Relative Controllability of ψ-Caputo Fractional Neutral Delay Differential System

Abstract

The aim of this work is to analyze the relative controllability and Ulamn–Hyers stability of the $\mathsf{\psi }$-Caputo fractional neutral delay differential system. We use neutral $\mathsf{\psi }$-delayed perturbation of the Mitttag–Leffler matrix function and Banach contraction principle to examine the Ulam–Hyers stability of our considered system. We formulate the Grammian matrix to establish the controllability results of the linear fractonal differential system. Further, we employ the fixed-point technique of Krasnoselskii’s type to establish the sufficient conditions for the relative controllability of a semilinear $\mathsf{\psi }$-Caputo neutral fractional system. Finally, the theoretical study is validated by providing an application.

1. Introduction

The principle of causality has been widely used to evaluate systems in previous applications. A casual system is a system with output and internal states that depends only on the current and previous input values. In fact, many models have been developed to include the effects of past states in their formulations. One approach to incorporate past states is through the use of differential equations. In the late 1930s, Volterra introduced a set of comprehensive differential equations that included past states in his works on viscoelasticity and the predator–prey model [1,2]. Minorskii [3] explored the impact of retroaction mechanisms on automated helm and vessel stability. Such models have a significant impact on the fields of control theory and differential equations with antecedent states. In addition to these traditional approaches, the field of fractional calculus has gained momentum and is now widely used in various domains, including electrochemistry, mathematical physics, biophysics, control theory, electrical engineering, and signal processing [4,5,6,7,8,9]. Fractional calculus extends the notion of traditional calculus by introducing fractional order derivatives and their associated fractional differential equations. These equations have been used to convert integer-order differential equations to fractional-order differential equations. The controllability of fractional differntial systems is concerned with the ability to steer the system from any initial state to any desired final state using a suitable input signal. This concept has been studied extensively for the stability and controllability of differential models. Recently, the controllability of control systems defined by impulsive functional inclusions and neutral integro-differential systems is well discussed in the research publications [10,11,12]. Delay differential systems (DDS) are a class of dynamical systems in which the state variables depend not only on their current values but also on their previous values through a time-delay term. These systems arise naturally in many applications, such as biology, economics, and control systems. The relative controllability of delay differential system is concerned with the ability to control the state of one DDS relative to another delay system, even if neither system is controllable in an absolute sense. Overall, the concept of relative controllability provides a useful framework for analyzing the controllability of DDS and has led to many advances in control theory and its applications. However, in [13], the author gave a solution for the following linear DDS by introducing the exponential delayed matrix.

$$\begin{array}{cc}\hfill {\omega }^{\prime }\left(\varsigma \right)& =A\omega \left(\varsigma \right)+B\omega (\varsigma -h),\varsigma >0,h>0,\hfill \\ \hfill \omega \left(\varsigma \right)& =\varphi \left(\varsigma \right),-h⩽\varsigma ⩽0.\hfill \end{array}$$

(1)

In the articles [14,15], the authors analyzed the following fractional order DDS:

$$\begin{array}{cc}\hfill {}^{C}D_{0^{+}}^{\alpha }\omega \left(\varsigma \right)& =A\omega \left(\varsigma \right)+B\omega (\varsigma -h)+f(\varsigma ,\omega (\varsigma \left)\right),\varsigma >0,h>0,\hfill \\ \hfill \omega \left(\varsigma \right)& =\varphi \left(\varsigma \right),-h⩽\varsigma ⩽0.\hfill \end{array}$$

(2)

The solution to a pure delay fractional differential Equation (2) was initially provided by Li and Wang. By introducing a delayed perturbation Mittag–Leffler matrix function, the solution of the system (2) is obtained by Mahmudov for the non-permutable matrices. Further, this result extended to multi-delay fractional differential equations [16]. The relative controllability of a fractional delay systems was examined by You et al. [17], using a solution that had been previously derived. The findings of the above solution representation have been effectively applied in stability analysis and control problems for time-delay systems [18,19,20,21,22,23,24,25,26,27,28,29,30,31].

On the other hand, Pospíšil and Škripková [32] analyzed the solution of the following neutral delay differential equations with coefficent matrices that are permutable:

$$\begin{array}{cc}\hfill {\omega }^{\prime }\left(\varsigma \right)-E{\omega }^{\prime }(\varsigma -h)& =A\omega \left(\varsigma \right)+B\omega (\varsigma -h)+g\left(\varsigma \right),\varsigma >0,h>0,\hfill \\ \hfill \omega \left(\varsigma \right)& =\varphi \left(\varsigma \right),-h⩽\varsigma ⩽0.\hfill \end{array}$$

(3)

Specifically, for $A=0$, the author [33] conducted a study of the relative controllability of the neutral delay differential system (3) with permutable matrices. You et al. [34] proved the relative controllability of neutral systems (3) by applying Krasnoselskii’s fixed-point theorem. Zhang et al. [35] looked into the representation of the solution to the neutral fractional linear differential system having a constant delay

$$\begin{array}{cc}\hfill {}^{C}D_{0^{+}}^{\alpha }[\omega \left(\varsigma \right)-E\omega (\varsigma -h)]& =A\omega \left(\varsigma \right)+B\omega (\varsigma -h)+g\left(\varsigma \right),\varsigma >0,h>0,\hfill \\ \hfill \omega \left(\varsigma \right)& =\varphi \left(\varsigma \right),-h⩽\varsigma ⩽0,\hfill \end{array}$$

(4)

where $A,B,E\in {\mathbb{M}}_{n\times n}\left(\mathbb{R}\right)$, $\varphi \in C\left([-h,0],{\mathbb{R}}^n\right)$, $f\in C(I\times {\mathbb{R}}^n,{\mathbb{R}}^n)$, $g\in C\left(I,{\mathbb{R}}^n\right)I=[0,T]$, ${}^{C}D_{0^{+}}^{\alpha }$ is the Caputo derivative of order $\alpha \in (0,1)$.

The $\mathsf{\psi }$-Caputo fractional derivative is a generalization of the classical Caputo derivative, which is commonly used in fractional calculus to describe the behavior of the fractional order systems. It has many applications, such as in control systems and signal processing, where the initial state of the system may not be zero. The concept of $\mathsf{\psi }$-Caputo fractional derivatives has many applications in different function spaces and physical meanings [36,37,38,39,40].

Motivated by the above papers, we analyze the following $\mathsf{\psi }$-Caputo neutral fractional delay differential system (NFDDS)

$$\begin{array}{cc}\hfill {}^{C}D_{\mathsf{\psi }\left(\varsigma \right)}^{\alpha }[\omega \left(\varsigma \right)-E\omega (\varsigma -h)]& =A\omega \left(\varsigma \right)+B\omega (\varsigma -h)+Cu\left(\varsigma \right)+f\left(\varsigma \right),T\ge \varsigma >0,h>0\hfill \\ \hfill \omega \left(\varsigma \right)& =\varphi \left(\varsigma \right),-h⩽\varsigma ⩽0.\hfill \end{array}$$

(5)

In the above, we define ${}^C{D}_{\mathsf{\psi }\left(\varsigma \right)}^{\alpha }$ as the $\mathsf{\psi }$-Caputo derivative of fractional order $\alpha \in (0,1)$, $\mathsf{\psi }$ is a function which is increasing and ${\mathsf{\psi }}^{\prime }\left(\varsigma \right)$ exists for all $\varsigma \in [-h,T]$, $\varphi \in C([-h,0],{\mathbb{R}}^n)$, the matrices $A,B,C,E$ are need not be commutative, $u(.)$ derives its value from $L^2(I,{\mathbb{R}}^n)$, $f\in C(I,{\mathbb{R}}^n)$, $I=[0,T],$ where $T=lh$ for a fixed $l\in \mathbb{N}$. Take into account the semilinear $\mathsf{\psi }$-Caputo neutral fractional delayed differential control system as well.

$$\begin{array}{cc}\hfill {}^C{D}_{\mathsf{\psi }\left(\varsigma \right)}^{\alpha }[\omega \left(\varsigma \right)-E\omega (\varsigma -h)]& =A\omega \left(\varsigma \right)+B\omega (\varsigma -h)+Cu\left(\varsigma \right)+f(\varsigma ,\omega (\varsigma \left)\right),\varsigma >0,h>0,\hfill \\ \hfill \omega \left(\varsigma \right)& =\varphi \left(\varsigma \right),-h⩽\varsigma ⩽0,\hfill \end{array}$$

(6)

where $f\in C([0,T]\times {\mathbb{R}}^n,{\mathbb{R}}^n)$ and the remaining are identical to (5).

