Approximate Controllability of Fractional Evolution Equations with ψ-Caputo Derivative

Abstract

The primary objective of this study is to investigate the concept of approximate controllability in fractional evolution equations that involve the ψ-Caputo derivative. Specifically, we examine the scenario where the semigroup is compact and analytic. The findings are based on the application of the theory of fractional calculus, semigroup theory, and the fixed-point method, mainly Schauder’s fixed-point theorem. In addition, we assume that the corresponding linear system is approximately controllable. An example is provided to illustrate the obtained theoretical results.

1. Introduction

Controllability is an essential concept in the design and analysis of control systems. It helps engineers determine the appropriate control inputs to achieve the desired behavior of a system. Additionally, controllability analyses can identify potential problems and limitations in a control system and guide the design of more effective control strategies. Approximate controllability is a property of a dynamical system that ensures that the system can be driven to a desired state in a finite amount of time with a sufficiently small control, starting from any initial state. For fractional evolution equations with Caputo derivatives, this property means that a control function u(t) exists that can drive the system from any initial state to a desired state within a finite time interval, with the control function u(t) having a sufficiently small magnitude. The concept of approximate controllability is essential in practical applications where it may not be possible or desirable to achieve exact controllability due to limitations in the control system or the presence of disturbances. Approximate controllability allows for a certain degree of error in the control signal while ensuring that the system can be controlled to within an acceptable level of accuracy. The concept of approximate controllability is particularly useful in situations where the control input is subject to noise or measurement errors. It can also be used in situations where determining the exact control input is difficult or impossible in practice. The study of approximate controllability of fractional evolution equations with Caputo derivatives is an active area of research in the field of control theory, with various methods and techniques being developed to tackle the problem. Controllability problems have attracted a lot of scientific attention, and many contributions have been made in recent years, such as [1,2,3,4,5,6,7,8,9]. Several authors [10,11,12,13,14] have focused effectively on the existence problem and certain types of controllability for systems modeled by non-linear evolution equations using fixed-point theory. Fractional dynamical systems have gained valuable popularity and importance since they have a wide range of applications in many fields, such as mathematics, physics, and various disciplines of engineering (see [15,16,17,18]).

Fractional derivation and integration are sciences cumulatively introduced by a group of researchers. Some of the mathematicians who presented valuable definitions in fractional calculus are Riemann–Liouville (RL), Caputo–Hadamard, Caputo, Hilfer, etc. Almeida [19] gave a generalization of the definition of Caputo-type fractional derivatives, where the author considered the Caputo-type fractional derivative of a function with respect to another function $\mathsf{\psi }$ and studied some helpful properties of fractional calculus. The advantage of the new fractional derivative definition is that it provides better certainty for the model with a suitable function $\mathsf{\psi }$.

Mahmudov and Zorlu [10] studied the approximate controllability of fractional evolution equations with Caputo-type fractional derivatives. The results are obtained with the aid of fractional calculus and semigroup theory, together with Schauder’s theorem, where they assume that the corresponding linear system is A-controllable.

The main aim of this study is to build new sufficient conditions for the approximate controllability of special classes of abstract fractional evolution control equations with the following form:

$$\left\{\begin{array}{l}{{}_0{}^{\mathrm{C}}\mathrm{D}}_{\mathsf{\psi }}^{\mathrm{q}}\mathrm{x}(\mathrm{t})=-\mathrm{A}\mathrm{x}(\mathrm{t})+\mathrm{B}\mathrm{u}(\mathrm{t})+\mathrm{f}(\mathrm{t},\mathrm{x}(\mathrm{t}),\mathrm{G}\mathrm{x}(\mathrm{t})),\mathrm{t}\in \left[0,\mathrm{T}\right]\\ \mathrm{x}(0)={\mathrm{x}}_0\end{array}\right.$$

