Controllability of Stochastic Delay Systems Driven by the Rosenblatt Process

Abstract

In this work, we consider dynamical systems of linear and nonlinear stochastic delay-differential equations driven by the Rosenblatt process. With the aid of the delayed matrix functions of these systems, we derive the controllability results as an application. By using a delay Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear stochastic delay systems. In addition, by employing Krasnoselskii’s fixed point theorem, we present some necessary criteria for the controllability of nonlinear stochastic delay systems. Our results improve and extend some existing ones. Finally, an example is given to illustrate the main results.

1. Introduction

Due to its effective modeling in numerous fields of science and engineering, including economics, biology, physics, medicine, finance, fluid dynamics, etc., stochastic delay-differential equations and their applications have received a great deal of attention; see, for instance, [1,2,3,4,5,6,7,8,9,10]. The concept of the controllability of systems is one of the most fundamental and important concepts in contemporary control theory, which involves figuring out the control parameters that directly a control system’s solutions from their initial state to their final state using the set of permissible controls, where initial and final states may vary across the entire space. The representation of time-delay system solutions has recently received attention. The seminal study in [11,12], in particular, yielded several novel results in the representation of the solutions, stability, and controllability of time-delay systems; see, for example, [13,14,15,16,17,18,19,20,21] and the references therein.

The Hermite process of order one is known as the fractional Brownian motion, while the Hermite process of order two is known as the Rosenblatt process. Rosenblatt first proposed the following distribution for $x\ge 0$

$$Z_U\left(x\right)=D\left(U\right){\int }_{{\mathbb{R}}^2}\left({\int }_0^x{\left(\vartheta -t_1\right)}_{+}^{-\left(1+U\right)/2}{\left(\vartheta -t_2\right)}_{+}^{-\left(1+U\right)/2}\mathrm{d}\vartheta \right)\mathrm{d}J\left(t_1\right)\mathrm{d}J\left(t_2\right),$$

where $U\in \left(0,\frac{1}{2}\right)$ and $\left\{J\left(t\right),t\in U\right\}$ are a standard Brownian motion. The process of $Z_U\left(1\right)$ is known as the ‘1 non-Gaussian limiting distribution’ (Rosenblatt distribution); for more details see [22]. The Rosenblatt process is a non-Gaussian process with many interesting properties, such as the stationarity of the increments, long-range dependence, and self-similarity. Therefore, it is important to study a new class of stochastic differential equations driven by the Rosenblatt process. Shen and Ren [23] investigated the existence and uniqueness of the mild solution for neutral stochastic partial differential equations with finite delay driven by the Rosenblatt process in a real separable Hilbert space. Maejima and Tudor [24] presented a technique for constructing self-similar processes in the second Wiener chaos using limit theorems. Shen et al. [25] used fixed point theory to examine controllability and stability in functional nonlinear neutral stochastic systems with delay driven by the Rosenblatt process; furthermore, we refer the reader to [26,27] for further details on the Rosenblatt process.

Elshenhab and Wang [14] have established a novel formula to solve the following linear delay-differential systems

$$\begin{array}{c}{z}^{″}\left(x\right)+\Xi z\left(x-\omega \right)=g\left(x\right),x\ge 0,\\ z\left(x\right)\equiv \Pi \left(x\right),z^{\prime }\left(x\right)\equiv {\Pi }^{\prime }\left(x\right),-\omega \le x\le 0,\end{array}$$

(1)

of the form

$$\begin{array}{cc}\hfill z\left(x\right)& ={\mathcal{H}}_{\omega }\left(\Xi \left(x-\omega \right)\right)\Pi \left(0\right)+{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega \right)\right){\Pi }^{\prime }\left(0\right)\hfill \\ \hfill & -\Xi {\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)g\left(\vartheta \right)\mathrm{d}\vartheta ,\hfill \end{array}$$

(2)

where ${\mathcal{H}}_{\omega }\left(\Xi x\right)$ and ${\mathcal{M}}_{\omega }\left(\Xi x\right)$ are the delayed matrix functions defined by

$${\mathcal{H}}_{\omega }\left(\Xi x\right):=\left\{\begin{array}{cc}\hspace{1em}\mathsf{\Theta },\hfill & -\infty <x<-\omega ,\\ \hspace{1em}\mathbb{I},\hfill & -\omega \le x<0,\\ \hspace{1em}\mathbb{I}-\Xi \frac{x^2}{2!},\hfill & 0\le x<\omega ,\\ \hspace{1em}\hspace{1em}\hspace{1em}⋮\hfill & ⋮\\ \hspace{1em}\mathbb{I}-\Xi \frac{x^2}{2!}+{\Xi }^2\frac{{\left(x-\omega \right)}^4}{4!}\hfill & \\ \hspace{1em}+\cdots +{(-1)}^{\varsigma }{\Xi }^{\varsigma }\frac{{\left(x-\left(\varsigma -1\right)\omega \right)}^{2\varsigma }}{\left(2\varsigma \right)!},\hfill & \left(\varsigma -1\right)\omega \le x<\varsigma \omega ,\end{array}\right.$$

(3)

and

$${\mathcal{M}}_{\omega }\left(\Xi x\right):=\left\{\begin{array}{cc}\hspace{1em}\mathsf{\Theta },\hfill & -\infty <x<-\omega ,\\ \hspace{1em}\mathbb{I}\left(x+\omega \right),\hfill & -\omega \le x<0,\\ \hspace{1em}\mathbb{I}\left(x+\omega \right)-\Xi \frac{x^3}{3!},\hfill & 0\le x<\omega ,\\ \hspace{1em}\hspace{1em}\hspace{1em}⋮\hfill & ⋮\\ \hspace{1em}\mathbb{I}\left(x+\omega \right)-\Xi \frac{x^3}{3!}+{\Xi }^2\frac{{\left(x-\omega \right)}^5}{5!}\hfill & \\ \hspace{1em}+\cdots +{(-1)}^{\varsigma }{\Xi }^{\varsigma }\frac{{\left(x-\left(\varsigma -1\right)\omega \right)}^{2\varsigma +1}}{\left(2\varsigma +1\right)!},\hfill & \left(\varsigma -1\right)\omega \le x<\varsigma \omega ,\end{array}\right.$$

(4)

respectively, where $\varsigma =0,1,2,\dots$, and $\mathbb{I}$ and $\mathsf{\Theta }$ are the $n\times n$ identity and null matrix, respectively.

Based on [14], they [17] used the solutions (2) of (1) to obtain the controllability of the linear delay control systems

$$\begin{array}{c}{z}^{″}\left(x\right)+\Xi z\left(x-\omega \right)=\mathsf{\Lambda }u\left(x\right),x\in \mp :=\left[0,x_1\right],\\ z\left(x\right)\equiv \Pi \left(x\right),z^{\prime }\left(x\right)\equiv {\Pi }^{\prime }\left(x\right),-\omega \le x\le 0,\end{array}$$

(5)

and the controllability of the corresponding nonlinear delay control systems

$$\begin{array}{c}{z}^{″}\left(x\right)+\Xi z\left(x-\omega \right)=g\left(x,z\left(x\right)\right)+\mathsf{\Lambda }u\left(x\right),x\in \mp ,\\ z\left(x\right)\equiv \Pi \left(x\right),z^{\prime }\left(x\right)\equiv {\Pi }^{\prime }\left(x\right),-\omega \le x\le 0,\end{array}$$

(6)

where $\omega >0$ is a delay, $\Xi \in {\mathbb{R}}^{n\times n}$ and $\mathsf{\Lambda }\in {\mathbb{R}}^{n\times m}$ are any matrices, $x_1>\left(n-1\right)\omega$, $z\left(x\right)\in {\mathbb{R}}^n$, $\Pi \in C\left(\left[-\omega ,0\right],{\mathbb{R}}^n\right)$, and $g\in C\left(\mp \times {\mathbb{R}}^n,{\mathbb{R}}^n\right)$ are given functions, and $u\left(x\right)\in {\mathbb{R}}^m$ shows control vector.

In light of the preceding work, we investigate the controllability of stochastic linear delay-differential systems driven by the Rosenblatt process,

$$\begin{array}{c}{z}^{″}\left(x\right)+\Xi z\left(x-\omega \right)=\mathsf{\Lambda }u\left(x\right)+\overline{\Delta }\left(x\right)\mathrm{d}{Z}_H\left(x\right),x\in \mp :=\left[0,x_1\right],\\ z\left(x\right)\equiv \Pi \left(x\right),z^{\prime }\left(x\right)\equiv {\Pi }^{\prime }\left(x\right),-\omega \le x\le 0,\end{array}$$

(7)

in addition to the controllability of the corresponding stochastic nonlinear delay differential systems driven by the Rosenblatt process

$$\begin{array}{c}{z}^{″}\left(x\right)+\Xi z\left(x-\omega \right)=\mathsf{\Lambda }u\left(x\right)+\Delta \left(x,z\left(x\right)\right)\mathrm{d}{Z}_H\left(x\right),x\in \mp ,\\ z\left(x\right)\equiv \Pi \left(x\right),z^{\prime }\left(x\right)\equiv {\Pi }^{\prime }\left(x\right),-\omega \le x\le 0,\end{array}$$

(8)

where the state vector $z\left(x\right)\in {\mathbb{R}}^n$, $\Xi \in {\mathbb{R}}^{n\times n}$, and $\mathsf{\Lambda }\in {\mathbb{R}}^{n\times m}$ are any matrices, $\Pi \in C\left(\left[-\omega ,0\right],{\mathbb{R}}^n\right)$ and $u\left(x\right)\in {\mathbb{R}}^m$ show control vectors, and $\overline{\Delta }\in C\left(\mp ,\mathcal{T}\left({\mathbb{R}}^n\right)\right)$ is the Thorin class, symbolized by $\mathcal{T}\left({\mathbb{R}}^n\right)$, and is the smallest distribution class on ${\mathbb{R}}^n$ that comprises all Gamma distributions and is closed under convolution and weak convergence. Let $z\left(·\right)$ take value in the separable Hilbert space ${\mathbb{R}}^n$ with inner product $〈·,·〉$ and norm $∥·∥$. $Z_H\left(x\right)$ is a Rosenblatt process with parameter $H\in \left(\frac{1}{2},1\right)$ on an another real separable Hilbert space $\left(K,{∥·∥}_K,{〈·,·〉}_K\right)$. Moreover, assume $\Delta \in C\left(\mp \times {\mathbb{R}}^n,L_2^0\right)$, where $L_2^0=L_2\left(Q^{\frac{1}{2}}K,{\mathbb{R}}^n\right)$.