The most significant contributions are listed below.

• The existence and uniqueness of solution to $\mathsf{\psi }$-Caputo NFDDS is analyzed.
• We investigate the Ulam–Hyers stability of $\mathsf{\psi }$-Caputo NFDDS.
• The neutral $\mathsf{\psi }$—delayed perturbation of Mitttag–Leffler matrix function exhibits the Grammain matrix, based on this matrix we derive the necessary and sufficient condition for the linear system which is relatively controllable.
• Via Krasnoselskii’s fixed-point technique, we scrutinize the relative controllability of the semi-linear $\mathsf{\psi }$-Caputo NFDDS.

Before finding the results, we introduce some notations such as the set of all continuous functions from $[\mathtt{a},\mathtt{b}]$ into ${\mathbb{R}}^n$ is denoted as $C([\mathtt{a},\mathtt{b}],{\mathbb{R}}^n)$, then

$${\parallel f\parallel }_{\infty }=\underset{\varsigma \in [\mathtt{a},\mathtt{b}]}{sup}\parallel f\left(\varsigma \right)\parallel ,$$

where $\parallel .\parallel$ is a norm on ${\mathbb{R}}^n$.

The set of all absolute continuous functions from $[\mathtt{a},\mathtt{b}]$ into ${\mathbb{R}}^n$ is denoted as $AC([\mathtt{a},\mathtt{b}],{\mathbb{R}}^n)$ and

$$A{C}^n[\mathtt{a},\mathtt{b}]=\left\{f:[\mathtt{a},\mathtt{b}]\to \mathbb{C}:f^{n-1}\left(\varsigma \right)\mathrm{exists},f^{n-1}\mathrm{is}\mathrm{absolute}\mathrm{continuous}\right\}.$$

2. Basic Preliminaries

Definition 1

([41]). Let a function f and an increasing function $\mathsf{\psi }$ on $[a,b]$ be integrable and continuously differentiable, respectively, and ${\mathsf{\psi }}^{\prime }\left(\varsigma \right)\ne 0,\varsigma \in [a,b]$, then the $\mathsf{\psi }$-Riemann–Liouville (RL) fractional integrals of f of order $\alpha >0$ is defined as

$$\begin{array}{c}\hfill {}_{{\mathtt{a}}^{+}}^{RL}{I}_{\mathsf{\psi }}^{\alpha }f\left(\varsigma \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_a^{\varsigma }{\left(\mathsf{\psi }\left(\varsigma \right)-\mathsf{\psi }\left(s\right)\right)}^{\alpha -1}f\left(s\right)d\mathsf{\psi }\left(s\right).\end{array}$$

The $\mathsf{\psi }$-RL derivative of f with order $\alpha >0$ is defined as

$$\begin{array}{c}\hfill {}_{{\mathtt{a}}^{+}}^{RL}{D}_{\mathsf{\psi }}^{\alpha }f\left(\varsigma \right):=\frac{1}{\Gamma (n-\alpha )}{\left(\frac{d}{d\mathsf{\psi }\left(\varsigma \right)}\right)}^n{\int }_a^{\varsigma }{(\mathsf{\psi }\left(\varsigma \right)-\mathsf{\psi }\left(s\right))}^{n-\alpha -1}f\left(s\right)d\mathsf{\psi }\left(s\right),\end{array}$$

where $n=\left[\alpha \right]+1$.

Definition 2

([41]). If $f,\mathsf{\psi }\in A{C}^n([a,b],\mathbb{R})$ with $\mathsf{\psi }$ is increasing and ${\mathsf{\psi }}^{\prime }\left(\varsigma \right)\ne 0$ for every $\varsigma \in [a,b]$, then the left $\mathsf{\psi }$-Caputo fractional derivative of f of order α is defined as

$$\begin{array}{c}\hfill {}_{{\mathtt{a}}^{+}}^C{D}_{\mathsf{\psi }}^{\alpha }f\left(\varsigma \right)={}_{{\mathtt{a}}^{+}}^{RL}{I}_{\mathsf{\psi }}^{n-\alpha }{\left(\frac{d}{d\mathsf{\psi }\left(\varsigma \right)}\right)}^{n}f\left(\varsigma \right)\end{array}$$

where $n=\left[\alpha \right]+1,\alpha \in {\mathbb{R}}^{+}$. We use the notation $f_{\mathsf{\psi }}^{\left[n\right]}\left(\varsigma \right)$ for ${\left(\frac{d}{d\mathsf{\psi }\left(\varsigma \right)}\right)}^{n}f\left(\varsigma \right).$

Theorem 1

([41]). Let $f\in A{C}^n([a,b],\mathbb{R})$ and $\alpha \in {\mathbb{R}}^{+}$, then

$$\begin{array}{c}\hfill {}_{{\mathtt{a}}^{+}}^C{D}_{\mathsf{\psi }}^{\alpha }f\left(\varsigma \right){=}_{{\mathtt{a}}^{+}}^{RL}{D}_{\mathsf{\psi }}^{\alpha }\left[f\left(\varsigma \right)-\sum _{i=0}^{n-1}\frac{{(\mathsf{\psi }\left(\varsigma \right)-\mathsf{\psi }\left(\mathtt{a}\right))}^k{f}_{\mathsf{\psi }}^{\left[i\right]}\left(\mathtt{a}\right)}{i!}\right]\end{array}$$

where $\alpha \in {\mathbb{R}}^{+}$.

Lemma 1

([41]). If $R\left(\alpha \right)⩾0$ and $R\left(\beta \right)>0$, then

$${}_{{\mathtt{a}}^{+}}^C{D}_{\mathsf{\psi }}^{\alpha }f\left(\varsigma \right)={}_{-{\mathtt{a}}^{+}}^{RL}{D}_{\mathsf{\psi }}^{\alpha }(f\left(\varsigma \right)-f\left(\mathtt{a}\right)),{}_{{\mathtt{a}}^{+}}^C{D}_{\mathsf{\psi }}^{\alpha }{[\mathsf{\psi }\left(\varsigma \right)]}^{\beta -1}=\frac{\Gamma \left(\beta \right)}{\Gamma (\beta -\alpha )}{(\mathsf{\psi }\left(\varsigma \right))}^{\beta -\alpha -1}$$

Definition 3

([42]). The matrix function $H_{\alpha ,\beta }:\mathbb{R}\to {\mathbb{R}}^n$ defined as

$$\begin{array}{cc}\hfill H_{\alpha ,\beta }\left(\varsigma \right)& ={\varsigma }^{\beta -1}\sum _{i=0}^{\infty }\frac{A^i{\varsigma }^{\alpha i}}{\Gamma (\alpha i+\beta )},\alpha ,\beta >0,\varsigma \in \mathbb{R}\hfill \end{array}$$

is called the Mittag–Leffler-type matrix function of two parameters.