(1)

where $\mathrm{x}$ is the state variable defined on the Hilbert space $\mathrm{X};{}_0^{\mathrm{C}}{\mathrm{D}}_{\mathsf{\psi }}^{\mathrm{q}}$ describes the $\mathsf{\psi }$-Caputo fractional derivative with order $0<\mathrm{q}<1;\mathrm{A}$ is the infinitesimal generator of a strongly continuous semigroup $\mathsf{\mu }(\mathrm{t})$ of bounded operators on $\mathrm{X}.$ Furthermore, $\mathrm{u}$, the control function, is defined in ${\mathrm{L}}_2(\left[0,\mathrm{T}\right],\mathrm{U})$, where $\mathrm{U}$ is a Hilbert space. ${\mathrm{L}}_2(\left[0,\mathrm{T}\right],\mathrm{U})$ is the space of square integrable functions with respect to the Lebesgue measure on the interval $\left[0,\mathrm{T}\right]$, taking values in the Hilbert space $\mathrm{U}$. In other words, u(t) is a measurable function that satisfies the condition ${\int }_0^{\mathrm{T}}{‖\mathrm{u}(\mathrm{t})‖}_{\mathrm{U}}^2\mathrm{d}\mathrm{t}<\mathrm{\infty }$.$\mathrm{B}$ is a linear bounded operator from $\mathrm{U}$ into ${\mathrm{X}}_{\mathsf{\alpha }};\mathrm{f}$ is a given non-linear term such that $\mathrm{f}\left(\mathrm{t},.,.\right):\left[0,\mathrm{T}\right]\times {\mathrm{X}}_{\mathsf{\alpha }}\times {\mathrm{X}}_{\mathsf{\alpha }}\to {\mathrm{X}}_{\mathsf{\beta }},\mathsf{\beta }\in \left[\mathsf{\alpha },1\right],0<\mathsf{\alpha }<1,\mathrm{T}<\mathrm{\infty },$ and the Volterra integral operator is as follows:

$$\mathrm{G}\mathrm{x}(\mathrm{t})={\int }_0^{\mathrm{t}}\mathrm{K}(\mathrm{t},\mathrm{s})\mathrm{x}(\mathrm{s})\mathrm{d}\mathrm{s}$$

with a kernel of $\mathrm{K}\in \mathrm{C}\left(\mathsf{\Delta },\left.(0,\mathrm{\infty }\right)\right),\mathsf{\Delta }=\left\{\left(\mathrm{t},\mathrm{s}\right):0\le \mathrm{s}\le \mathrm{t}\le \mathrm{T}\right\}$ (see [20,21,22,23,24,25,26,27,28,29]).

As mentioned above, this study is based on the lack of existing research on the approximate controllability of special classes of abstract fractional evolution control equations in the framework of the $\mathsf{\psi }$-Caputo fractional derivative with order $0<\mathrm{q}<1$. Additionally, the presence of the non-linear terms f and the Volterra integral operator $\mathrm{G}\mathrm{x}(\mathrm{t})$ further complicate the analysis of the system. To address these challenges, the study proposes new sufficient conditions for the approximate controllability of the system and utilizes fractional calculus and semigroup theory together with Schauder’s fixed-point theorem. Furthermore, the study aims to demonstrate the practical application of the results through an illustrative example.

The structure of this paper is designed to facilitate the exploration of semilinear fractional control dynamical systems. The second section presents preliminary definitions and results of fractional calculus that will be relevant throughout the study. Section 3 focuses on the theory of approximate controllability conditions for these systems. Finally, the last section provides an illustrative example to demonstrate the practical application of the results.