The following is how the rest of this paper is structured: In Section 2, we provide some introductions, fundamental notations, and definitions, as well as some relevant lemmas. In Section 3, using a delay Gramian matrix, we obtain the sufficient and necessary conditions for the controllability of (7). In Section 4, by applying Krasnoselskii’s fixed point theorem, we establish sufficient conditions for the controllability of (8). Finally, to illustrate the theoretical findings, we provide numerical examples.

2. Preliminaries

Throughout the paper, let $\left(\Omega ,\mathfrak{F},\mathbb{P}\right)$ be the complete probability space with probability measure $\mathbb{P}$ on $\Omega$ with a filtration $\left\{{\mathfrak{F}}_x|x\in \mp \right\}$ generated by $\left\{Z_H\left(s\right)|s\in \left[0,x\right]\right\}$. Let $\mathcal{X}$ and $\mathcal{Y}$ be two Banach spaces and $L_b\left(\mathcal{X},\mathcal{Y}\right)$ be the space of bounded linear operators from $\mathcal{X}$ to $\mathcal{Y}$ and $Q\in L_b\left(\mathcal{X},\mathcal{X}\right)$ represent a non-negative self-adjoint trace class operator on $\mathcal{X}$. Let $L_2^0=L_2\left(Q^{\frac{1}{2}}\mathcal{X},\mathcal{Y}\right)$ be the space of all Q-Hilbert–Schmidt operators from $Q^{\frac{1}{2}}\mathcal{X}$ into $\mathcal{Y}$, equipped with the norm

$${∥\phi ∥}_{L_2^0}^2={∥\phi Q^{\frac{1}{2}}∥}^2=\mathrm{Tr}\left(\phi Q{\phi }^T\right).$$

Now, for some $1<e<\infty$, let $L^e\left(\Omega ,{\mathfrak{F}}_{x_1},{\mathbb{R}}^n\right)$ be the Hilbert space of all ${\mathfrak{F}}_{x_1}$-measurable eth-integrable variables with values in ${\mathbb{R}}^n$ with norm ${∥z∥}_{L^e}^e=\mathbf{E}{∥z\left(x\right)∥}^e$, where the expectation $\mathbf{E}$ is defined by $\mathbf{E}z={\int }_{\Omega }z\mathrm{d}\mathbb{P}$. Let $L_{\mathfrak{F}}^e\left(\mp ,{\mathbb{R}}^n\right)$ be the Banach space of all functions $g:\mp ⟶{\mathbb{R}}^n$ that are Bochner integrable normed by ${∥g∥}_{L_{\mathfrak{F}}^e\left(\mp ,{\mathbb{R}}^n\right)}$- and ${\mathfrak{F}}_{x_1}$-measurable processes with values in ${\mathbb{R}}^n$. Let $\mathcal{F}:=C\left(\left[-\omega ,0\right]\right.$, $\left.L^e\left(\Omega ,{\mathfrak{F}}_{x_1},\mathbb{P},{\mathbb{R}}^n\right)\right)$ be the Banach space of all eth-integrable and ${\mathfrak{F}}_{x_1}$-adapted processes $\varphi$ endowed with the norm ${∥\varphi ∥}_C={\left({sup}_{x\in \left[-\omega ,0\right]}\mathbf{E}{∥\varphi \left(x\right)∥}^e\right)}^{1/e}$. Additionally, we denote $C\left(\mp ,L^e\left(\Omega ,{\mathfrak{F}}_{x_1},\mathbb{P},{\mathbb{R}}^n\right)\right)$ as the Banach space of continuous function from $\mp ⟶L^e\left(\Omega ,{\mathfrak{F}}_{x_1},\mathbb{P},{\mathbb{R}}^n\right)$ equipped with the norm ${∥z∥}_{C\left(\mp \right)}={\left({sup}_{x\in \mp }\mathbf{E}{∥z\left(x\right)∥}^e\right)}^{1/e}$ for a norm $∥·∥$ on ${\mathbb{R}}^n$. We describe a space

$$\begin{array}{cc}\hfill & C^1\left(\mp ,L^e\left(\Omega ,{\mathfrak{F}}_{x_1},\mathbb{P},{\mathbb{R}}^n\right)\right)\hfill \\ \hfill & =\left\{z\in C\left(\mp ,L^e\left(\Omega ,{\mathfrak{F}}_{x_1},\mathbb{P},{\mathbb{R}}^n\right)\right):z^{\prime }\in C\left(\mp ,L^e\left(\Omega ,{\mathfrak{F}}_{x_1},\mathbb{P},{\mathbb{R}}^n\right)\right)\right\},\hfill \end{array}$$

and we use the matrix norm (column sum)

$$∥\Xi ∥=max\left\{\sum _{i=1}^n\left|a_{i1}\right|,\sum _{i=1}^n\left|a_{i2}\right|,\dots ,\sum _{i=1}^n\left|a_{in}\right|\right\},$$

where $\Xi :{\mathbb{R}}^n⟶{\mathbb{R}}^n$. Furthermore, we let

$${∥\Pi ∥}_C={\left(\underset{s\in \left[-\omega ,0\right]}{sup}\mathbf{E}{∥\Pi \left(s\right)∥}^e\right)}^{1/e}\mathrm{and}{∥{\Pi }^{\prime }∥}_C={\left(\underset{s\in \left[-\omega ,0\right]}{sup}\mathbf{E}{∥{\Pi }^{\prime }\left(s\right)∥}^e\right)}^{1/e}.$$

The Wiener–Ito multiple integral of order k with respect to the standard Wiener process ${\left(\mathcal{G}\left(\rho \right)\right)}_{\rho \in \mathbb{R}}$ is defined by

$$Z_H^k\left(x\right)=c\left(H,k\right){\int }_{R^k}\left({\int }_0^x{\displaystyle \prod _{j=1}^k}{\left(\vartheta -{\rho }_j\right)}_{+}^{-\left(\frac{1}{2}+\frac{1-H}{k}\right)}\mathrm{d}\vartheta \right)\mathrm{d}\mathcal{G}\left({\rho }_1\right)\dots \mathrm{d}\mathcal{G}\left({\rho }_k\right),$$

(9)

where $c\left(H,k\right)$ is a normalizing constant such that $\mathbf{E}{\left(Z_H^k\left(1\right)\right)}^2=1$ and ${\rho }_{+}=max\left(\rho ,0\right)$. The process ${\left(Z_H^k\left(x\right)\right)}_{x\ge 0}$ is called the Hermite process. If $k=1$, the Hermite process given by (9) is the fBm with Hurst parameter $H\in \left(\frac{1}{2},1\right)$; furthermore, the process is not Gaussian for $k=2$. Moreover, for $k=2$, the Hermite process given by (9) is called the Rosenblatt process.

We provide some fundamental concepts and lemmas used in this work.

 Lemma 1 

([28]). If $\sigma :\mp ⟶L_2^0$ satisfies

$${\int }_0^{x_1}{∥\sigma \left(\vartheta \right)∥}_{L_2^0}^2\mathrm{d}\vartheta <\infty ,$$

then, for a, $b\in \mp$ with $b>a$, we have

$$\mathbf{E}{∥{\int }_0^x\sigma \left(\vartheta \right)\mathrm{d}{Z}_H\left(\vartheta \right)∥}^2\le 2H{x}^{2H-1}{\int }_0^x{∥\sigma \left(\vartheta \right)∥}_{L_2^0}^2\mathrm{d}\vartheta .$$

 Definition 1 

([29]). If there is a control function $u\in L_{\mathfrak{F}}^e\left(\mp ,{\mathbb{R}}^m\right)$ such that (7) or (8) have a solution $z:\left[-\omega ,x_1\right]⟶{\mathbb{R}}^n$ with $z\left(x\right)=\Pi \left(x\right)$, $z^{\prime }\left(x\right)={\Pi }^{\prime }\left(x\right)$ satisfies $z\left(x_1\right)=z_1$ for all $x\in \left[-\omega ,0\right]$, $z_1\in {\mathbb{R}}^n$, then, the systems (7) or (8) are controllable on $\mp =\left[0,x_1\right].$

 Definition 2 

([30]). The two-parameter Mittag-Leffler function is provided by

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

In the case of $\gamma =1$, then

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

 Lemma 2 

([17] (Lemmas 1 and 2)). For any $x\in \left[\left(m-1\right)\omega ,m\omega \right]$, $m=1,2,...$, we have

$$∥{\mathcal{H}}_{\omega }\left(\Xi \left(x\right)\right)∥\le {\mathbb{E}}_2\left(∥\Xi ∥x^2\right),$$

and

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

 Lemma 3 

(Krasnoselskii’s fixed point theorem [31]). Let M be a closed, bounded, and convex subset of a real Banach space $\mathcal{K}$, and let $J_1$ and $J_2$ be operators on M satisfying the following conditions:

 (i) 

$J_{1}x+J_{2}z\in M$ for x, $z\in M$;

 (ii) 

$J_1$ is compact and continuous;

 (iii) 

$J_2$ is a contraction mapping.