Definition 4

([43]). The matrix function $H_{h,\alpha ,\beta }^B:\mathbb{R}\to {\mathbb{R}}^{n\times n}$ defined by

$$H_{h,\alpha ,\beta }^B\left(\varsigma \right)=\left\{\begin{array}{cc}\Theta ,\hfill & -\infty <\varsigma ⩽-h,\hfill \\ I\frac{{(h+\varsigma )}^{\beta -1}}{\Gamma \left(\beta \right)},\hfill & -h<\varsigma ⩽0,\hfill \\ \sum _{k=0}^p\frac{B^k{(\varsigma -(k-1)h)}^{k\alpha +\beta -1}}{\Gamma (k\alpha +\beta )},\hfill & ph<\varsigma ⩽(p+1)h,\hfill \end{array}\right.$$

is called the delayed Mittag–Leffere type matrix function, where $l\in \mathbb{N}$, Θ is the null matrix, the identity matrix is denoted with the notion I.

Definition 5

([43]). The function $H_{h,\alpha ,\beta }^{A,B}(.):[0,\infty )\to {\mathbb{R}}^n$ is as follows:

$$H_{h,\alpha ,\beta }^{A,B}\left(\varsigma \right)=\left\{\begin{array}{cc}\Theta ,\hfill & -\infty <\varsigma ⩽-h,\hfill \\ I,\hfill & \varsigma =0,\hfill \\ \sum _{i=0}^{\infty }\sum _{k=0}^p{Q}_{i+1}\left(kh\right)\frac{{(\varsigma -kh)}^{i\alpha +\beta -1}}{\Gamma (i\alpha +\beta )},\hfill & ph<\varsigma ⩽(p+1)h,\hfill \end{array}\right.$$

(7)

is called the delayed perturbation Mittag–Leffler-type matrix function, where

$$Q_0\left(s\right)=Q_i(-h)=\Theta ,Q_1\left(0\right)=I,$$

$$Q_{i+1}\left(kh\right)=A{Q}_i\left(kh\right)+B{Q}_i(kh-h),$$

$$fori=0,1,2,\dots ,k=0,1,2,\dots \mathit{and}s=0,h,2h\dots .$$

Definition 6

([43]). The matrix function $H_{h,\alpha ,\beta ,\mathsf{\psi }}^{A,B}(.):\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}^n$ defined by

$$\begin{array}{c}\hfill H_{h,\alpha ,\beta ,\mathsf{\psi }}^{A,B}(\varsigma ,s)=\left\{\begin{array}{cc}\Theta ,\hfill & -h⩽\varsigma -s<0,\hfill \\ I,\hfill & \varsigma =s,\hfill \\ \sum _{i=0}^{\infty }\sum _{k=0}^p{Q}_{i+1}\left(kh\right)\frac{{[\mathsf{\psi }\left(\varsigma \right)-\mathsf{\psi }(s+kh)]}^{i\alpha +\beta -1}}{\Gamma (i\alpha +\beta )},\hfill & ph<\varsigma -s<(p+1)h,\hfill \end{array}\right.\end{array}$$

(8)

is called $\mathsf{\psi }$-delayed perturbation Mittag–Leffler-type matrix function, where

$$Q_0\left(s\right)=Q_i(-h)=\Theta ,Q_1\left(0\right)=I,$$

$$Q_{i+1}\left(kh\right)=A{Q}_i\left(kh\right)+B{Q}_i(kh-h),$$

$$\mathit{for}i=0,1,2,\dots ,k=0,1,2,\dots \mathit{and}s=0,h,2h\dots .$$

Definition 7

([43]). The matrix function $H_{\alpha ,\beta }^{A,B,E}(.):\mathbb{R}\to {\mathbb{R}}^n$ defined by

$$\begin{array}{c}\hfill H_{h,\alpha ,\beta }^{A,B,E}\left(\varsigma \right)=\left\{\begin{array}{c}\Theta ,-\infty <\varsigma ⩽-h,\hfill \\ I,\varsigma =0,\hfill \\ {\displaystyle \sum _{i=0}^{\infty }}{\displaystyle \sum _{nk=0}^p}{Q}_{i+1}\left(kh\right)\frac{{(\varsigma -kh)}^{i\alpha +\beta -1}}{\Gamma (i\alpha +\beta )},ph⩽\varsigma ⩽(p+1)h,\hfill \end{array}\right.\end{array}$$

(9)

is called the neutral delayed perturbation Mittag–Leffler-type matrix function, where

$$Q_0\left(s\right)=Q_i(-h)=\Theta ,Q_1\left(0\right)=I,$$

$$Q_{i+1}\left(kh\right)=A{Q}_i\left(kh\right)+B{Q}_i(kh-h)+E{Q}_{i+1}(kh-h),$$

$$\mathit{for}i=0,1,2,\dots ,k=0,1,2,\dots \mathit{and}s=0,h,2h\dots .$$

Definition 8

([43]). The matrix function $H_{h,\alpha ,\beta ,\mathsf{\psi }}^{A,B,E}(.):\mathbb{R}\to {\mathbb{R}}^n$ defined by

$$\begin{array}{c}\hfill H_{h,\alpha ,\beta ,\mathsf{\psi }}^{A,B,E}\left(\varsigma \right)=\left\{\begin{array}{c}\Theta ,-\infty <\varsigma -s⩽-h,\hfill \\ I,\varsigma -s=0,\hfill \\ {\displaystyle \sum _{i=0}^{\infty }}{\displaystyle \sum _{nk=0}^p}{Q}_{i+1}\left(kh\right)\frac{{(\mathsf{\psi }\left(\varsigma \right)-\mathsf{\psi }\left(kh\right))}^{i\alpha +\beta -1}}{\Gamma (i\alpha +\beta )},ph⩽\varsigma -s⩽(p+1)h,\hfill \end{array}\right.\end{array}$$

(10)

is called the $\mathsf{\psi }$-neutral delayed perturbation Mittag–Leffler-type matrix function. where

$$Q_0\left(s\right)=Q_i(-h)=\Theta ,Q_1\left(0\right)=I$$

$$Q_{i+1}\left(kh\right)=A{Q}_i\left(kh\right)+B{Q}_i(kh-h)+E{Q}_{i+1}(kh-h),$$

$$\mathit{for}i=0,1,2,\dots ,k=0,1,2,\dots \mathit{and}s=0,h,2h\dots .$$

Remark 1.

If $\mathsf{\psi }\left(\varsigma \right)=\varsigma$ and $s=0$ then

• The $\mathsf{\psi }$—neutral delayed perturbation Mittag–Leffler-type matrix function reduces to the Mittag–Leffler-type matrix function [44], i.e., $H_{h,\alpha ,\beta ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)={\varsigma }^{\beta -1}{H}_{\alpha ,\beta }\left(A{\varsigma }^{\alpha }\right)$, when $E=B=0$.
• When $E=0$, the $\mathsf{\psi }$—neutral delayed perturbation Mittag–Leffler-type matrix function corresponds to the delayed perturbation Mittag–Leffler-type matrix function [45].
• For $E=0,A=0,$$H_{\alpha ,\beta ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)$ reduces to delayed Mittag–Leffler-type matrix function [45], where $\mathsf{\psi }\left(\varsigma \right)=\varsigma$.