2. Preliminaries

In this section, we will provide fundamental facts that will be utilized throughout this paper. Unless otherwise specified, the following notations will be used consistently: $\mathrm{X}$ is a Hilbert space with the norm given by $‖ ‖:=\sqrt{⟨,⟩}$. $\mathrm{C}\left(\left[0,\mathrm{T}\right],\mathrm{X}\right)$ denotes the Banach space of continuous functions from $\left[0,\mathrm{T}\right]$ into $\mathrm{X}$, where the norm of $\mathrm{x}$ is given by $‖\mathrm{x}‖={\sup }_{\mathrm{t}\in \left[0,\mathrm{T}\right]}‖\mathrm{x}(\mathrm{t})‖$ for $\mathrm{x}\in \mathrm{C}\left(\left[0,\mathrm{T}\right],\mathrm{X}\right)$. It is assumed that $-\mathrm{A}:\mathrm{D}\left(\mathrm{A}\right)\subset \mathrm{X}\to \mathrm{X}$ is the infinitesimal generator of an analytic compact semigroup $\mathsf{\mu }\left(\mathrm{t}\right),\mathrm{t}\ge 0$ of uniformly linear bounded operators in $\mathrm{X},$ where there exists a constant $\mathrm{M}>1$ such that $‖\mathsf{\mu }\left(\mathrm{t}\right)‖\le \mathrm{M}$ for all $\mathrm{t}\ge 0$. Without loss of generality, we also assume that $0\in \mathsf{\rho }\left(\mathrm{A}\right),$ where $\mathsf{\rho }\left(\mathrm{A}\right)$ is the resolvent set of $\mathrm{A}.$ Thus, for any $\mathsf{\alpha }>0,$ we can define ${\mathrm{A}}^{-\mathsf{\alpha }}$ as follows:

$${\mathrm{A}}^{-\mathsf{\alpha }}:=\frac{1}{\mathsf{\Gamma }\left(\mathsf{\alpha }\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{t}}^{\mathsf{\alpha }-1}\mathsf{\mu }\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}$$

Additionally, we define $-\mathrm{A}$ by the following:

$$-\mathrm{A}:=\underset{\mathrm{t}\to 0^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\mathsf{\mu }\left(\mathrm{t}\right)\mathrm{x}-\mathrm{x}}{\mathrm{x}}, \mathrm{where} \mathrm{D}\left(\mathrm{A}\right):=\left\{\mathrm{x}\in \mathrm{X}:\underset{\mathrm{t}\to 0^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\mathsf{\mu }\left(\mathrm{t}\right)\mathrm{x}-\mathrm{x}}{\mathrm{x}}\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{s}\right\}.$$

The Hilbert space of $\mathrm{D}\left({\mathrm{A}}^{\mathsf{\alpha }}\right)$ is endowed with the norm.

$${‖\mathrm{x}‖}_{\mathsf{\alpha }}:=‖{\mathrm{A}}^{\mathsf{\alpha }}\mathrm{x}‖=\sqrt{⟨{\mathrm{A}}^{\mathsf{\alpha }}\mathrm{x},{\mathrm{A}}^{\mathsf{\alpha }}\mathrm{x}⟩} \mathrm{for} \mathrm{x}\in \mathrm{D}\left({\mathrm{A}}^{\mathsf{\alpha }}\right)$$

is denoted by ${\mathrm{X}}_{\mathsf{\alpha }}$, which coincides with the graph norm of ${\mathrm{A}}_{\mathsf{\alpha }}.$ Hence, we have ${\mathrm{X}}_{\mathsf{\beta }}\mapsto {\mathrm{X}}_{\mathsf{\alpha }}$ for $0\le \mathsf{\alpha }\le \mathsf{\beta }$ (with ${\mathrm{X}}_0=\mathrm{X}$), implying the continuity of the embedding [30,31,32,33,34,35,36].

Definition 1.

[19] ($\psi$-Riemann–Liouville Fractional Integral)

Let $q>0, f$ be an integrable function on $\left[a,b\right]$ and $\psi \in C^1\left(\left[a,b\right]\right)$ be an increasing function with ${\psi }^{\prime }\left(t\right)\ne 0,$ $\forall$ $t\in \left[a,b\right]$. The $\psi$-Riemann–Liouville fractional integral operator of order $q$ of a function $f$ is defined as follows:

$$\mathrm{a}{\mathrm{I}}_{\mathsf{\psi }}^{\mathrm{q}}\mathrm{f}(\mathrm{t})=\frac{1}{\mathsf{\Gamma }(\mathrm{q})}{\int }_{\mathrm{a}}^{\mathrm{t}}(\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(\mathrm{s}){)}^{\mathrm{q}-1}\mathrm{f}(\mathrm{s}){\mathsf{\psi }}^{\prime }(\mathrm{s})\mathrm{d}\mathrm{s}$$

(2)

Indeed, when $\psi (t)=t$, Equation (2) reduces to the classical Riemann–Liouville fractional integral.