Then, there exists $m\in M$ such that $m=J_{1}m+J_{2}m$.

We define the operator $Q_{x_1}\in L_b\left(L_{\mathfrak{F}}^e\left(\mp ,{\mathbb{R}}^m\right),L^e\left(\Omega ,{\mathfrak{F}}_{x_1},{\mathbb{R}}^n\right)\right)$ as

$$Q_{x_1}u={\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }u\left(\vartheta \right)\mathrm{d}\vartheta ,$$

and its adjoint operator $Q_{x_1}^T\in L_b\left(L^e\left(\Omega ,{\mathfrak{F}}_{x_1},{\mathbb{R}}^n\right),L_{\mathfrak{F}}^e\left(\mp ,{\mathbb{R}}^m\right)\right)$ is defined as

$$Q_{x_1}^{T}u={\mathsf{\Lambda }}^T{\mathcal{M}}_{\omega }\left({\Xi }^T\left(x_1-\omega -x\right)\right)\mathbf{E}\left\{t|{\mathfrak{F}}_x\right\}.$$

Consider the linear controllability operator

$$\begin{array}{cc}\hfill {\mathsf{\Gamma }}_{\omega }^{x_1}\left\{·\right\}& =Q_{x_1}{Q}_{x_1}^T\left\{·\right\}\hfill \\ \hfill & ={\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }{\mathsf{\Lambda }}^T{\mathcal{M}}_{\omega }\left({\Xi }^T\left(x_1-\omega -\vartheta \right)\right)\mathbf{E}\left\{·|{\mathfrak{F}}_{\vartheta }\right\}\mathrm{d}\vartheta ,\hfill \end{array}$$

and the delayed Gramian matrix $W_{\omega }^{\mathcal{M}}\left[0,x_1\right]\in L_b\left({\mathbb{R}}^n,{\mathbb{R}}^n\right)$ defined by

$$W_{\omega }^{\mathcal{M}}\left[0,x_1\right]={\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }{\mathsf{\Lambda }}^T{\mathcal{M}}_{\omega }\left({\Xi }^T\left(x_1-\omega -\vartheta \right)\right)\mathrm{d}\vartheta .$$

(10)

Here, T denotes the transpose.

3. Controllability of Linear Stochastic Delay Systems

In this section, we derive the controllability results of (7) using the delayed Gramian matrix $W_{\omega }^{\mathcal{M}}\left[0,x_1\right]$ defined by (10).

 Theorem 1. 

The stochastic system (7) is controllable if and only if $W_{\omega }^{\mathcal{M}}\left[0,x_1\right]$ is positive definite.

 Proof. Sufficiency. 

Assume that $W_{\omega }^{\mathcal{M}}\left[0,x_1\right]$ is positive definite; then, it is invertible. Consequently, for any finite terminal conditions $z_1$, $z_1^{\prime }\in {\mathbb{R}}^n$, we can determine the related control input $u\left(x\right)$ as

$$u\left(x\right)={\mathsf{\Lambda }}^T{\mathcal{M}}_{\omega }\left({\Xi }^T\left(x_1-\omega -x\right)\right){\left(W_{\omega }^{\mathcal{M}}\right)}^{-1}\left[0,x_1\right]\beta ,$$

(11)

where

$$\begin{array}{cc}\hfill \beta & =z_1-{\mathcal{H}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right)\Pi \left(0\right)-{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right){\Pi }^{\prime }\left(0\right)\hfill \\ \hfill & +\Xi {\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & -{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\overline{\Delta }\left(\vartheta \right)\mathrm{d}{Z}_H\left(\vartheta \right).\hfill \end{array}$$

(12)

Applying the Formula (2), the solution of (7) can be expressed as

$$\begin{array}{cc}\hfill z\left(x\right)& ={\mathcal{H}}_{\omega }\left(\Xi \left(x-\omega \right)\right)\Pi \left(0\right)+{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega \right)\right){\Pi }^{\prime }\left(0\right)\hfill \\ \hfill & -\Xi {\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\mathsf{\Lambda }u\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\overline{\Delta }\left(\vartheta \right)\mathrm{d}{Z}_H\left(\vartheta \right).\hfill \end{array}$$

(13)

From (13), the solution $z\left(x_1\right)$ of (7) can be expressed as

$$\begin{array}{cc}\hfill z\left(x_1\right)& ={\mathcal{H}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right)\Pi \left(0\right)+{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right){\Pi }^{\prime }\left(0\right)\hfill \\ \hfill & -\Xi {\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }u\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\overline{\Delta }\left(\vartheta \right)\mathrm{d}{Z}_H\left(\vartheta \right).\hfill \end{array}$$

(14)

Substituting (11) into (14), we obtain

$$\begin{array}{cc}\hfill z\left(x_1\right)& ={\mathcal{H}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right)\Pi \left(0\right)+{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right){\Pi }^{\prime }\left(0\right)\hfill \\ \hfill & -\Xi {\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }{\mathsf{\Lambda }}^T{\mathcal{M}}_{\omega }\left({\Xi }^T\left(x_1-\omega -\vartheta \right)\right)\mathrm{d}\vartheta \hfill \\ \hfill & \times {\left(W_{\omega }^{\mathcal{M}}\right)}^{-1}\left[0,x_1\right]\beta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\overline{\Delta }\left(\vartheta \right)\mathrm{d}{Z}_H\left(\vartheta \right).\hfill \\ \hfill & \hfill \end{array}$$

(15)

From (10), (12), and (15), we obtain

$$\begin{array}{cc}\hfill z\left(x_1\right)& ={\mathcal{H}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right)\Pi \left(0\right)+{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right){\Pi }^{\prime }\left(0\right)\hfill \\ \hfill & -\Xi {\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta +\beta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\overline{\Delta }\left(\vartheta \right)\mathrm{d}{Z}_H\left(\vartheta \right).\hfill \\ \hfill & =z_1.\hfill \end{array}$$

We can see from (3), (4), and (13) that the boundary conditions $z\left(x\right)\equiv \Pi \left(x\right)$, $z^{\prime }\left(x\right)\equiv {\Pi }^{\prime }\left(x\right)$, $-\omega \le x\le 0$ are fulfilled. Therefore, (7) is controllable.

Necessity. Let (7) be controllable. Assume, for the sake of a contradiction, that $W_{\omega }^{\mathcal{M}}\left[0,x_1\right]$ is not positive definite and there exists at least a nonzero vector $\rho \in {\mathbb{R}}^n$ such that ${\rho }^T{W}_{\omega }^{\mathcal{M}}\left[0,x_1\right]\rho =0$, which leads to

$$\begin{array}{cc}\hfill 0& ={\rho }^T{W}_{\omega }^{\mathcal{M}}\left[0,x_1\right]\rho \hfill \\ \hfill & ={\int }_0^{x_1}{\rho }^T{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }{\mathsf{\Lambda }}^T{\mathcal{M}}_{\omega }\left({\Xi }^T\left(x_1-\omega -\vartheta \right)\right)\rho \mathrm{d}\vartheta \hfill \\ \hfill & ={\int }_0^{x_1}\left[{\rho }^T{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }\right]{\left[{\rho }^T{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }\right]}^T\mathrm{d}\vartheta \hfill \\ \hfill & ={\int }_0^{x_1}∥{\rho }^T{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }∥\mathrm{d}\vartheta .\hfill \end{array}$$

Hence,

$${\rho }^T{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }=\left(0,\dots ,0\right):={\mathbf{0}}^T,\mathrm{for}\mathrm{all}\vartheta \in \mp ,$$

(16)

where $\mathbf{0}$ denotes the n dimensional zero vector. From Definition 1, there is a control function $u_1\left(x\right)$ that steers the initial state to $z_1=0$ at $x=x_1$. Then,

$$\begin{array}{cc}\hfill z\left(x_1\right)& ={\mathcal{H}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right)\Pi \left(0\right)+{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right){\Pi }^{\prime }\left(0\right)\hfill \\ \hfill & -\Xi {\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }{u}_1\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\overline{\Delta }\left(\vartheta \right)\mathrm{d}{Z}_H\left(\vartheta \right)\hfill \\ \hfill & =\mathbf{0}.\hfill \end{array}$$

(17)

Similarly, there is a control function $u_2\left(x\right)$ that steers the initial state to $z_1=\rho$ at $x=x_1$. Thus,

$$\begin{array}{cc}\hfill z\left(x_1\right)& ={\mathcal{H}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right)\Pi \left(0\right)+{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right){\Pi }^{\prime }\left(0\right)\hfill \\ \hfill & -\Xi {\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }{u}_2\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\overline{\Delta }\left(\vartheta \right)\mathrm{d}{Z}_H\left(\vartheta \right)\hfill \\ \hfill & =\rho .\hfill \end{array}$$

(18)

Combining (17) with (18), we have

$$\rho ={\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }\left[u_2\left(\vartheta \right)-u_1\left(\vartheta \right)\right]\mathrm{d}\vartheta$$

(19)

Multiplying (19) by ${\rho }^T$ and using (16), we have ${\rho }^T\rho =0$. This is a contradiction to $\rho \ne \mathbf{0}$. Thus, $W_{\omega }^{\mathcal{M}}\left[0,x_1\right]$ is positive definite. This completes the proof. □

 Corollary 1. 

