\begin{array}{cc}4.1667\times {10}^{-2}& 8.3333\times {10}^{-2}\\ 8.3333\times {10}^{-2}& 0.16667\end{array}\right).$$

Then, we obtain

$$W_{0.5}^{\mathcal{M}}\left[0,1\right]=O_1+O_2=\left(\begin{array}{cc}0.32409& 0.65729\\ 0.65729& 1.3334\end{array}\right),$$

and

$${\left(W_{0.5}^{\mathcal{M}}\right)}^{-1}\left[0,1\right]=\left(\begin{array}{cc}11962.865& -5897.01\\ -5897.01& 2907.638\end{array}\right).$$

Therefore, we see that $W_{0.5}^{\mathcal{M}}\left[0,1\right]$ is positive definite. Furthermore, for any finite terminal conditions $y_1$, $y_1^{\prime }\in {\mathbb{R}}^2$ such that $y\left(x_1\right)=y_1={\left(y_{11},y_{12}\right)}^T$, $y^{\prime }\left(x_1\right)=y_1^{\prime }={\left(y_{11}^{\prime },y_{12}^{\prime }\right)}^T$; as a result, we can establish the corresponding control as follows:

$$u\left(x\right)={\mathbb{B}}^T{\mathcal{M}}_{0.5}\left({\mathbb{A}}^T\left(0.5-x\right)\right){\left(W_{0.5}^{\mathcal{M}}\right)}^{-1}\left[0,1\right]\beta ,$$

where

$$\begin{array}{cc}\hfill \beta & =y_1-{\mathcal{M}}_{0.5}\left(\mathbb{A}\left(0.5\right)\right){\psi }^{\prime }\left(0\right)+\mathbb{A}{\int }_{-0.5}^0{\mathcal{M}}_{0.5}\left(\mathbb{A}\left(-\vartheta \right)\right)\psi \left(\vartheta \right)d\vartheta \hfill \\ \hfill & =\left(\begin{array}{c}{y}_{11}-2.1042\\ y_{12}-1\end{array}\right).\hfill \end{array}$$

Hence, the system (27) is controllable on $\left[0,1\right]$ by Theorem 1.

Example 2.

Consider the following nonlinear delay differential controlled system:

$$\begin{array}{c}{y}^{″}\left(x\right)+\mathbb{A}y\left(x-0.6\right)=f\left(x,y\left(x\right)\right)+\mathbb{B}u\left(x\right),\mathrm{for}x\in {{\Omega }}_1:=\left[0,1.2\right],\\ y\left(x\right)\equiv \psi \left(x\right),y^{\prime }\left(x\right)\equiv {\psi }^{\prime }\left(x\right)\mathrm{for}-0.6\le x\le 0,\end{array}$$

(28)

where

$$\mathbb{A}=\mathbb{B}={\mathbb{I}}^{2\times 2},\psi \left(x\right)=\left(\begin{array}{c}3x+1\\ x^2\end{array}\right),{\psi }^{\prime }\left(x\right)=\left(\begin{array}{c}3\\ 2x\end{array}\right),$$

$$f\left(x,y\left(x\right)\right)=\left(\begin{array}{c}0.5\left(x-0.6\right)cos\left[y_1\left(x\right)\right]\\ 0.5\left(x-0.6\right)cos\left[y_2\left(x\right)\right]\end{array}\right).$$

Now, we set $u\left(x\right)=\tilde{y}$, where $\tilde{y}\in {\mathbb{R}}^2$. From the definition of Q in $\left(\mathbf{G}\mathbf{2}\right)$, we obtain

$$\begin{array}{cc}\hfill Q& ={\int }_0^{1.2}{\mathcal{M}}_{0.6}\left(\mathbb{A}\left(0.6-\vartheta \right)\right)\mathbb{B}d\vartheta \tilde{y}\hfill \\ \hfill & ={\int }_0^{0.6}{\mathcal{M}}_{0.6}\left(\mathbb{A}\left(0.6-\vartheta \right)\right)d\vartheta \tilde{y}+{\int }_{0.6}^{1.2}{\mathcal{M}}_{0.6}\left(\mathbb{A}\left(0.6-\vartheta \right)\right)d\vartheta \tilde{y}\hfill \\ \hfill & ={\int }_0^{0.6}\left[\mathbb{I}\left(1.2-\vartheta \right)-\mathbb{I}\frac{{\left(0.6-\vartheta \right)}^3}{3!}\right]d\vartheta \tilde{y}+{\int }_{0.6}^{1.2}\mathbb{I}\left(1.2-\vartheta \right)d\vartheta \tilde{y}\hfill \\ \hfill & =\left(\begin{array}{cc}0.5346& 0\\ 0& 0.5346\end{array}\right)\tilde{y}+\left(\begin{array}{cc}0.18& 0\\ 0& 0.18\end{array}\right)\tilde{y}\hfill \\ \hfill & =\left(\begin{array}{cc}0.7146& 0\\ 0& 0.7146\end{array}\right)\tilde{y},\hfill \end{array}$$