Definition 2.

[35] ($\psi$-Riemann–Liouville Fractional Derivative)

Let $n-1<q<n, f$ be an integrable function defined on $\left[a,b\right]$, and $\psi \in C^1\left(\left[a,b\right]\right)$ be an increasing function with ${\psi }^{\prime }\left(t\right)\ne 0,$ $\forall$ $t\in \left[a,b\right].$ The $\psi$-Riemann–Liouville fractional derivative of order $q$ of a function $f$ is defined as follows:

$$\mathrm{a}{\mathrm{D}}_{\mathsf{\psi }}^{\mathrm{q}}\mathrm{f}(\mathrm{t})=(\frac{1}{{\mathsf{\psi }}^{\prime }\left(\mathrm{t}\right)}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{{)}^{\mathrm{n}}}_{\mathrm{a}}{\mathrm{I}}_{\mathsf{\psi }}^{\mathrm{n}-\mathrm{q}}\mathrm{f}(\mathrm{t})=\frac{(\frac{1}{{\mathsf{\psi }}^{\prime }(\mathrm{t})}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{)}^{\mathrm{n}}}{\mathsf{\Gamma }(\mathrm{n}-\mathrm{q})}{\int }_{\mathrm{a}}^{\mathrm{t}}(\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(\mathrm{s}){)}^{\mathrm{n}-\mathrm{q}-1}\mathrm{f}(\mathrm{s}){\mathsf{\psi }}^{\prime }(\mathrm{s})\mathrm{d}\mathrm{s}$$

(3)

where $n=\left[q\right]+1$.

Definition 3.

[19] ($\psi$-Caputo Fractional Derivative)

Let $n-1<q<n, f\in C^n(\left[a,b\right])$ and $\psi \in C^n(\left[a,b\right])$ be an increasing function with ${\psi }^{\prime }(t)\ne 0$ $\forall t\in \left[a,b\right].$ The $\psi$-Caputo fractional derivative of order $q$ of a function $f$ is defined by the following:

$${{}_{\mathrm{a}}{}^{\mathrm{c}}\mathrm{D}}_{\mathsf{\psi }}^{\mathrm{q}}\mathrm{f}\left(\mathrm{t}\right){=}_{\mathrm{a}}{\mathrm{I}}_{\mathsf{\psi }}^{\mathrm{n}-\mathrm{q}}{\mathrm{f}}^{\left[\mathrm{n}\right]}\left(\mathrm{t}\right)=\frac{1}{\mathsf{\Gamma }(\mathrm{n}-\mathrm{q})}{\int }_{\mathrm{a}}^{\mathrm{t}}(\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(\mathrm{s}){)}^{\mathrm{n}-\mathrm{q}-1}\mathrm{f}(\mathrm{s}){\mathsf{\psi }}^{\prime }(\mathrm{s})\mathrm{d}\mathrm{s}$$

(4)

where $n=\left[q\right]+1$ and $f^{\left[n\right]}(t):=(\frac{1}{{\psi }^{\prime }(t)}\frac{d}{dt}{)}^{n}f(t)$ on $\left[a,b\right]$.

According to Definitions 1 and 3, we can rewrite (1) in the following equivalent integral form:

$$\mathrm{x}\left(\mathrm{t}\right)={\mathrm{x}}_0+\frac{1}{\mathsf{\Gamma }(\mathrm{q})}{\int }_0^{\mathrm{t}}(\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(\mathrm{s}){)}^{\mathrm{q}-1}\left[\mathrm{A}\mathrm{x}(\mathrm{s})+\mathrm{B}\mathrm{u}(\mathrm{s})+\mathrm{f}(\mathrm{s},\mathrm{x}(\mathrm{s}),\mathrm{G}\mathrm{x}(\mathrm{s}))\right]{\mathsf{\psi }}^{\prime }(\mathrm{s})\mathrm{d}\mathrm{s},$$

(5)

where $\mathrm{t}\in \left[0,\mathrm{T}\right]$.