Theorem 1 is satisfied in the case of $\Xi ={\Xi }^2$ in (7).

 Corollary 2. 

In the case of Ξ being a nonsingular $n\times n$ matrix, and $\Xi ={\Xi }^2$ in (7), the system (7) is controllable if and only if $W_{\omega }\left[0,x_1\right]$ is nonsingular, where $W_{\omega }\left[0,x_1\right]$ is defined as

$$W_{\omega }\left[0,x_1\right]={\Xi }^{-1}{\int }_0^{x_1}{sin}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }{\mathsf{\Lambda }}^T{sin}_{\omega }\left({\Xi }^T\left(x_1-\omega -\vartheta \right)\right)\mathrm{d}\vartheta ,$$

and ${sin}_{\omega }\left(\Xi x\right)$ and ${cos}_{\omega }\left(\Xi x\right)$ are known as the delayed matrices of the sine and cosine types, respectively, which are given in [12].

Proof. 

When $\Xi ={\Xi }^2$ and using the definition of ${\mathcal{H}}_{\omega }\left(\Xi x\right)$ and ${\mathcal{M}}_{\omega }\left(\Xi x\right)$, we have

$${\mathcal{H}}_{\omega }\left({\Xi }^{2}x\right)={cos}_{\omega }\left(\Xi x\right),{\mathcal{M}}_{\omega }\left({\Xi }^{2}x\right)={\Xi }^{-1}{sin}_{\omega }\left(\Xi x\right),$$

which implies that

$$\begin{array}{cc}\hfill W_{\omega }^{\mathcal{M}}\left[0,x_1\right]& ={\int }_0^{x_1}{\mathcal{M}}_{\omega }\left({\Xi }^2\left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }{\mathsf{\Lambda }}^T{\mathcal{M}}_{\omega }\left({\left({\Xi }^2\right)}^T\left(x_1-\omega -\vartheta \right)\right)\mathrm{d}\vartheta \hfill \\ \hfill & ={\int }_0^{x_1}{\Xi }^{-1}{sin}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }{\mathsf{\Lambda }}^T{sin}_{\omega }\left({\Xi }^T\left(x_1-\omega -\vartheta \right)\right){\left({\Xi }^{-1}\right)}^T\mathrm{d}\vartheta \hfill \\ \hfill & ={\Xi }^{-1}{\int }_0^{x_1}{sin}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\mathsf{\Lambda }{\mathsf{\Lambda }}^T{sin}_{\omega }\left({\Xi }^T\left(x_1-\omega -\vartheta \right)\right)\mathrm{d}\vartheta {\left({\Xi }^T\right)}^{-1}\hfill \\ \hfill & =W_{\omega }\left[0,x_1\right]{\left({\Xi }^T\right)}^{-1}.\hfill \end{array}$$

Hence,

$$W_{\omega }\left[0,x_1\right]=W_{\omega }^{\mathcal{M}}\left[0,x_1\right]{\Xi }^T.$$

(20)

From the conclusion of Theorem 1, we determine that $W_{\omega }^{\mathcal{M}}\left[0,x_1\right]$ is invertible. Thus, from (20), we find that $W_{\omega }\left[0,x_1\right]$ is also invertible. This completes the proof. □

4. Controllability of Nonlinear Stochastic Delay Systems

In this section, we present sufficient conditions of the controllability of (8).

The following hypotheses are made:

 (S1) 

The function $\Delta :\mp \times {\mathbb{R}}^n⟶L_2^0$ is continuous, and there exists a constant $L_{\Delta }\in L^q\left(\mp ,{\mathbb{R}}^{+}\right)$ and $q>1$ such that

$$\mathbf{E}{∥\Delta \left(x,z_1\right)-\Delta \left(x,z_2\right)∥}_{L_2^0}^e\le L_{\Delta }\left(x\right){∥z_1-z_2∥}^e,\mathrm{for}\mathrm{all}x\in \mp ,z_1,z_2\in {\mathbb{R}}^n.$$

Let $e\in \left[2,\infty \right)$ and ${sup}_{x\in \mp }\mathbf{E}{∥\Delta \left(x,0\right)∥}_{L_2^0}^e=N_{\Delta }<\infty$.

 (S2) 

The linear stochastic delay system (7) is controllable on $\mp$.

Under the assumption of $\left(\mathbf{S}\mathbf{2}\right)$, for some $\eta >0$, we have $\mathbf{E}〈{\mathsf{\Gamma }}_{\omega }^{x_1}z,z〉\ge \eta \mathbf{E}{∥z∥}^e$ for all $z\in L^e\left(\Omega ,{\mathfrak{F}}_{x_1},{\mathbb{R}}^n\right)$ (see [32] [Lemma 2]). Furthermore, ${∥{\left({\mathsf{\Gamma }}_{\omega }^{x_1}\right)}^{-1}∥}^e\le 1/\eta :=N_1$ (see [33]) and set $N:=max\left\{{∥W_{\omega }^{\mathcal{M}}\left[\vartheta ,x_1\right]∥}^e:\vartheta \in \mp \right\}$.

 Theorem 2. 

Let $\left(\mathbf{S}\mathbf{1}\right)$ and $\left(\mathbf{S}\mathbf{2}\right)$ be satisfied. Then, the nonlinear stochastic system (8) is controllable on Υ if there exists a constant ${\tau }_e>0$ such that

$$N_2\left[1+5^{e-1}N{N}_1\right]<1,$$

(21)

where

$$N_2:=\frac{5^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{e\left(H+1\right)-\frac{1}{q}}}{{\left(ep+1\right)}^{\frac{1}{p}}}{\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_1^2\right)\right)}^e{∥L_{\Delta }∥}_{L^q\left(\mp ,{\mathbb{R}}^{+}\right)},$$

and $\frac{1}{p}+\frac{1}{q}=1$, p, $q>1$.

Proof. 

We consider the following set before beginning to prove this theorem

$${\mathcal{B}}_{\nearrow }=\left\{z\in \mathcal{F}:{∥z∥}_{\mathcal{F}}^e=\underset{x\in \left[-\omega ,x_1\right]}{sup}\mathbf{E}{∥z\left(x\right)∥}^e\le \nearrow \right\},$$

for each postive number $\nearrow$. Let $x\in \left[0,x_1\right]$. Applying Formula (2), the solution of (8) can be expressed as

$$\begin{array}{cc}\hfill z\left(x\right)& ={\mathcal{H}}_{\omega }\left(\Xi \left(x-\omega \right)\right)\Pi \left(0\right)+{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega \right)\right){\Pi }^{\prime }\left(0\right)\hfill \\ \hfill & -\Xi {\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\mathsf{\Lambda }{u}_z\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right),\hfill \end{array}$$

and its control function $u_z$ is defined as

$$\begin{array}{cc}\hfill u_z\left(x\right)& ={\mathsf{\Lambda }}^T{\mathcal{M}}_{\omega }\left({\Xi }^T\left(x_1-\omega -x\right)\right)\hfill \\ \hfill & \times \mathbf{E}\left\{{\left({\mathsf{\Gamma }}_{\omega }^{x_1}\right)}^{-1}\left[z_1-{\mathcal{H}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right)\Pi \left(0\right)-{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right){\Pi }^{\prime }\left(0\right)\right.\right.\hfill \\ \hfill & +\Xi {\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & \left.\left.-{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right)\right]|{\mathfrak{F}}_x\right\}\hfill \end{array}$$

(22)

for $x\in \mp$. Additionally, we define the following operators ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ on ${\mathcal{B}}_{\nearrow }$ of the form

$$\begin{array}{cc}\hfill \left({\mathcal{L}}_{1}z\right)\left(x\right)& ={\mathcal{H}}_{\omega }\left(\Xi \left(x-\omega \right)\right)\Pi \left(0\right)+{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega \right)\right){\Pi }^{\prime }\left(0\right)\hfill \\ \hfill & -\Xi {\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & +{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\mathsf{\Lambda }{u}_z\left(\vartheta \right)\mathrm{d}\vartheta ,\hfill \end{array}$$

(23)

$$\left({\mathcal{L}}_{2}z\right)\left(x\right)={\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right).$$

(24)

At this point, it is evident that ${\mathcal{B}}_{\nearrow }$ is a closed, bounded, and convex set of $\mathcal{F}$. Therefore, there are three essential steps to our proof:

Step 1. We prove that there exists a $\nearrow >0$ such that ${\mathcal{L}}_{1}z+{\mathcal{L}}_2\rho \in {\mathcal{B}}_{\nearrow }$ for all z, $\rho \in {\mathcal{B}}_{\nearrow }$.