Mustafa Aydin and Nazim I. Mahmudov investigated the solution of $\mathsf{\psi }$-Caputo neutral linear fractional order system (5); it can be expressed as

$$\begin{array}{cc}\hfill \omega \left(\varsigma \right)& =\left[H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,0)-H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,h)E\right]\phi \left(0\right)+{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)\left[B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)Cu\left(s\right)d\mathsf{\psi }\left(s\right)+{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)f\left(s\right)d\mathsf{\psi }\left(s\right).\hfill \end{array}$$

(11)

Further, the solution of the semi-linear system (6) was stated as

$$\begin{array}{cc}\hfill \omega \left(\varsigma \right)& =\left[H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,0)-H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,h)E\right]\phi \left(0\right)+{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)\left[B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)Cu\left(s\right)d\mathsf{\psi }\left(s\right)+{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right).\hfill \end{array}$$

(12)

3. Existence and Uniqueness

Upon scrutinizing the attributes of each term in system (6), such as the continuity of $f(\varsigma ,\omega (\varsigma \left)\right)$, we can deduce an explicit solution. However, these attributes alone do not suffice to establish explicit solution uniqueness. Hence, we introduce an additional criterion for the continuous function $f(\varsigma ,\omega (\varsigma \left)\right)$ to guarantee the uniqueness. To meet the requirements, it is necessary that $f(\varsigma ,\omega (\varsigma \left)\right)$ satisfies the Lipschitz condition, i.e., that there exists a Lipschitz constant $L_f$ s.t $\parallel f(\varsigma ,\omega \left(\varsigma \right))-f(\varsigma ,u\left(\varsigma \right))\parallel ⩽L_f\parallel \omega \left(\varsigma \right)-u\left(\varsigma \right)\parallel .$

Lemma 2.

Let $H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)$ be as in (8), then

$${\int }_0^{\varsigma }\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)\parallel d\mathsf{\psi }\left(s\right)⩽(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)$$

holds true.

Proof. 

It is sufficient to utilize $[\mathsf{\psi }(\varsigma )-\mathsf{\psi }(s+kh\left)\right]⩽[\mathsf{\psi }(\varsigma )-\mathsf{\psi }(kh\left)\right]$ along with the expansion $H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)$ to determine whether the inequality above is valid. Hence, we leave it for readers to analyze. □

Theorem 2.

For the integral equation expressed in (12), a unique solution in the interval $[-h,T]$ is guaranteed if and only if the function $f(\varsigma ,\omega (\varsigma \left)\right)$ satisfies the condition of type, with a Lipschitz constant $L_f$ that fulfills the inequality

$$L_f(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)<1.$$

Proof. 

Let $G:C([-h,T],{\mathbb{R}}^n)\to C([-h,T],{\mathbb{R}}^n)$ as

$$\begin{array}{cc}\hfill G\omega \left(\varsigma \right)& =\left[H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,0)-H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,h)E\right]\phi \left(0\right)\hfill \\ & +{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right)+{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)\left[B\phi \left(s\right)+E({}^{C}D_{\mathsf{\psi }}^{\alpha }\phi \left(s\right)\right]d\mathsf{\psi }\left(s\right)\hfill \end{array}$$

Given arbitrary functions $\omega$ and v from $C([-h,T],{\mathbb{R}}^n),$ then from Lemma 2, we obtain

$$\begin{array}{cc}\hfill \parallel G\omega \left(\varsigma \right)-Gv\left(\varsigma \right)\parallel & ⩽{\int }_0^{\varsigma }\parallel H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)\parallel \parallel f(s,\omega \left(s\right))-f(s,v\left(s\right))\parallel d\mathsf{\psi }\left(s\right)\hfill \\ & ⩽L_f(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))\parallel \omega -v\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\hfill \end{array}$$

The verification of this theorem assures the contractive nature of G. By applying the Banach contraction principle, it follows that G possesses a unique fixed point on $[-h,T]$, which can be denoted as ${\omega }_0\in C([-h,T],{\mathbb{R}}^n)$ such that ${\omega }_0\left(\varsigma \right)=G{\omega }_0\left(\varsigma \right)$. □

Next, we proced to examine the Ulam–Hyers stability of the system (6).

Definition 9.

Given $ϵ>0$, then (6) is Ulam–Hyers type stable if for all $\omega \in C([0,T],{\mathbb{R}}^n)$ satisfying

$$\begin{array}{c}\hfill \parallel {}^C{D}_{0^{+}}^{\alpha }[\omega \left(\varsigma \right)-E\omega (\varsigma -h)]-A\omega \left(\varsigma \right)-B\omega (\varsigma -h)-Cu\left(\varsigma \right)-f(\varsigma ,\omega \left(\varsigma \right))\parallel ⩽ϵ,\end{array}$$

(13)

there exists a ${\omega }_0\in C([0,T],{\mathbb{R}}^n)$ of (6) and $u_{{}_h}>0$ such that $\parallel \omega \left(\varsigma \right)-{\omega }_0\left(\varsigma \right)\parallel ⩽u_{{}_h}ϵ$ for all $\varsigma \in [0,T]$.

Remark 2.

A function $\omega \in C^1([0,T],{\mathbb{R}}^n)$ is a solution of (13) if and only if there exists $z\in C([0,T],{\mathbb{R}}^n)$ such that

• $\parallel z\left(\varsigma \right)\parallel <ϵ$
• ${}^C{D}_{0^{+}}^{\alpha }[\omega \left(\varsigma \right)-E\omega (\varsigma -h)]=A\omega \left(\varsigma \right)+B\omega (\varsigma -h)+Cu\left(\varsigma \right)+f(\varsigma ,\omega \left(\varsigma \right))+z\left(\varsigma \right)$.

Theorem 3.

The Ulam–Hyers stability of (6) can be established given that all the assertions made in Theorem 2 are true.

Proof. 

Suppose that $\omega$ is a function in the space $C([0,T],{\mathbb{R}}^n)$ that satisfies the inequality (13). Let ${\omega }_0\in C([0,T],{\mathbb{R}}^n)$ be the unique solution of the initial value problem (6), subject to the initial condition ${\omega }_0\left(\varsigma \right)=\omega \left(\varsigma \right)$ for each $\varsigma \in [-h,0]$. Combining Remark 2 and inequality (13), we can conclude that there exists $z\in C([0,T],{\mathbb{R}}^n)$ such that $\parallel z\left(\varsigma \right)\parallel <ϵ$ and

$$\begin{array}{c}\hfill {}^C{D}_{0^{+}}^{\alpha }[\omega \left(\varsigma \right)-E\omega (\varsigma -h)]=A\omega \left(\varsigma \right)+B\omega (\varsigma -h)+Cu\left(\varsigma \right)+f(\varsigma ,\omega \left(\varsigma \right))+z\left(\varsigma \right),\end{array}$$

for all $\varsigma \in [0,T]$. So, using (12) and the function G, we determine

$$\omega \left(\varsigma \right)=G\omega \left(\varsigma \right)+{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)z\left(s\right)d\mathsf{\psi }\left(s\right).$$

Consequently, we can estimate as follows:

$$\begin{array}{cc}\hfill \parallel G\omega \left(\varsigma \right)-\omega \left(\varsigma \right)\parallel & ⩽{\int }_0^{\varsigma }\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)\parallel \parallel z\left(s\right)\parallel d\mathsf{\psi }\left(s\right)\hfill \\ & ⩽(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)ϵ.\hfill \end{array}$$

By Theorem 2,

$$\begin{array}{cc}\hfill \parallel {\omega }_0\left(\varsigma \right)-\omega \left(\varsigma \right)\parallel & ⩽\parallel G{\omega }_0\left(\varsigma \right)-G\omega \left(\varsigma \right)\parallel +\parallel G\omega \left(\varsigma \right)-\omega \left(\varsigma \right)\parallel \hfill \\ & ⩽L_f(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel {\omega }_0-\omega \parallel \hfill \\ & +(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)ϵ.\hfill \end{array}$$

This implies that

$$\begin{array}{c}\hfill \left[1-L_f(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\right]\parallel {\omega }_0-\omega \parallel ⩽(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)ϵ.\end{array}$$

The desired outcome is then achieved.