Lemma 1.

If (5) holds, then we have the following:

$$\mathrm{x}(\mathrm{t})={\int }_0^{\mathrm{\infty }}{\mathsf{\xi }}_{\mathrm{q}}(\mathsf{\theta })\mathsf{\mu }((\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(0){)}^{\mathrm{q}}\mathsf{\theta }){\mathrm{x}}_0\mathrm{d}\mathsf{\theta }+\mathrm{q}{\int }_0^{\mathrm{t}}{\int }_0^{\mathrm{\infty }}\mathsf{\theta }(\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(\mathrm{s}){)}^{\mathrm{q}-1}{\mathsf{\xi }}_{\mathrm{q}}(\mathsf{\theta })\mathsf{\mu }((\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(0){)}^{\mathrm{q}}\mathsf{\theta })\left[\mathrm{B}\mathrm{u}\left(\mathrm{s}\right)+\mathrm{f}(\mathrm{s},\mathrm{x}(\mathrm{s}),\mathrm{G}\mathrm{x}(\mathrm{s}))\right]{\mathsf{\psi }}^{\prime }(\mathrm{s})\mathrm{d}\mathsf{\theta }\mathrm{d}\mathrm{s}$$

where

$$\begin{array}{c}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)=\frac{1}{\mathrm{q}}{\mathsf{\theta }}^{-1-\frac{1}{\mathrm{q}}}{\mathsf{\varpi }}_{\mathrm{q}}\left({\mathsf{\theta }}^{-\frac{1}{\mathrm{q}}}\right)\ge 0,\\ {\mathsf{\varpi }}_{\mathrm{q}}(\mathsf{\theta })=\frac{1}{\mathsf{\pi }}{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{\infty }}}\left((-1{)}^{\mathrm{n}-1}{\mathsf{\theta }}^{-\mathrm{q}\mathrm{n}-1}\frac{\mathsf{\Gamma }\left(\mathrm{n}\mathrm{q}+1\right)}{\mathrm{n}!}\sin (\mathrm{n}\mathsf{\pi }\mathrm{q})\right),\mathsf{\theta }\in \left(0,\mathrm{\infty }\right).\end{array}$$

Proof. 

Assuming $\mathsf{\gamma }>0,$ we apply Laplace transform [37] to (5) as follows:

$${\mathrm{L}}_{\mathsf{\psi }}\left\{\mathrm{x}\left(\mathrm{t}\right)\right\}\left(\mathsf{\gamma }\right)={\mathrm{L}}_{\mathsf{\psi }}\left\{{\mathrm{x}}_0\right\}\left(\mathsf{\gamma }\right)+{\mathrm{L}}_{\mathsf{\psi }}\left\{\frac{1}{\mathsf{\Gamma }(\mathrm{q})}{\int }_0^{\mathrm{t}}(\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(\mathrm{s}){)}^{\mathrm{q}-1}\left[\mathrm{A}\mathrm{x}\left(\mathrm{s}\right)+\mathrm{B}\mathrm{u}\left(\mathrm{s}\right)+\mathrm{f}(\mathrm{s},\mathrm{x}(\mathrm{s}),\mathrm{G}\mathrm{x}(\mathrm{s}))\right]{\mathsf{\psi }}^{\prime }(\mathrm{s})\mathrm{d}\mathrm{s}\right\}(\mathsf{\gamma })$$