Using (23) and (24), we obtain

$$\begin{array}{cc}\hfill & {∥{\mathcal{L}}_{1}z+{\mathcal{L}}_2\rho ∥}_{\mathcal{F}}^e\hfill \\ \hfill & =\underset{x\in \left[-\omega ,x_1\right]}{sup}\mathbf{E}{∥\left({\mathcal{L}}_{1}z+{\mathcal{L}}_2\rho \right)\left(x\right)∥}^e\hfill \\ \hfill & \le 5^{e-1}\left[{∥{\mathcal{H}}_{\omega }\left(\Xi \left(x-\omega \right)\right)∥}^e\mathbf{E}{∥\Pi \left(0\right)∥}^e+{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega \right)\right)∥}^e\mathbf{E}{∥{\Pi }^{\prime }\left(0\right)∥}^e\right.\hfill \\ \hfill & +{∥\Xi ∥}^e\mathbf{E}{∥{\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta ∥}^e\hfill \\ \hfill & +\mathbf{E}{∥{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\mathsf{\Lambda }{u}_z\left(\vartheta \right)\mathrm{d}\vartheta ∥}^e\hfill \\ \hfill & \left.+\mathbf{E}{∥{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right)∥}^e\right]\hfill \\ \hfill & ={\sum }_{n=1}^5{\mathbf{I}}_n,\hfill \end{array}$$

(25)

for each $x\in \mp$ and z, $\rho \in {\mathcal{B}}_{\nearrow }$. From Lemma 2, we have

$$\begin{array}{cc}\hfill {\mathbf{I}}_1& =5^{e-1}{∥{\mathcal{H}}_{\omega }\left(\Xi \left(x-\omega \right)\right)∥}^e\mathbf{E}{∥\Pi \left(0\right)∥}^e\hfill \\ \hfill & \le 5^{e-1}{\left({\mathbb{E}}_2\left(∥\Xi ∥{\left(x-\omega \right)}^2\right)\right)}^e\mathbf{E}{∥\Pi ∥}_C^e,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathbf{I}}_2& =5^{e-1}{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega \right)\right)∥}^e\mathbf{E}{∥{\Pi }^{\prime }\left(0\right)∥}^e\hfill \\ \hfill & \le 5^{e-1}{\left(x{\mathbb{E}}_{2,2}\left(∥\Xi ∥x^2\right)\right)}^e\mathbf{E}{∥{\Pi }^{\prime }∥}_C^e,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathbf{I}}_3& =5^{e-1}{∥\Xi ∥}^e\mathbf{E}{∥{\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta ∥}^e\hfill \\ \hfill & \le 5^{e-1}{∥\Xi ∥}^e{\omega }^{e-1}\mathbf{E}{∥\Pi ∥}_C^e{\int }_{-\omega }^0{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x-2\omega -\vartheta \right)\right)∥}^e\mathrm{d}\vartheta \hfill \\ \hfill & \le 5^{e-1}{∥\Xi ∥}^e{\omega }^e{\left(x{\mathbb{E}}_{2,2}\left(∥\Xi ∥x^2\right)\right)}^e\mathbf{E}{∥\Pi ∥}_C^e,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathbf{I}}_4& =5^{e-1}\mathbf{E}{∥{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right)∥}^e\hfill \\ \hfill & =5^{e-1}\mathbf{E}{\left\{{∥{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right)∥}^2\right\}}^{e/2},\hfill \end{array}$$

and by employing Lemma 1, the Kahane–khintchine inequality, and Hölder’s inequality, there exists a constant ${\tau }_e$ such that

$$\begin{array}{cc}\hfill {\mathbf{I}}_4& \le 5^{e-1}{\tau }_e{\left\{\mathbf{E}{∥{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right)∥}^2\right\}}^{e/2}\hfill \\ \hfill & \le 5^{e-1}{\tau }_e{\left\{2H{x}^{2H-1}{\int }_0^x\mathbf{E}{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)∥}_{L_2^0}^2\mathrm{d}\vartheta \right\}}^{e/2}\hfill \\ \hfill & \le 5^{e-1}{\tau }_e{\left(2H{x}^{2H-1}\right)}^{e/2}{\left\{{\int }_0^x\mathbf{E}{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)∥}_{L_2^0}^2\mathrm{d}\vartheta \right\}}^{e/2}\hfill \\ \hfill & \le 5^{e-1}{\tau }_e{\left(2H{x}^{2H-1}\right)}^{e/2}\hfill \\ \hfill & \times {\left\{{\left({\int }_0^x{\left(\mathbf{E}{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)∥}_{L_2^0}^2\right)}^{e/2}\mathrm{d}\vartheta \right)}^{2/e}{\left({\int }_0^x\mathrm{d}\vartheta \right)}^{\frac{e-2}{e}}\right\}}^{e/2}\hfill \\ \hfill & \le 5^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{eH-1}{\int }_0^x\mathbf{E}{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)∥}_{L_2^0}^e\mathrm{d}\vartheta ,\hfill \end{array}$$

and by employing Lemma 2 and $\left(\mathbf{S}\mathbf{1}\right)$, we obtain

$$\begin{array}{cc}\hfill {\mathbf{I}}_4& \le 5^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{eH-1}{\int }_0^x{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\Xi ∥{\left(x-\vartheta \right)}^2\right)\right)}^e\mathbf{E}{∥\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)∥}_{L_2^0}^e\mathrm{d}\vartheta \hfill \\ & \le 5^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{eH-1}\hfill \\ & \times 2^{e-1}\left\{{\int }_0^x{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\Xi ∥{\left(x-\vartheta \right)}^2\right)\right)}^e\mathbf{E}{∥\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)-\Delta \left(\vartheta ,0\right)∥}_{L_2^0}^e\mathrm{d}\vartheta \right.\hfill \\ & \left.+{\int }_0^x{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\Xi ∥{\left(x-\vartheta \right)}^2\right)\right)}^e\mathbf{E}{∥\Delta \left(\vartheta ,0\right)∥}_{L_2^0}^e\mathrm{d}\vartheta \right\}\hfill \\ & \le 5^{e-1}{2}^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{eH-1}\left\{{\int }_0^x{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\Xi ∥{\left(x-\vartheta \right)}^2\right)\right)}^e{L}_{\Delta }\left(\vartheta \right){∥\rho \left(\vartheta \right)∥}^e\mathrm{d}\vartheta \right.\hfill \\ & \left.+N_{\Delta }{\int }_0^x{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\Xi ∥{\left(x-\vartheta \right)}^2\right)\right)}^e\mathrm{d}\vartheta \right\}\hfill \\ & \le 5^{e-1}{2}^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{eH-1}\{{∥\rho ∥}_{\mathcal{F}}^e{\int }_0^x{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\Xi ∥{\left(x-\vartheta \right)}^2\right)\right)}^e{L}_{\Delta }\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ & +\frac{x_1^{e+1}{N}_{\Delta }}{e+1}({\mathbb{E}}_{2,2}{\left(∥\Xi ∥x_1^2\right))}^e\}\hfill \end{array}$$

(26)

Moreover, from $\left(\mathbf{S}\mathbf{1}\right)$ and the Hölder inequality, we have

$$\begin{array}{cc}\hfill & {\int }_0^x{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\Xi ∥{\left(x-\vartheta \right)}^2\right)\right)}^e{L}_{\Delta }\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & \le {\left({\int }_0^x{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\Xi ∥{\left(x-\vartheta \right)}^2\right)\right)}^{ep}\mathrm{d}\vartheta \right)}^{\frac{1}{p}}{\left({\int }_0^x{L}_{\Delta }^q\left(\vartheta \right)\mathrm{d}\vartheta \right)}^{\frac{1}{q}}\hfill \\ \hfill & \le {\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_1^2\right)\right)}^e{\left({\int }_0^x{\left(x-\vartheta \right)}^{ep}\mathrm{d}\vartheta \right)}^{\frac{1}{p}}{\left({\int }_0^x{L}_{\Delta }^q\left(\vartheta \right)\mathrm{d}\vartheta \right)}^{\frac{1}{q}}\hfill \\ \hfill & \le \frac{x_1^{e+\frac{1}{p}}}{{\left(ep+1\right)}^{\frac{1}{p}}}{\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_1^2\right)\right)}^e{∥L_{\Delta }∥}_{L^q\left(\mp ,{\mathbb{R}}^{+}\right)}.\hfill \end{array}$$

(27)

Substituting (27) into (26), we obtain

$$\begin{array}{cc}\hfill {\mathbf{I}}_4& \le 5^{e-1}{2}^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{eH-1}\hfill \\ \hfill & \times \left\{\frac{\nearrow x_1^{e+\frac{1}{p}}}{{\left(ep+1\right)}^{\frac{1}{p}}}{\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_1^2\right)\right)}^e{∥L_{\Delta }∥}_{L^q\left(\mp ,{\mathbb{R}}^{+}\right)}+\frac{x_1^{e+1}{N}_{\Delta }}{e+1}{\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_1^2\right)\right)}^e\right\}\hfill \\ \hfill & =2^{e-1}{N}_2\nearrow +\frac{{\left(10\right)}^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{e\left(H+1\right)}{N}_{\Delta }}{e+1}{\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_1^2\right)\right)}^e.\hfill \end{array}$$