where

$${\mathcal{M}}_{0.6}\left(\mathbb{A}\left(x\right)\right):=\left\{\begin{array}{c}{\Theta },-\infty <x<-0.6,\\ \mathbb{I}\left(x+0.6\right),-0.6\le x<0,\\ \mathbb{I}\left(x+0.6\right)-\mathbb{A}\frac{x^3}{3!},0\le x<0.6,\\ \mathbb{I}\left(x+0.6\right)-\mathbb{A}\frac{x^3}{3!}+{\mathbb{A}}^2\frac{{\left(x-0.6\right)}^5}{5!},0.6\le x<1.\end{array}\right.$$

Define the inverse $Q^{-1}:{\mathbb{R}}^2\to L^2\left({{\Omega }}_1,{\mathbb{R}}^2\right)$ by

$$\left(Q^{-1}\tilde{y}\right)\left(x\right):=\left(\begin{array}{cc}1.3994& 0\\ 0& 1.3994\end{array}\right)\tilde{y}.$$

Then, we obtain

$$∥\left(Q^{-1}\tilde{y}\right)\left(x\right)∥\le ∥\left(\begin{array}{cc}1.3994& 0\\ 0& 1.3994\end{array}\right)∥∥\tilde{y}∥=1.3994∥\tilde{y}∥,$$

and thus, we obtain $∥Q^{-1}∥\le 1.3994=:M_1$. Hence, the assumption $\left(\mathbf{G}\mathbf{2}\right)$ is satisfied by Q.

Next, keep in mind that $\left|cos\lambda -cos\delta \right|\le \left|\lambda -\delta \right|$, for all λ, $\delta \in \mathbb{R}$, we have

$$\begin{array}{cc}\hfill & ∥f\left(x,y\right)-f\left(x,z\right)∥\hfill \\ \hfill & =\left|0.5\left(x-0.6\right)\right|\sqrt{{\left(cos\left[y_1\left(x\right)\right]-cos\left[z_1\left(x\right)\right]\right)}^2+{\left(cos\left[y_2\left(x\right)\right]-cos\left[z_2\left(x\right)\right]\right)}^2}\hfill \\ \hfill & \le \left|0.5\left(x-0.6\right)\right|\sqrt{{\left(y_1\left(x\right)-z_1\left(x\right)\right)}^2+{\left(y_2\left(x\right)-z_2\left(x\right)\right)}^2}\hfill \\ \hfill & =\left|0.5\left(x-0.6\right)\right|∥y-z∥,\hfill \end{array}$$

for all $x\in {{\Omega }}_1$, and $y\left(x\right)$, $z\left(x\right)\in {\mathbb{R}}^2$. We set $L_f\left(x\right)=\left|0.5\left(x-0.6\right)\right|$ such that $L_f\in L^q\left({{\Omega }}_1,{\mathbb{R}}^{+}\right)$ in $\left(\mathbf{G}\mathbf{1}\right)$. By choosing $p=q=2$, we have

$${∥L_f∥}_{L^2\left({{\Omega }}_1,{\mathbb{R}}^{+}\right)}={\left({\int }_0^{1.2}{\left[0.5\left(\vartheta -0.6\right)\right]}^{2}d\vartheta \right)}^{\frac{1}{2}}=0.18974.$$

Then, we obtain

$$M_2=\frac{{\left(1.2\right)}^{1+\frac{1}{2}}}{{\left(3\right)}^{\frac{1}{2}}}{\mathbb{E}}_{2,2}\left({\left(1.2\right)}^2\right){∥L_f∥}_{L^q\left({\Omega },{\mathbb{R}}^{+}\right)}=0.18114.$$

Finally, we calculate that

$$M_2\left[1+\frac{M_1{\left(1.2\right)}^{1.8}}{1.8}{\mathbb{E}}_{1.8,1.8}\left(∥\mathbb{A}∥{\left(1.2\right)}^2\right)∥\mathbb{B}∥\right]=0.41072<1,$$

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 $\mathbb{A}$ is used in [17] rather than $\mathbb{A}$, and the systems (27) and (28) are considered with matrix $\mathbb{A}$ rather than ${\mathbb{A}}^2$. That is, the term ${\mathbb{A}}^{2}y\left(x-h\right)$ is replaced by $\mathbb{A}y\left(x-h\right)$; then, the definition of ${sin}_h\mathbb{A}x$ and ${cos}_h\mathbb{A}x$ must be modified by using the square root $\sqrt{\mathbb{A}}$ instead of $\mathbb{A}$. However, $\sqrt{\mathbb{A}}$, 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.

7. 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 examples 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 $\alpha \in \left(1,2\right]$. Another challenge is to find out if similar results can be derived in the case of variable delays in (1) and (2).