Hence,

$$\begin{array}{ll}\mathrm{X}(\mathsf{\gamma })& ={\mathsf{\gamma }}^{\mathrm{q}-1}({\mathsf{\gamma }}^{\mathrm{q}}\mathrm{I}-\mathrm{A}{)}^{-1}{\mathrm{x}}_0+({\mathsf{\gamma }}^{\mathrm{q}}\mathrm{I}-\mathrm{A}{)}^{-1}\left[\mathrm{B}\mathrm{U}(\mathsf{\gamma })+\mathrm{F}(\mathsf{\gamma })\right]\\ & ={\mathsf{\gamma }}^{\mathrm{q}-1}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-{\mathsf{\gamma }}^{\mathrm{q}}\mathrm{s}}\mathsf{\mu }(\mathrm{s}){\mathrm{x}}_0\mathrm{d}\mathrm{s}+{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-{\mathsf{\gamma }}^{\mathrm{q}}\mathrm{s}}\mathsf{\mu }(\mathrm{s})\left[\mathrm{B}\mathrm{U}(\mathsf{\gamma })+\mathrm{F}(\mathsf{\gamma })\right]\mathrm{d}\mathrm{s}\end{array}$$

Now, let $\mathrm{s}={\mathsf{\tau }}^{\mathrm{q}},\mathrm{d}\mathrm{s}=\mathrm{q}{\mathsf{\tau }}^{\mathrm{q}-1}\mathrm{d}\mathsf{\tau }$. Then

$$\begin{array}{ll}\mathrm{X}\left(\mathsf{\gamma }\right)& ={\mathsf{\gamma }}^{\mathrm{q}-1}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-{\left(\mathsf{\gamma }\mathsf{\tau }\right)}^{\mathrm{q}}}\mathsf{\mu }\left({\mathsf{\tau }}^{\mathrm{q}}\right){\mathrm{x}}_0\mathrm{q}{\mathsf{\tau }}^{\mathrm{q}-1}\mathrm{d}\mathsf{\tau }\\ & +{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-{\left(\mathsf{\gamma }\mathsf{\tau }\right)}^{\mathrm{q}}}\mathsf{\mu }\left({\mathsf{\tau }}^{\mathrm{q}}\right)\left[\mathrm{B}\mathrm{U}\left(\mathsf{\gamma }\right)+\mathrm{F}\left(\mathsf{\gamma }\right)\right]\mathrm{q}{\mathsf{\tau }}^{\mathrm{q}-1}\mathrm{d}\mathsf{\tau }\\ & =\mathrm{q}{{\int }_0^{\mathrm{\infty }}\left(\mathsf{\gamma }\mathsf{\tau }\right)}^{\mathrm{q}-1}{\mathrm{e}}^{-(\mathsf{\gamma }\mathsf{\tau }{)}^{\mathrm{q}}}\mathsf{\mu }\left({\mathsf{\tau }}^{\mathrm{q}}\right){\mathrm{x}}_0\mathrm{d}\mathsf{\tau }\\ & +\mathrm{q}{{\int }_0^{\mathrm{\infty }}\mathsf{\tau }}^{\mathrm{q}-1}{\mathrm{e}}^{-(\mathsf{\gamma }\mathsf{\tau }{)}^{\mathrm{q}}}\mathsf{\mu }({\mathsf{\tau }}^{\mathrm{q}})\left[\mathrm{B}\mathrm{U}(\mathsf{\gamma })+\mathrm{F}(\mathsf{\gamma })\right]\mathrm{d}\mathsf{\tau }\end{array}$$

Taking $\mathsf{\tau }=\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right),\mathrm{d}\mathsf{\tau }={\mathsf{\psi }}^{\prime }(\mathrm{t})\mathrm{d}\mathrm{t},$ we obtain the following:

$$\begin{array}{ll}\mathrm{X}\left(\mathsf{\gamma }\right)& =\mathrm{q}{{\int }_0^{\mathrm{\infty }}\mathsf{\gamma }}^{\mathrm{q}-1}{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}-1}{\mathrm{e}}^{-(\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right){)}^{\mathrm{q}}}\mathsf{\mu }\left({\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}\right){\mathrm{x}}_0{\mathsf{\psi }}^{\prime }\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}\\ & +\mathrm{q}{{\int }_0^{\mathrm{\infty }}\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}-1}{\mathrm{e}}^{-(\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right){)}^{\mathrm{q}}}\mathsf{\mu }\left({\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}\right)\left[\mathrm{B}\mathrm{U}\left(\mathsf{\gamma }\right)+\mathrm{F}\left(\mathsf{\gamma }\right)\right]{\mathsf{\psi }}^{\prime }\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}\\ & ={\int }_0^{\mathrm{\infty }}-\frac{1}{\mathsf{\gamma }}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\left[{\mathrm{e}}^{-(\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right){)}^{\mathrm{q}}}\right]\mathsf{\mu }\left({\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}\right){\mathrm{x}}_0\mathrm{d}\mathrm{t}\\ & +{{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\mathrm{q}\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}-1}{\mathrm{e}}^{-(\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right){)}^{\mathrm{q}}}\mathsf{\mu }\left({\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}\right){\mathrm{e}}^{-\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}\\ & \times \left[\mathrm{B}\mathrm{u}(\mathrm{s})+\mathrm{f}(\mathrm{s},\mathrm{x}(\mathrm{s}),\mathrm{G}\mathrm{x}(\mathrm{s})\right]{\mathsf{\psi }}^{\prime }(\mathrm{s}){\mathsf{\psi }}^{\prime }(\mathrm{t})\mathrm{d}\mathrm{s}\mathrm{d}\mathrm{t}\\ & ={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\mathsf{\theta }{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right){\mathrm{e}}^{-\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}\mathsf{\mu }\left({\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}\right){\mathrm{x}}_0{\mathsf{\psi }}^{\prime }\left(\mathrm{t}\right)\mathrm{d}\mathsf{\theta }\mathrm{d}\mathrm{t}\\ & +{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\mathrm{q}{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}-1}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right){\mathrm{e}}^{-\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)\mathsf{\theta }}\mathsf{\mu }\left({\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}\right){\mathrm{e}}^{-\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}\\ & \times \left[\mathrm{B}\mathrm{u}(\mathrm{s})+\mathrm{f}(\mathrm{s},\mathrm{x}(\mathrm{s}),\mathrm{G}\mathrm{x}(\mathrm{s})\right]{\mathsf{\psi }}^{\prime }(\mathrm{s}){\mathsf{\psi }}^{\prime }(\mathrm{t})\mathrm{d}\mathsf{\theta }\mathrm{d}\mathrm{s}\mathrm{d}\mathrm{t}={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}\left({\int }_0^{\mathrm{\infty }}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)\mathsf{\mu }\left(\frac{{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}}{{\mathsf{\theta }}^{\mathrm{q}}}\right){\mathrm{x}}_0\mathrm{d}\mathsf{\theta }\right){\mathsf{\psi }}^{\prime }\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}\\ & +{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\mathrm{q}{\mathrm{e}}^{-\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)+\mathsf{\psi }\left(\mathrm{t}\right)-2\mathsf{\psi }\left(0\right)\right)}\frac{{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}-1}}{{\mathsf{\theta }}^{\mathrm{q}}}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)\mathsf{\mu }\left(\frac{{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}}{{\mathsf{\theta }}^{\mathrm{q}}}\right)\\ & \times \left[\mathrm{B}\mathrm{u}\left(\mathrm{s}\right)+\mathrm{f}\left(\mathrm{s},\mathrm{x}\left(\mathrm{s}\right),\mathrm{G}\mathrm{x}\left(\mathrm{s}\right)\right)\right]{\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right){\mathsf{\psi }}^{\prime }\left(\mathrm{t}\right)\mathrm{d}\mathsf{\theta }\mathrm{d}\mathrm{s}\mathrm{d}\mathrm{t}={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}\left({\int }_0^{\mathrm{\infty }}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)\mathsf{\mu }\left(\frac{{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}}{{\mathsf{\theta }}^{\mathrm{q}}}\right){\mathrm{x}}_0\mathrm{d}\mathsf{\theta }\right){\mathsf{\psi }}^{\prime }\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}\\ & +{\int }_0^{\mathrm{\infty }}{\int }_{\mathrm{t}}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\mathrm{q}{\mathrm{e}}^{-\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathsf{\tau }\right)-\mathsf{\psi }\left(0\right)\right)}\frac{{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}-1}}{{\mathsf{\theta }}^{\mathrm{q}}}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)\mathsf{\mu }\left(\frac{{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}}{{\mathsf{\theta }}^{\mathrm{q}}}\right)\\ & \times \left[Bu\left({\mathsf{\psi }}^{-1}\left(\mathsf{\psi }\left(\mathsf{\tau }\right)-\mathsf{\psi }\left(\mathrm{t}\right)+\mathsf{\psi }\left(0\right)\right)\right)\right]+\mathrm{f}\left({\mathsf{\psi }}^{-1}\left(\mathsf{\psi }\left(\mathsf{\tau }\right)-\mathsf{\psi }\left(\mathrm{t}\right)+\mathsf{\psi }\left(0\right)\right),\mathrm{x}{\mathsf{\psi }}^{-1}\left(\left(\mathsf{\psi }\left(\mathsf{\tau }\right)-\mathsf{\psi }\left(\mathrm{t}\right)+\mathsf{\psi }\left(0\right)\right)\right),\mathrm{G}\mathrm{x}\left({\mathsf{\psi }}^{-1}\left(\mathsf{\psi }\left(\mathsf{\tau }\right)-\mathsf{\psi }\left(\mathrm{t}\right)+\mathsf{\psi }\left(0\right)\right)\right)\right){\mathsf{\psi }}^{\prime }(\mathsf{\tau }){\mathsf{\psi }}^{\prime }(\mathrm{t})\mathrm{d}\mathsf{\theta }\mathrm{d}\mathsf{\tau }\mathrm{d}\mathrm{t}\end{array}$$