Furthermore, using (22), we obtain

$$\begin{array}{cc}\hfill {\mathbf{I}}_5& =5^{e-1}\mathbf{E}{∥{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\mathsf{\Lambda }{u}_z\left(\vartheta \right)\mathrm{d}\vartheta ∥}^e\hfill \\ \hfill & \le 5^{e-1}{∥W_{\omega }^{\mathcal{M}}\left[0,x_1\right]∥}^e\hfill \\ \hfill & \times \left\{{∥{\left({\mathsf{\Gamma }}_{\omega }^{x_1}\right)}^{-1}∥}^e{5}^{e-1}\left[\mathbf{E}{∥z_1∥}^e+{∥{\mathcal{H}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right)∥}^e\mathbf{E}{∥\Pi \left(0\right)∥}^e\right.\right.\hfill \\ \hfill & +{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-\omega \right)\right)∥}^e\mathbf{E}{∥{\Pi }^{\prime }\left(0\right)∥}^e\hfill \\ \hfill & +{∥\Xi ∥}^e\mathbf{E}{∥{\int }_{-\omega }^0{\mathcal{M}}_{\omega }\left(\Xi \left(x_1-2\omega -\vartheta \right)\right)\Pi \left(\vartheta \right)\mathrm{d}\vartheta ∥}^e\hfill \\ \hfill & \left.\left.+\mathbf{E}{∥{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right)∥}^e\right]\right\}\hfill \\ \hfill & \le 5^{2\left(e-1\right)}N{N}_1\left[\mathbf{E}{∥z_1∥}^e+\theta \left(x_1\right)+{\left(\frac{2}{5}\right)}^{e-1}{N}_2\nearrow \right],\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \theta \left(x\right)& :={\left({\mathbb{E}}_2\left(∥\Xi ∥{\left(x-\omega \right)}^2\right)\right)}^e\mathbf{E}{∥\Pi ∥}_C^e+{\left(x{\mathbb{E}}_{2,2}\left(∥\Xi ∥x^2\right)\right)}^e\mathbf{E}{∥{\Pi }^{\prime }∥}_C^e\hfill \\ \hfill & +{∥\Xi ∥}^e{\omega }^e{\left(x{\mathbb{E}}_{2,2}\left(∥\Xi ∥x^2\right)\right)}^e\mathbf{E}{∥\Pi ∥}_C^e\hfill \\ \hfill & +\frac{2^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{e\left(H+1\right)}{N}_{\Delta }}{e+1}{\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_1^2\right)\right)}^e.\hfill \end{array}$$

From ${\mathbf{I}}_1$ to ${\mathbf{I}}_5$, (25) becomes

$$\begin{array}{cc}\hfill & {∥{\mathcal{L}}_{1}z+{\mathcal{L}}_2\rho ∥}_{\mathcal{F}}^e\hfill \\ \hfill & \le 5^{e-1}\left\{{\left({\mathbb{E}}_2\left(∥\Xi ∥{\left(x-\omega \right)}^2\right)\right)}^e\mathbf{E}{∥\Pi ∥}_C^e\right.\hfill \\ \hfill & +{\left(x{\mathbb{E}}_{2,2}\left(∥\Xi ∥x^2\right)\right)}^e\mathbf{E}{∥{\Pi }^{\prime }∥}_C^e\hfill \\ \hfill & +{∥\Xi ∥}^e{\omega }^e\mathbf{E}{∥\Pi ∥}_C^e{\left(x{\mathbb{E}}_{2,2}\left(∥\Xi ∥x^2\right)\right)}^e\hfill \\ \hfill & +{\left(\frac{2}{5}\right)}^{e-1}{N}_2\nearrow +\frac{2^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{e\left(H+1\right)}{N}_{\Delta }}{e+1}{\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_1^2\right)\right)}^e\hfill \\ \hfill & \left.+5^{e-1}N{N}_1\left[\mathbf{E}{∥z_1∥}^e+\theta \left(x_1\right)+{\left(\frac{2}{5}\right)}^{e-1}{N}_2\nearrow \right]\right\}\hfill \\ \hfill & \le 5^{e-1}\left\{\theta \left(x_1\right)\left(1+5^{e-1}N{N}_1\right)\right.\hfill \\ \hfill & \left.+5^{e-1}N{N}_1\mathbf{E}{∥z_1∥}^e+{\left(\frac{2}{5}\right)}^{e-1}\nearrow N_2\left(1+5^{e-1}N{N}_1\right)\right\}.\hfill \end{array}$$

Thus, some $\nearrow$ are sufficiently large, and from (21), we have ${\mathcal{L}}_{1}z+{\mathcal{L}}_2\rho \in {\mathcal{B}}_{\nearrow }$.

Step 2. We prove ${\mathcal{L}}_1:{\mathcal{B}}_{\nearrow }⟶\mathcal{F}$ is a contraction.

Using (22), we obtain

$$\begin{array}{cc}\hfill & \mathbf{E}{∥\left({\mathcal{L}}_{1}z\right)\left(x\right)-\left({\mathcal{L}}_1\rho \right)\left(x\right)∥}^e\hfill \\ \hfill & =\mathbf{E}{∥{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\mathsf{\Lambda }\left[u_z\left(\vartheta \right)-u_{\rho }\left(\vartheta \right)\right]\mathrm{d}\vartheta ∥}^e\hfill \\ \hfill & \le {∥W_{\omega }^{\mathcal{M}}\left[0,x_1\right]∥}^e{∥{\left({\mathsf{\Gamma }}_{\omega }^{x_1}\right)}^{-1}∥}^e\hfill \\ \hfill & \times \mathbf{E}{∥{\int }_0^{x_1}{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\left[\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)-\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\right]\mathrm{d}{Z}_H\left(\vartheta \right)∥}^e\hfill \\ \hfill & \le {\tau }_{e}N{N}_1{\left(2H\right)}^{e/2}{x}_1^{eH-1}\hfill \\ \hfill & \times {\int }_0^x\mathbf{E}{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\left[\Delta \left(\vartheta ,\rho \left(\vartheta \right)\right)-\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\right]∥}_{L_2^0}^e\mathrm{d}\vartheta \hfill \\ \hfill & \le {\tau }_{e}N{N}_1{\left(2H\right)}^{e/2}{x}_1^{eH-1}\mathbf{E}{∥z-\rho ∥}_{\mathcal{F}}^e{\int }_0^x{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left({∥\Xi ∥}^e{\left(x-\vartheta \right)}^2\right)\right)}^e{L}_{\Delta }\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & \le {\tau }_{e}N{N}_1\frac{{\left(2H\right)}^{e/2}{x}_1^{e\left(H+1\right)-\frac{1}{q}}}{{\left(ep+1\right)}^{\frac{1}{p}}}{\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_1^2\right)\right)}^e{∥L_{\Delta }∥}_{L^q\left(\mp ,{\mathbb{R}}^{+}\right)}\mathbf{E}{∥z-\rho ∥}_{\mathcal{F}}^e\hfill \\ \hfill & \le \frac{N{N}_1{N}_2}{5^{e-1}}\mathbf{E}{∥z-\rho ∥}_{\mathcal{F}}^e\hfill \\ \hfill & \le \mu \mathbf{E}{∥z-\rho ∥}_{\mathcal{F}}^e,\hfill \end{array}$$

for each $x\in \mp$ and z, $\rho \in {\mathcal{B}}_{\nearrow }$, where $\mu :=N{N}_1{N}_2/5^{e-1}$. We may deduce from (21), noting $\mu <1$, that ${\mathcal{L}}_1$ is a contraction mapping.

Step 3. We prove ${\mathcal{L}}_2:{\mathcal{B}}_{\nearrow }⟶\mathcal{F}$ is a continuous compact operator.

Firstly, we show that ${\mathcal{L}}_2$ is continuous. Let $\left\{z_n\right\}$ be a sequence such that $z_n⟶z$ as $n⟶\infty$ in ${\mathcal{B}}_{\nearrow }$. Hence, using (24) and Lebesgue’s dominated convergence theorem, we obtain

$$\begin{array}{cc}\hfill & \mathbf{E}{∥\left({\mathcal{L}}_2{z}_n\right)\left(x\right)-\left({\mathcal{L}}_{2}z\right)\left(x\right)∥}^e\hfill \\ \hfill & \le {\tau }_e{\left(2H\right)}^{e/2}{x}_1^{eH-1}{\int }_0^x{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)∥}^e\mathbf{E}{∥\Delta \left(\vartheta ,z_n\left(\vartheta \right)\right)-\Delta \left(\vartheta ,z\left(\vartheta \right)\right)∥}_{L_2^0}^e\mathrm{d}\vartheta \hfill \\ \hfill & \le {\tau }_e{\left(2H\right)}^{e/2}{x}_1^{eH-1}{\int }_0^x{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\Xi ∥{\left(x-\vartheta \right)}^2\right)\right)}^e{L}_{\Delta }\left(\vartheta \right)\hfill \\ \hfill & \mathbf{E}{∥z_n\left(\vartheta \right)-z\left(\vartheta \right)∥}^e\mathrm{d}\vartheta ⟶0,\mathrm{as}n⟶\infty ,\hfill \end{array}$$

for each $x\in \mp$. Thus ${\mathcal{L}}_2:{\mathcal{B}}_{\nearrow }⟶\mathcal{F}$ is continuous.

After that, we prove that ${\mathcal{L}}_2$ is uniformly bounded on ${\mathcal{B}}_{\nearrow }$. For each $x\in \mp$, $z\in {\mathcal{B}}_{\nearrow }$, we obtain

$$\begin{array}{cc}\hfill {∥{\mathcal{L}}_{2}z∥}_{\mathcal{F}}^e& =\underset{x\in \mp }{sup}\mathbf{E}{∥\left({\mathcal{L}}_{2}z\right)\left(x\right)∥}^e\hfill \\ \hfill & \le \underset{x\in \mp }{sup}\left\{\mathbf{E}{∥{\int }_0^x{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right)∥}^e\right\}\hfill \\ \hfill & \le {\left(\frac{2}{5}\right)}^{e-1}{N}_2\nearrow +\frac{2^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{e\left(H+1\right)}{N}_{\Delta }}{e+1}{\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_1^2\right)\right)}^e,\hfill \end{array}$$

which leads to ${\mathcal{L}}_2$ being uniformly bounded on ${\mathcal{B}}_{\nearrow }$.