$\parallel \omega -{\omega }_0\parallel ⩽\eta ϵ$, $\mathrm{where}\eta =\frac{(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)}{\left[1-L_f(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\right]}>0$. □

4. Relative Controllability

We begin this section by examining the relative controllability of (6) under both linearity and semi-linearity. In order to present our main findings, we must first provide some preliminary definitions and notations. We define the norm of the matrix A as $\parallel A\parallel ={max}_{1⩽i⩽n}{\sum }_{j=1}^n\left|a_{ij}\right|$, where $a_{ij}$ denotes the elements of A. Additionally, let $Y_1$, $Y_2$ be the Banach spaces and $B(Y_1,Y_2)$ be any linear bounded operator from $Y_1$ to $Y_2$. If I is a closed bounded interval, then $L^{\infty }(I,Y_2)$ is the Banach space with the norm ${\parallel ·\parallel }_{L^{\infty }(I,Y_2)}$

Definition 10

([46]). For any initial function $\phi \in \left(C^1[-h,0],{\mathbb{R}}^n\right)$, time T and a given final state ${\omega }_{{}_T}\in {\mathbb{R}}^n$, the system (6) is considered relatively controllable if there exists a control $u\in L^{\infty }(I,{\mathbb{R}}^n)$ such that the system (6) has a solution $\omega \in \left(C^1[-h,T],{\mathbb{R}}^n\right)$ satisfying the initial condition φ and $\omega \left(T\right)={\omega }_{{}_T}$.

Lemma 3

([47]). (Krasnoselskii’s fixed-point theorem) Let Y be a Banach space and let $D\subseteq Y$ be a convex, closed, and bounded subset of Y. If $G_1,G_2$ are maps D into Y such that

• $G_{1}z+G_{2}y\in D$$\forall z,y\in D$;
• $G_2$ are continuous and compact;
• $G_1$ is a contractive;

then the equation $G_{1}z+G_{2}z=z$ is of a solution on D.

We explore the system’s relative controllability with linearity and semilinearity in this part.

Case 1. If $f(\varsigma ,\omega \left(\varsigma \right))\equiv {(0,0,\dots 0)}^T=0$, $\varsigma \in I$, then (6) can be reduced as

$$\begin{array}{ccc}\hfill {}^C{D}_{0^{+}}^{\alpha }[\omega \left(\varsigma \right)-E\omega (\varsigma -h)]& =& A\omega \left(\varsigma \right)+B\omega (\varsigma -h)+Cu\left(\varsigma \right),\varsigma >0,h>0,\hfill \\ \hfill \omega \left(\varsigma \right)& =& \varphi \left(\varsigma \right),-h⩽\varsigma ⩽0,\hfill \end{array}$$

(14)

with the solution

$$\begin{array}{cc}\hfill \omega \left(\varsigma \right)& =\left[H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,0)-H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,h)E\right]\phi \left(0\right)+{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)u\left(s\right)d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)\left[B\phi \left(s\right)+E({}^{C}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right).\hfill \end{array}$$

The delayed Grammian matrix $W_{h,\alpha }[0,T]$ is

$$\begin{array}{ccc}\hfill W_{h,\alpha }[0,T]& =& {\int }_0^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s)C{C}^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A^T,B^T,E^T}(T,s)d\mathsf{\psi }\left(s\right).\hfill \end{array}$$

(15)

Theorem 4.

The necessary and sufficient condition for the system (14) to be relatively controllable is that $W_{h,\alpha }[0,T]$ is non-singular.

Proof. 

Sufficient conditions: Since $W_{h,\alpha }[0,T]$ is non-singular, then $W_{h,\alpha }^{-1}[0,T]$ is well-defined. A control function can be chosen as follows:

$$u\left(\varsigma \right)=C^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A^T,B^T,E^T}(T,\varsigma )W_{h,\alpha }^{-1}[0,T]\eta ,$$

where $\eta ={\omega }_{{}_T}-\left[H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(T,0)-H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(T,h)E\right]\phi \left(0\right)+{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s+h)\left[B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right)$.

Then

$$\begin{array}{cc}\hfill \omega \left(T\right)& =\left[H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(T,0)-H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(T,h)E\right]\phi \left(0\right)+{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s+h)\left[B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_0^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s)C{C}^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A^T,B^T,E^T}(T,s)W_{h,\alpha }^{-1}[0,T]\eta d\mathsf{\psi }\left(s\right).\hfill \\ & ={\omega }_{{}_T}.\hfill \end{array}$$

Necessity: Suppose that $W_{h,\alpha }[0,T]$ is singular, that is, that there exists a nonzero state $\overline{h}\in {\mathbb{R}}^n$ such that $W_{h,\alpha }[0,T]\overline{h}=0$.

Subsequently,

$$\begin{array}{cc}\hfill 0& ={\overline{h}}^T{W}_{h,\alpha }[0,T]\overline{h}\hfill \\ & ={\int }_0^T{\overline{h}}^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s)C{C}^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A^T,B^T,E^T}(T,s)\overline{h}d\mathsf{\psi }\left(s\right),\hfill \\ & ={\int }_0^T{\parallel {\overline{h}}^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s)C\parallel }^{2}d\mathsf{\psi }\left(s\right),\hfill \end{array}$$

or

$${\overline{h}}^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s)C=0,0⩽s⩽T.$$

Since the system (14) is relatively controllable, which implies the existence of a control input $u_{{}_1}$ derived as initial state to zero at time T,

$$\begin{array}{cc}\hfill \omega \left(T\right)& =\left[H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(T,0)-H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(T,h)E\right]\phi \left(0\right)+{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s+h)\left[B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_0^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s)C{u}_{{}_1}\left(s\right)d\mathsf{\psi }\left(s\right)=\mathbf{0}.\hfill \end{array}$$

Analogously, there exists a control input $u_{{}_2}$ that can be employed to achieve a nonzero value of $\overline{h}$ at time T starting from the initial state, i.e.,

$$\begin{array}{cc}\hfill \omega \left(T\right)& =\left[H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(T,0)-H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(T,h)E\right]\phi \left(0\right)+{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s+h)\left[B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_0^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s)C{u}_{{}_2}\left(s\right)d\mathsf{\psi }\left(s\right)=\overline{h}.\hfill \end{array}$$

It follows that

$$\overline{h}={\int }_0^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s)C[u_{{}_2}\left(s\right)-u_1\left(s\right)]d\mathsf{\psi }\left(s\right)$$

and

$${\overline{h}}^T\overline{h}={\int }_0^T\overline{h}{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s)C[u_2\left(s\right)-u_1\left(s\right)]d\mathsf{\psi }\left(s\right)=0.$$

Hence, $\overline{h}=0$, which is antithetical to the supposition of $\overline{h}$ being nonzero. As a result, $W_{h,\alpha }[0,T]$ is nonsingular hence proved. □

Case (ii): For $f(\varsigma ,\omega (\varsigma \left)\right)\ne 0,\varsigma \in I=[0,T]$ the solution of the following system:

$$\begin{array}{cc}\hfill {}^{C}D_{0+}^{\alpha }[\omega \left(\varsigma \right)-E\omega (\varsigma -h)]& =A\omega \left(\varsigma \right)+B\omega (\varsigma -h)+Cu\left(\varsigma \right)+f(\varsigma ,\omega (\varsigma \left)\right),\varsigma >0,h>0,\hfill \\ \hfill \omega \left(\varsigma \right)& =\mu \left(\varsigma \right),-h⩽\varsigma ⩽0,\hfill \end{array}$$

(16)

can be expressed as

$$\begin{array}{c}\omega \left(\varsigma \right)=\left[H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,0)-H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,h)E\right]\phi \left(0\right)+{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)\left[B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right)\hfill \\ \hfill +{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)u\left(s\right)d\mathsf{\psi }\left(s\right)+{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right).\end{array}$$

(17)

The following hypothesis is assumed as

 ${\mathbf{H}}_{\mathbf{1}}$

Define a map $W:L^2(I,{\mathbb{R}}^n)$ into ${\mathbb{R}}^n$ by

$$Wu={\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s)Cu\left(s\right)d\mathsf{\psi }\left(s\right),$$

Then, $W^{-1}$ exist and takes values in $L^2(I,{\mathbb{R}}^n)/ker\left(W\right)$.

 ${\mathbf{H}}_{\mathbf{2}}$

The function $f:I\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuous and $L_f(.)\in L^{\infty }(I,{\mathbb{R}}^n)$ such that for arbitrary $\omega ,v\in {\mathbb{R}}^n$

$\parallel f(\varsigma ,\omega \left(\varsigma \right))-f(\varsigma ,z\left(\varsigma \right))\parallel ⩽L_f\parallel \omega \left(\varsigma \right)-z\left(\varsigma \right)\parallel$, $\varsigma \in I.$

Now, let us introduce the following notations: $H= \parallel W^{-1}{\parallel }_{L_b\left({\mathbb{R}}^n,L^2(I,{\mathbb{R}}^n)/ker\left(W\right)\right)}$.

$$\begin{array}{cc}\hfill H_1& =H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel \phi \left(0\right)\parallel +\parallel E\parallel H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,h)\parallel \phi \left(0\right)\parallel \hfill \\ & +{\int }_{-h}^0{H}_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\left[B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right)\hfill \\ & +N_{{}_f}{H}_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right)),\hfill \end{array}$$

$H_2=H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0){\parallel L_f\parallel }_{L^{\infty }(I,{\mathbb{R}}^n)}(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))$.

Theorem 5.

Let α be such that $1>\alpha ⩾0.5,$ and suppose that $\left({\mathbf{H}}_{\mathbf{1}}\right)$ and $\left({\mathbf{H}}_{\mathbf{2}}\right)$ hold. Then, system (16) evinces relative controllability provided that

$$\begin{array}{c}\hfill H_2(1+H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel C\parallel H)<1.\end{array}$$

(18)

Proof. 

From by (${\mathbf{H}}_{\mathbf{1}}$), for every $\omega \in \mathcal{C}=C(I,{\mathbb{R}}^n),$ we establish the control function $u_{\omega }\left(\varsigma \right)$ as

$$\begin{array}{c}{u}_{{}_{\omega }}\left(\varsigma \right)=W^{-1}[{\omega }_{{}_T}-H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(T,0)\phi \left(0\right)-H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(T,0)\phi \left(0\right)E\phi \left(0\right)-{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s+h)\hfill \\ \hfill \left[B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right)-{\int }_0^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(T,s)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right)]\left(\varsigma \right)\end{array}$$

(19)

Define $K:\mathcal{C}⟶\mathcal{C}$ as

$$\begin{array}{cc}\hfill K\omega \left(\varsigma \right)& =\left[H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,0)+E{H}_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,h)\right]\phi \left(0\right)+{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)\left[B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)Cu\left(s\right)d\mathsf{\psi }\left(s\right)+{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right).\hfill \end{array}$$

(20)

Specifically, the mild solution of (16) is represented by the fixed-point $\omega$. The relative controllability of system (16) with (19) is determined by the existence of a solution $\omega$ belonging to $C([-h,T],{\mathbb{R}}^n)$ that fulfills (20) and meets the initial conditions $\omega \left(T\right)={\omega }_{{}_T}$ and $\omega \left(\varsigma \right)=\phi \left(\varsigma \right)$ for $\varsigma \in [-h,T]$, as per the definition.

We know that for $ϵ>0$

$$D_{ϵ}=\left\{\omega \in \mathcal{C}: \parallel \omega \parallel ⩽ϵ\right\}\subseteq \mathcal{C}$$

is a bounded, convex, closed subset of $\mathcal{C}$. This proof is split into three steps in an effort to make it easier to understand.

Step 1: We will prove that there is a positive constant $ϵ>0$ such that $K\left(D_{ϵ}\right)\subseteq D_{ϵ}.$ We may now evaluate $\parallel u_{\omega }\left(\varsigma \right)\parallel$ by utilizing (${\mathbf{H}}_{\mathbf{1}}$), (${\mathbf{H}}_{\mathbf{2}}$) and Holder’s inequality.

$$\begin{array}{cc}\hfill \parallel u_{{}_{\omega }}\left(\varsigma \right)\parallel & ⩽ \parallel W^{-1}\parallel {\displaystyle [}\parallel {\omega }_{{}_T}\parallel +\parallel H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,0)\parallel \parallel \phi \left(0\right)\parallel +\parallel H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,h)\parallel \parallel E\parallel \parallel \phi \left(0\right)\parallel \hfill \\ & +{\int }_{-h}^0\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)\parallel \parallel [B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)]\parallel d\mathsf{\psi }\left(s\right)+{\int }_0^{\varsigma }\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)\parallel \parallel f(s,\omega \left(s\right))\parallel d\mathsf{\psi }(s){\displaystyle ]}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel u_{{}_{\omega }}\left(\varsigma \right)\parallel & ⩽H{\displaystyle [}\parallel {\omega }_{{}_T}\parallel +H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel \phi \left(0\right)\parallel +H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel E\parallel \parallel \phi \left(0\right)\parallel \hfill \\ & +{\int }_{-h}^0\parallel H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel [B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)]\parallel d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_0^T{H}_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel [f(s,\omega \left(s\right))-f(s,0)]\parallel d\mathsf{\psi }\left(s\right)+{\int }_0^T{H}_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel f(s,0)\parallel d\mathsf{\psi }\left(s\right)]\hfill \\ & ⩽H{\displaystyle [}\parallel {\omega }_{{}_T}\parallel +H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel \phi \left(0\right)\parallel +H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel E\parallel \parallel \phi \left(0\right)\parallel \hfill \\ & +{\int }_{-h}^0\parallel H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel [B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)]\parallel d\mathsf{\psi }\left(s\right)\hfill \\ & +H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel L_f{\parallel }_{L^{\infty }(I,{\mathbb{R}}^n)}\parallel \omega \parallel (\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))+N_{{}_f}{H}_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))]\hfill \\ & ⩽H\parallel {\omega }_{{}_T}\parallel +H{H}_1+H{H}_2{\parallel \omega \parallel }_C.\hfill \end{array}$$

To determine $ϵ>0$ such that $K\omega \left(\varsigma \right)\in D_{ϵ}$.