It remains to be shown that ${\mathcal{L}}_2$ is equicontinuous. For each $x_2$, $x_3\in \mp$, $0<x_2<x_3\le x_1$ and $z\in {\mathcal{B}}_{\nearrow }$, using (24), we obtain

$$\begin{array}{cc}\hfill & \left({\mathcal{L}}_{2}z\right)\left(x_3\right)-\left({\mathcal{L}}_{2}z\right)\left(x_2\right)\hfill \\ \hfill & ={\int }_0^{x_3}{\mathcal{M}}_{\omega }\left(\Xi \left(x_3-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right)\hfill \\ \hfill & -{\int }_0^{x_2}{\mathcal{M}}_{\omega }\left(\Xi \left(x_2-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right)\hfill \\ \hfill & ={\mathsf{\Psi }}_1+{\mathsf{\Psi }}_2,\hfill \end{array}$$

where

$${\mathsf{\Psi }}_1={\int }_{x_2}^{x_3}{\mathcal{M}}_{\omega }\left(\Xi \left(x_3-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right),$$

and

$${\mathsf{\Psi }}_2={\int }_0^{x_2}\left[{\mathcal{M}}_{\omega }\left(\Xi \left(x_3-\omega -\vartheta \right)\right)-{\mathcal{M}}_{\omega }\left(\Xi \left(x_2-\omega -\vartheta \right)\right)\right]\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right).$$

Thus,

$$\begin{array}{cc}\hfill \mathbf{E}{∥\left({\mathcal{L}}_{2}z\right)\left(x_3\right)-\left({\mathcal{L}}_{2}z\right)\left(x_2\right)∥}^e& =\mathbf{E}{∥{\mathsf{\Psi }}_1+{\mathsf{\Psi }}_2∥}^e\hfill \\ \hfill & \le 2^{e-1}\left\{\mathbf{E}{∥{\mathsf{\Psi }}_1∥}^e+\mathbf{E}{∥{\mathsf{\Psi }}_2∥}^e\right\}.\hfill \end{array}$$

(28)

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

$$\begin{array}{cc}\hfill \mathbf{E}{∥{\mathsf{\Psi }}_1∥}^e& =\mathbf{E}{∥{\int }_{x_2}^{x_3}{\mathcal{M}}_{\omega }\left(\Xi \left(x_3-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right)∥}^e\hfill \\ \hfill & \le {\tau }_e{\left(2H\right)}^{e/2}{\left(x_3-x_2\right)}^{eH-1}{\int }_{x_2}^{x_3}\mathbf{E}{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)\Delta \left(\vartheta ,z\left(\vartheta \right)\right)∥}_{L_2^0}^e\mathrm{d}\vartheta \hfill \\ \hfill & \le 2^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{\left(x_3-x_2\right)}^{eH-1}\hfill \\ \hfill & \times \left\{{∥z∥}_{\mathcal{F}}^e{\int }_{x_2}^{x_3}{\left(\left(x-\vartheta \right){\mathbb{E}}_{2,2}\left(∥\Xi ∥{\left(x-\vartheta \right)}^2\right)\right)}^e{L}_{\Delta }\left(\vartheta \right)\mathrm{d}\vartheta \right.\hfill \\ \hfill & \left.+\frac{{\left(x_3-x_2\right)}^{e+1}{N}_{\Delta }}{e+1}{\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_3^2\right)\right)}^e\right\}⟶0,\mathrm{as}{x}_2⟶x_3.\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill & \mathbf{E}{∥{\mathsf{\Psi }}_2∥}^e\hfill \\ \hfill & =\mathbf{E}{∥{\int }_0^{x_2}\left[{\mathcal{M}}_{\omega }\left(\Xi \left(x_3-\omega -\vartheta \right)\right)-{\mathcal{M}}_{\omega }\left(\Xi \left(x_2-\omega -\vartheta \right)\right)\right]\Delta \left(\vartheta ,z\left(\vartheta \right)\right)\mathrm{d}{Z}_H\left(\vartheta \right)∥}^e\hfill \\ \hfill & \le {\tau }_e{\left(2H\right)}^{e/2}{x}_2^{eH-1}\hfill \\ \hfill & \times {\int }_0^{x_2}\mathbf{E}{∥\left[{\mathcal{M}}_{\omega }\left(\Xi \left(x_3-\omega -\vartheta \right)\right)-{\mathcal{M}}_{\omega }\left(\Xi \left(x_2-\omega -\vartheta \right)\right)\right]\Delta \left(\vartheta ,z\left(\vartheta \right)\right)∥}_{L_2^0}^e\mathrm{d}\vartheta \hfill \\ \hfill & \le 2^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_2^{eH-1}\hfill \\ \hfill & \times \left\{\nearrow {\int }_0^{x_2}{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x_3-\omega -\vartheta \right)\right)-{\mathcal{M}}_{\omega }\left(\Xi \left(x_2-\omega -\vartheta \right)\right)∥}^e{L}_{\Delta }\left(\vartheta \right)\mathrm{d}\vartheta \right.\hfill \\ \hfill & \left.+N_{\Delta }{\int }_0^{x_2}{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x_3-\omega -\vartheta \right)\right)-{\mathcal{M}}_{\omega }\left(\Xi \left(x_2-\omega -\vartheta \right)\right)∥}^e\mathrm{d}\vartheta \right\}\hfill \\ \hfill & \le 2^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_2^{eH-1}\hfill \\ \hfill & \times \left\{\nearrow {∥L_{\Delta }∥}_{L^q\left(\mp ,{\mathbb{R}}^{+}\right)}\right.\hfill \\ \hfill & \times {\left({\int }_0^{x_2}{\left({∥{\mathcal{M}}_{\omega }\left(\Xi \left(x_3-\omega -\vartheta \right)\right)-{\mathcal{M}}_{\omega }\left(\Xi \left(x_2-\omega -\vartheta \right)\right)∥}^e\right)}^p\right)}^{1/p}\mathrm{d}\vartheta \hfill \\ \hfill & \left.+N_{\Delta }{\int }_0^{x_2}{∥{\mathcal{M}}_{\omega }\left(\Xi \left(x_3-\omega -\vartheta \right)\right)-{\mathcal{M}}_{\omega }\left(\Xi \left(x_2-\omega -\vartheta \right)\right)∥}^e\mathrm{d}\vartheta \right\}\hfill \end{array}$$

From (4), we know that ${\mathcal{M}}_{\omega }\left(\Xi x\right)$ is uniformly continuous for $x\in \mp$. Hence,

$$∥{\mathcal{M}}_{\omega }\left(\Xi \left(x_3-\omega -\vartheta \right)\right)-{\mathcal{M}}_{\omega }\left(\Xi \left(x_2-\omega -\vartheta \right)\right)∥⟶0,\mathrm{as}{x}_2⟶x_3.$$

Therefore, we have $∥{\mathsf{\Psi }}_i∥⟶0$ as $x_2⟶x_3$, $i=1$, 2, which implies that, using (28),

$$\mathbf{E}{∥\left({\mathcal{L}}_{2}z\right)\left(x_3\right)-\left({\mathcal{L}}_{2}z\right)\left(x_2\right)∥}^e⟶0,\mathrm{as}{x}_2⟶x_3,$$

for all $z\in {\mathcal{B}}_{\nearrow }$. As a result, ${\mathcal{L}}_2$ is compact on ${\mathcal{B}}_{\nearrow }$ by applying the Arzelà–Ascoli theorem. Thus, ${\mathcal{L}}_1+{\mathcal{L}}_2$ has a fixed point z on ${\mathcal{B}}_{\nearrow }$ via Krasnoselskii’s fixed point theorem (Lemma 3). Moreover, z is also a solution of (8), and $\left({\mathcal{L}}_{1}z+{\mathcal{L}}_{2}z\right)\left(x_1\right)=z_1$. This indicates that $u_z$ steers the system (8) from $z_0$ to $z_1$ in finite time $x_1$, implying that (8) is controllable on $\mp$. This completes the proof. □

 Corollary 3. 

Theorem 2 is satisfied in the case of $\Xi ={\Xi }^2$ in (8).

 Corollary 4. 

When Ξ is a nonsingular $n\times n$ matrix and $\Xi ={\Xi }^2$ in (8). Theorem 2 coincides with Theorem 3.2 in [16].

Proof. 

Since ${\mathcal{M}}_{\omega }\left({\Xi }^{2}x\right)={\Xi }^{-1}{sin}_{\omega }\left(\Xi x\right)$. From $\left(\mathbf{S}\mathbf{1}\right)$ and the Hölder inequality, we have

$$\begin{array}{cc}\hfill & {\int }_0^x∥{\mathcal{M}}_{\omega }\left({\Xi }^2\left(x-\omega -\vartheta \right)\right)∥L_{\Delta }\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & =∥{\Xi }^{-1}∥{\int }_0^x∥\Xi {\mathcal{M}}_{\omega }\left({\Xi }^2\left(x-\omega -\vartheta \right)\right)∥L_{\Delta }\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & \le ∥{\Xi }^{-1}∥{\int }_0^x∥{sin}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)∥L_{\Delta }\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & \le ∥{\Xi }^{-1}∥{\int }_0^{x}sinh\left(∥\Xi ∥\left(x-\vartheta \right)\right)L_{\Delta }\left(\vartheta \right)\mathrm{d}\vartheta \hfill \\ \hfill & \le ∥{\Xi }^{-1}∥{\left({\int }_0^x{\left(sinh\left(∥\Xi ∥\left(x-\vartheta \right)\right)\right)}^p\mathrm{d}\vartheta \right)}^{\frac{1}{p}}{\left({\int }_0^x{L}_{\Delta }^q\left(\vartheta \right)\mathrm{d}\vartheta \right)}^{\frac{1}{q}}\hfill \\ \hfill & =∥{\Xi }^{-1}∥{\left({\int }_0^x\frac{exp\left(∥\Xi ∥p\left(x-\vartheta \right)\right)}{2^p}\mathrm{d}\vartheta \right)}^{\frac{1}{p}}{\left({\int }_0^x{L}_{\Delta }^q\left(\vartheta \right)\mathrm{d}\vartheta \right)}^{\frac{1}{q}}\hfill \\ \hfill & =∥{\Xi }^{-1}∥{\left(\frac{1}{2^p∥\Xi ∥p}\left(exp\left(∥\Xi ∥px-1\right)\right)\right)}^{\frac{1}{p}}{∥L_{\Delta }∥}_{L^q\left(\mp ,{\mathbb{R}}^{+}\right)}.\hfill \end{array}$$