Using $\left({\mathbf{H}}_{\mathbf{1}}\right)$, $\left({\mathbf{H}}_{\mathbf{2}}\right)$ we obtain

$$\begin{array}{cc}\hfill \parallel \left(K\omega \right)\left(\varsigma \right)\parallel & ⩽H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel \phi \left(0\right)\parallel +\parallel H_{h,\alpha ,1,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel E\parallel \parallel \phi \left(0\right)\parallel \hfill \\ & +{\int }_{-h}^0\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)\parallel \parallel [B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)]\parallel d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_0^{\varsigma }\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)\parallel \parallel f(s,\omega \left(s\right))\parallel d\mathsf{\psi }\left(s\right)+{\int }_0^{\varsigma }\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)\parallel \parallel C\parallel \parallel u_{{}_{\omega }}\left(s\right)\parallel d\mathsf{\psi }\left(s\right)\hfill \end{array}$$

or

$$\begin{array}{cc}\hfill \parallel \left(K\omega \right)\left(\varsigma \right)\parallel & ⩽H_1+H_2\parallel \omega \parallel +H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel C\parallel [H\parallel {\omega }_{{}_T}\parallel +H{H}_1+H{H}_2{\parallel \omega \parallel }_{\mathcal{C}}](\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))\hfill \\ \hfill & ⩽\left(1+H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel C\parallel H(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))\right)H_1\hfill \\ & +H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel }(T,0)\parallel C\parallel H\parallel {\omega }_{{}_T}\parallel (\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))\hfill \\ & +H_2\left(1+H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel \parallel C\parallel H(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))\right)ϵ\hfill \\ \hfill & ⩽ϵ\hfill \end{array}$$

for

$$\begin{array}{c}\hfill ϵ=\frac{\left(1+H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel C\parallel H(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))\right)H_1+H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel \mathcal{C}\parallel H\parallel {\omega }_{t_1}\parallel (\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))}{1-H_2\left(1+H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel C\parallel H(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))\right)}.\end{array}$$

Thus, $K\left(B_{ϵ}\right)\subseteq D_{ϵ}$.

Consider the maps $F_1$ and $F_2$ defined on $D_{ϵ}$, which are as follows:

$$\begin{array}{cc}\hfill \left(F_1\omega \right)\left(\varsigma \right)& =\left[H_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,0)+E{H}_{h,\alpha ,1,\mathsf{\psi }}^{A,B,E}(\varsigma ,h)\right]\phi \left(0\right)+{\int }_{-h}^0{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)\left[B\phi \left(s\right)+E({}^{\mathrm{C}}D_{\mathsf{\psi }}^{\alpha }\phi )\left(s\right)\right]d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)C{u}_{\omega }\left(s\right)d\mathsf{\psi }\left(s\right)\hfill \end{array}$$

and

$$\left(F_2\omega \right)\left(\varsigma \right)={\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right)$$

where $\varsigma \in I$.

Step 2:

To prove that $F_1$ is a contraction mapping, take $\omega ,y\in D_{ϵ},$ By $\left({\mathbf{H}}_{\mathbf{1}}\right)$–$\left({\mathbf{H}}_{\mathbf{2}}\right)$, for any $\varsigma \in I$

$$\begin{array}{cc}\hfill \parallel u_{{}_{\omega }}\left(\varsigma \right)-u_{{}_y}\left(\varsigma \right)\parallel & ⩽{\int }_0^T{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel f(s,\omega \left(s\right))-f(s,y\left(s\right))\parallel d\mathsf{\psi }\left(s\right)\hfill \\ & ⩽H{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)L_f\parallel \omega -y\parallel (\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))\hfill \\ \hfill & ⩽H{H}_2\parallel \omega -y\parallel .\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \parallel \left(F_1\omega \right)\left(\varsigma \right)-\left(F_{1}y\right)\left(\varsigma \right)\parallel & ⩽{\int }_0^{\varsigma }\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel \parallel C\parallel \parallel u_{{}_{\omega }}\left(s\right)-u_{{}_y}\left(s\right)\parallel d\mathsf{\psi }\left(s\right)\hfill \\ \hfill & ⩽\parallel C\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)\parallel {\int }_0^{\varsigma }\parallel u_{{}_{\omega }}\left(s\right)-u_{{}_y}\left(s\right)\parallel d\mathsf{\psi }\left(s\right)\hfill \\ \hfill & ⩽\parallel C\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)H{H}_2(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))\parallel u_{\omega }-u_y\parallel \hfill \\ \hfill & ⩽P\parallel u_{\omega }-u_y\parallel ,\hfill \end{array}$$

where

$$P=\parallel C\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)H{H}_2(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right)).$$

In view of (18), we obtain that $P<1$. Thus $F_1$ is a contraction.

Step 3: Next, we demonstrate the compactness and continuity of the operator $F_2$.

Let ${\omega }_n\in D_{ϵ}$ such that ${\omega }_n\to \omega$ in $D_{ϵ}.$

By $\left({\mathbb{H}}_2\right),$

$$f(s,{\omega }_n\left(s\right))\to f(s,\omega \left(s\right))$$

in $\mathcal{C}$ and by using dominant convergence theorem, we obtain

$$\begin{array}{cc}\hfill \parallel \left(F_2{\omega }_n\right)\left(\varsigma \right)-\left(F_2\omega \right)\left(\varsigma \right)\parallel & ⩽{\int }_0^{\varsigma }\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)\parallel \parallel f(s,{\omega }_n\left(s\right))-f(s,\omega \left(s\right))\parallel d\mathsf{\psi }\left(s\right)\hfill \\ \hfill & ⩽{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)L_f\parallel {\omega }_n-\omega \parallel d\mathsf{\psi }\left(s\right)\hfill \\ \hfill & ⩽H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)L_f\parallel {\omega }_n-\omega \parallel (\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))\to 0\mathrm{as}n\to \infty ,\hfill \end{array}$$

thus, $F_2$ is continuous on $D_{ϵ}$

In order to establish $F_2,$ we demonstrate that $F_2\left(D_{ϵ}\right)\in \mathcal{C}$ exhibits equicontinuity and uniform boundedness. Now, we have for every $\omega \in D_{ϵ},0<\varsigma <\varsigma +h⩽T$, we have

$$\begin{array}{cc}\hfill \left(F_2\omega \right)(\varsigma +h)-\left(F_2\omega \right)\left(\varsigma \right)& ={\int }_0^{\varsigma +h}{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right)-{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right),\hfill \\ \hfill & ={\int }_0^{\varsigma +h}{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)f(s,\omega \left(s\right))-{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right)-{\int }_0^{\varsigma }{H}_{h,\alpha ,\alpha }^{A,B,\mathsf{\psi }}(\varsigma ,s)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right),\hfill \\ & ={\int }_{\varsigma }^{\varsigma +h}{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right)\hfill \\ & +{\int }_0^{\varsigma }[H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)-H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)]f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right).\hfill \end{array}$$

Denote

$$W_1={\int }_0^{\varsigma }[H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)-H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)]f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right),$$

$$W_2={\int }_{\varsigma }^{\varsigma +h}{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)f(s,\omega \left(s\right))d\mathsf{\psi }\left(s\right).$$

Now,

$$\parallel \left(F_2\omega \right)(\varsigma +h)-\left(F_2\omega \right)\left(\varsigma \right)\parallel ⩽\parallel W_1\parallel +\parallel W_2\parallel .$$

We will check $\parallel W_i\parallel \to 0$ as $h\to 0,i=1,2.$

$$\begin{array}{cc}\hfill \parallel W_1\parallel & ⩽{\int }_0^{\varsigma }\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)-H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)\parallel \parallel f(s,\omega \left(s\right))\parallel d\mathsf{\psi }\left(s\right)\hfill \\ \hfill & ⩽L_f{\int }_0^{\varsigma }\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)-H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)\parallel \parallel \omega \parallel d\mathsf{\psi }\left(s\right)\hfill \\ & +N_f{\int }_0^{\varsigma }\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s+h)-H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)\parallel d\mathsf{\psi }\left(s\right)\to 0\mathrm{as}h\to 0\hfill \end{array}$$

Similar to that of $\parallel W_2\parallel \to 0$ as $h\to 0.$

Thus,

$$\parallel \left(F_2\omega \right)(\varsigma +h)-\left(F_2\omega \right)\left(\varsigma \right)\parallel \to 0.$$

Consequently, $F_2\left(D_{ϵ}\right)$ is equicontinuous.