(29)

and

$$\begin{array}{cc}\hfill & {\int }_0^x∥{\mathcal{M}}_{\omega }\left({\Xi }^2\left(x-\omega -\vartheta \right)\right)∥∥\Delta \left(\vartheta ,0\right)∥\mathrm{d}\vartheta \hfill \\ \hfill & =∥{\Xi }^{-1}∥{\int }_0^x∥{sin}_{\omega }\left(\Xi \left(x-\omega -\vartheta \right)\right)∥∥\Delta \left(\vartheta ,0\right)∥\mathrm{d}\vartheta \hfill \\ \hfill & \le N_{\Delta }∥{\Xi }^{-1}∥{\int }_0^{x}sinh\left(∥\Xi ∥\left(x-\vartheta \right)\right)\mathrm{d}\vartheta \hfill \\ \hfill & =\frac{N_{\Delta }∥{\Xi }^{-1}∥}{∥\Xi ∥}\left[cosh\left(∥\Xi ∥x\right)-1\right].\hfill \end{array}$$

(30)

Similar to how the Theorem 2 was proved at $\Xi ={\Xi }^2$, and as a result of (29) and (30), Theorem 3.2 in [16] yields the same result. This completes the proof. □

 Remark 1. 

We note that Theorem 1 extends Theorem 3.1 in [16] by choosing matrix Ξ, which is an arbitrary, not necessarily squared, matrix, and Corollaries 2 and 4 coincide with Theorems 3.1 and 3.2 in [16]. Therefore, our results in Corollaries 1–4 extend and improve Theorems 3.1 and 3.2 in [16] by removing the condition that Ξ is a nonsingular matrix.

5. An Example

Consider the following linear stochastic delay system:

$$\begin{array}{c}{z}^{″}\left(x\right)+\Xi z\left(x-0.5\right)=\mathsf{\Lambda }u\left(x\right)+\overline{\Delta }\left(x\right)\mathrm{d}{Z}_H\left(x\right),\mathrm{for}x\in \mp :=\left[0,1\right],\\ z\left(x\right)\equiv \Pi \left(x\right),z^{\prime }\left(x\right)\equiv {\Pi }^{\prime }\left(x\right)\mathrm{for}-0.5\le x\le 0,\end{array}$$

(31)

where

$$\Xi =\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),\mathsf{\Lambda }=\left(\begin{array}{c}1\\ 2\end{array}\right),\overline{\Delta }\left(x\right)=\left(\begin{array}{c}\frac{\sqrt{x}{e}^{-x}}{4}\\ \frac{\sqrt{x}{e}^{-x}}{4}\end{array}\right),$$

and

$$\Pi \left(x\right)=\left(\begin{array}{c}2x\\ x\end{array}\right),{\Pi }^{\prime }\left(x\right)=\left(\begin{array}{c}2\\ 1\end{array}\right).$$

Generating the corresponding delay Gramian matrix of (31) using (10), we have

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

where

$$O_1={\int }_0^{0.5}{\mathcal{M}}_{0.5}\left(\Xi \left(0.5-\vartheta \right)\right)\mathsf{\Lambda }{\mathsf{\Lambda }}^T{\mathcal{M}}_{0.5}\left({\Xi }^T\left(0.5-\vartheta \right)\right)\mathrm{d}\vartheta ,$$

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

$$O_2={\int }_{0.5}^1{\mathcal{M}}_{0.5}\left(\Xi \left(0.5-\vartheta \right)\right)\mathsf{\Lambda }{\mathsf{\Lambda }}^T{\mathcal{M}}_{0.5}\left({\Xi }^T\left(0.5-\vartheta \right)\right)\mathrm{d}\vartheta ,$$

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

$${\mathcal{H}}_{0.5}\left(\Xi x\right):=\left\{\begin{array}{cc}\hspace{1em}\mathsf{\Theta },\hfill & -\infty <x<-0.5,\\ \hspace{1em}\mathbb{I},\hfill & -0.5\le x<0,\\ \hspace{1em}\mathbb{I}-\Xi \frac{x^2}{2},\hfill & 0\le x<0.5,\\ \hspace{1em}\mathbb{I}-\Xi \frac{x^2}{2}+{\Xi }^2\frac{{\left(x-0.5\right)}^4}{4!},\hfill & 0.5\le x<1,\end{array}\right.$$

and

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

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. Therefore, by Theorem 1, the system (31) is controllable on $\left[0,1\right]$, which implies that the assumption $\left(\mathbf{S}\mathbf{2}\right)$ is satisfied.

Furthermore, consider the corresponding nonlinear stochastic delay system of (31) as follows:

$$\begin{array}{c}{z}^{″}\left(x\right)+\Xi z\left(x-0.5\right)=\mathsf{\Lambda }u\left(x\right)+\Delta \left(x,z\left(x\right)\right)\mathrm{d}{Z}_H\left(x\right),\mathrm{for}x\in \mp :=\left[0,1\right],\\ z\left(x\right)\equiv \Pi \left(x\right),z^{\prime }\left(x\right)\equiv {\Pi }^{\prime }\left(x\right)\mathrm{for}-0.5\le x\le 0,\end{array}$$

(32)

where

$$\Delta \left(x,z\left(x\right)\right)=\left(\begin{array}{c}\frac{\sqrt{x}{e}^{-x}}{4}{z}_1\left(x\right)\\ \frac{\sqrt{x}{e}^{-x}}{4}{z}_2\left(x\right)\end{array}\right).$$

Next, by selecting $e=p=q=2$, we obtain

$$\begin{array}{cc}\hfill \mathbf{E}{∥\Delta \left(x,z\right)-\Delta \left(x,\rho \right)∥}_{L_2^0}^2& ={\left(\frac{\sqrt{x}{e}^{-x}}{4}\right)}^2\left[{\left(z_1\left(x\right)-{\rho }_1\left(x\right)\right)}^2+{\left(z_2\left(x\right)-{\rho }_2\left(x\right)\right)}^2\right]\hfill \\ \hfill & =\frac{x{e}^{-2x}}{16}{∥z-\rho ∥}_{L_2^0}^2.\hfill \end{array}$$

for all $x\in \mp$, $z\left(x\right)$, and $\rho \left(x\right)\in {\mathbb{R}}^2$. We set $L_{\Delta }\left(x\right)=xexp(-2x)/16$ such that $L_{\Delta }\in L^2\left(\mp ,{\mathbb{R}}^{+}\right)$ in $\left(\mathbf{S}\mathbf{1}\right)$, so we have

$${∥L_{\Delta }∥}_{L^2\left(\mp ,{\mathbb{R}}^{+}\right)}={\left({\int }_0^1{\left[\frac{\vartheta exp(-2\vartheta )}{16}\right]}^2\mathrm{d}\vartheta \right)}^{\frac{1}{2}}=0.00964.$$

Then, by choosing ${\tau }_e=0.23$ and $H=0.75$, we obtain

$$N_2:=\frac{5^{e-1}{\tau }_e{\left(2H\right)}^{e/2}{x}_1^{e\left(H+1\right)-\frac{1}{q}}}{{\left(ep+1\right)}^{\frac{1}{p}}}{\left({\mathbb{E}}_{2,2}\left(∥\Xi ∥x_1^2\right)\right)}^e{∥L_{\Delta }∥}_{L^q\left(\mp ,{\mathbb{R}}^{+}\right)}=0.01.$$

Furthermore, we have

$$\mathbf{E}〈W_{0.5}^{\mathcal{M}}\left[0,1\right]z,z〉=\left(\begin{array}{cc}0.32409z_1^2& 0.65729z_2^2\\ 0.65729z_1^2& 1.3334z_2^2\end{array}\right)\ge \eta \mathbf{E}{∥z∥}^2,$$

where $0<\eta \le 0.32409$, and thus $N_1=3.0856$ and $N=3.98$. Finally, we calculate that

$$N_2\left[1+5^{e-1}N{N}_1\right]=0.62403<1,$$

which implies that all the conditions of Theorem 2 are met. Therefore, the system (32) is controllable.

 Remark 2. 

We see that the controllability of systems (31) and (32) cannot determined using Theorems 3.1 and 3.2 in [16] because the matrix Ξ is singular and the square of matrix Ξ is used in [16] instead of Ξ; furthermore, systems (31) and (32) are considered with matrix Ξ instead of ${\Xi }^2$.

6. Conclusions

In this work, first via a delay Gramian matrix and the solutions of these systems, we developed some sufficient and necessary criteria for the controllability of linear stochastic delay systems. Moreover, by employing Krasnoselskii’s fixed point theorem and the solutions of these systems, we derived some sufficient criteria of controllability of nonlinear stochastic delay systems. As a result, the results apply to any matrix, not necessarily just those that are nonsingular and squared.