Using the aforementioned procedure, we now have

$$\begin{array}{cc}\hfill \parallel \left(F_2\omega \right)\left(\varsigma \right)\parallel & ⩽{\int }_0^{\varsigma }\parallel H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{A,B,E}(\varsigma ,s)\parallel \parallel f(s,\omega \left(s\right))\parallel d\mathsf{\psi }\left(s\right)\hfill \\ \hfill & ⩽H_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)L_f\parallel \omega \parallel (\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right))+N_{{}_g}{H}_{h,\alpha ,\alpha ,\mathsf{\psi }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(T,0)(\mathsf{\psi }\left(T\right)-\mathsf{\psi }\left(0\right)).\hfill \end{array}$$

Hence, $F_2\left(D_{ϵ}\right)$ is bounded and by the Arzela–Ascoli theorem, $F_2\left(D_{ϵ}\right)\subseteq \mathcal{C}$ is relatively compact in $\mathcal{C}.$ Consequently, the operator $F_2$ is both continuous and compact. By virtue of Lemma 3, it follows that F has a fixed-point $\omega$ on $D_{ϵ}$. Evidently, $\omega$ represents the solution of (6) with $\omega \left(T\right)={\omega }_{{}_T}$ and $\omega \left(\varsigma \right)=\varphi \left(\varsigma \right),-h⩽\varsigma ⩽0$ holds by (16). Hence, the proof is complete. □

5. Illustrative Examples

Example 1.

Consider the following neutral $\sqrt{\varsigma }$-Caputo fractional differential system with delay.

$$\begin{array}{ccccc}\hfill & \hfill & \hfill {}_{0^{+}}^C{D}_{\sqrt{\varsigma }}^{\frac{3}{4}}[\omega \left(\varsigma \right)-E\omega (\varsigma -2)]& =A\omega \left(\varsigma \right)+B\omega (\varsigma -0.2)+Cu\left(\varsigma \right)+f(\varsigma ,\omega (\varsigma \left)\right)\hfill & \hfill 0<\varsigma ⩽6,\\ \hfill & \hfill & \hfill \omega \left(\varsigma \right)& =\phi \left(\varsigma \right),\hfill & \hfill -2⩽\varsigma ⩽0,\end{array}$$

(21)

where $A=\left(\begin{array}{cc}0& 0.4\\ 0.9& 0.8\end{array}\right),B=\left(\begin{array}{cc}0.1& 0.7\\ 0.3& 0\end{array}\right),E=\left(\begin{array}{cc}0.3& 0\\ 0.2& 0.4\end{array}\right),C=\left(\begin{array}{cc}1& 0.4\\ 2& 0.9\end{array}\right),$ $\phi \left(\varsigma \right)=\left(\begin{array}{c}1\\ 5\end{array}\right),f(\varsigma ,\omega \left(\varsigma \right))={\left(\begin{array}{c}\frac{ta{n}^{-1}\left(\varsigma \right)}{{\left({\pi }^{2}t\right)}^2}\frac{cos\omega \left(\varsigma \right)}{{\pi }^{6}t}\end{array}\right)}^T.$

Clearly the matrices $A,B,C,E$ satisfy

$$AB\ne BA,AE\ne EA,AC\ne CA,BE\ne EB,EC\ne CE.$$

The solution $\omega \left(\varsigma \right)\in C([-2,6],{\mathbb{R}}^2)$ of the system (21) can be obtained from the Equation (17) and the Grammian matrix $W_{2,\frac{3}{4}}[0,6]$, $W_{2,\frac{3}{4}}^{-1}[0,6]$ is derived from (14). We know that

$$H=\parallel W^{-1}{\parallel }_{L_b({\mathbb{R}}^n,L^2(I,{\mathbb{R}}^n/ker\left(W\right)))}=\sqrt{\parallel W_{2,\frac{3}{4}}^{-1}[0,6]\parallel }.$$

Furthermore, for $\omega ,y\in {\mathbb{R}}^n$

$$\parallel f(\varsigma ,\omega \left(\varsigma \right)-f(\varsigma ,y\left(\varsigma \right))\parallel ⩽L_f\parallel \omega \left(\varsigma \right)-y\left(\varsigma \right)\parallel ,\varsigma \in [0,6],$$

where $L_f\in {\mathbb{R}}^{+}.$ So $\left({\mathbf{H}}_{\mathbf{1}}\right)$–$\left({\mathbf{H}}_{\mathbf{2}}\right)$ are true for the system (21). Moreover, the inequality (18) is satisfied, i.e.,

$$\begin{array}{c}\hfill H_2(1+H_{2,\frac{3}{4},\frac{3}{4},\sqrt{\varsigma }}^{\parallel A\parallel ,\parallel B\parallel ,\parallel E\parallel }(6,0)\parallel C\parallel H)<1.\end{array}$$

Thus, the fulfillment of each condition in Theorem 5 demonstrates that the system (21) is relatively controllable with the use of the control function

$$\begin{array}{c}{u}_{\omega }\left(\varsigma \right)=W^{-1}[{\omega }_{{}_T}-(H_{2,\frac{3}{4},1,\sqrt{\varsigma }}^{A,B,E}(\varsigma ,0)+E{H}_{2,\frac{3}{4},1,\sqrt{\varsigma }}^{A,B,E}(\varsigma ,2)){{\displaystyle (}\begin{array}{c}15\end{array}{\displaystyle )}}^T\hfill \\ \hfill -{\int }_{-2}^0{H}_{2,\frac{3}{4},\frac{3}{4},\sqrt{\varsigma }}^{A,B,E}(\varsigma ,s+2){{\displaystyle (}\begin{array}{c}15\end{array}{\displaystyle )}}^{T}B\frac{1}{2\sqrt{s}}ds-{\int }_0^{\varsigma }{H}_{2,\frac{3}{4},\frac{3}{4},\sqrt{\varsigma }}^{A,B,E}(\varsigma ,s){\left(\begin{array}{c}\frac{{tan}^{-1}\left(s\right)}{{\pi }^{4}s}\frac{cos\omega \left(s\right)}{{\pi }^{6}s}\end{array}\right)}^T\frac{1}{2\sqrt{s}}ds]\left(\varsigma \right).\end{array}$$

6. Conclusions

This paper explores the qualitative notions associated with $\mathsf{\psi }$-Caputo neutral fractional delay differential systems that feature non-commutative coefficient matrices. We analyzed the problem of solution existence, uniqueness, and Ulam–Hyers stability of using the Banach contraction principle applied to the nonlinear $\mathsf{\psi }$-Caputo neutral fractional delay differential system. The neutral $\mathsf{\psi }$-Caputo fractional delayed Grammian matrix is introduced as a means of determining the equivalent condition for relative controllability of neutral delayed homogeneous systems. Additionally, the application of Krasnoselskii’s fixed point theorem is employed to examine the relative controllability outcome of neutral delayed semi-linear systems. We trust that this work will inspire future research in this area. Exploring the approximate controllability and the stability analyses of linear and nonlinear $\mathsf{\psi }$-Caputo NFDDSs with non-commutative coefficient matrices represents one avenue for extending the findings of the presented article.