Therefore, we obtain the following:

$$\begin{array}{ll}\mathrm{X}\left(\mathsf{\gamma }\right)=& {\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}\left({\int }_0^{\mathrm{\infty }}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)\mathsf{\mu }\left(\frac{{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}}{{\mathsf{\theta }}^{\mathrm{q}}}\right){\mathrm{x}}_0\mathrm{d}\mathsf{\theta }\right){\mathsf{\psi }}^{\prime }\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}\\ & +{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\mathsf{\gamma }\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}\left({\int }_0^{\mathsf{\tau }}{\int }_0^{\mathrm{\infty }}\mathrm{q}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)\frac{{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}}{{\mathsf{\theta }}^{\mathrm{q}}}\mathsf{\mu }\left(\frac{{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}}}{{\mathsf{\theta }}^{\mathrm{q}}}\right)\left[\mathrm{B}\mathrm{u}\left(\mathrm{s}\right)+\mathrm{f}\left(\mathrm{s},\mathrm{x}\left(\mathrm{s}\right),\mathrm{G}\mathrm{x}\left(\mathrm{s}\right)\right)\right]\mathrm{d}\mathsf{\theta }\mathrm{d}\mathrm{s}\right){\mathsf{\psi }}^{\prime }\left(\mathsf{\tau }\right)\mathrm{d}\mathsf{\tau }.\end{array}$$

Applying the inverse Laplace transform, we reach the following:

$$\begin{array}{ll}\mathrm{x}\left(\mathrm{t}\right)& ={\int }_0^{\mathrm{\infty }}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)\mathsf{\mu }\left(\frac{{\left(\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(0)\right)}^{\mathrm{q}}}{{\mathsf{\theta }}^{\mathrm{q}}}\right){\mathrm{x}}_0\mathrm{d}\mathsf{\theta }\\ & +\mathrm{q}{\int }_0^{\mathrm{t}}{\int }_0^{\mathrm{\infty }}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)\frac{{\left(\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(\mathrm{s})\right)}^{\mathrm{q}-1}}{{\mathsf{\theta }}^{\mathrm{q}}}\mathsf{\mu }\left(\frac{{\left(\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(\mathrm{s})\right)}^{\mathrm{q}}}{{\mathsf{\theta }}^{\mathrm{q}}}\right)\left[\mathrm{B}\mathrm{u}\left(\mathrm{s}\right)+\mathrm{f}\left(\mathrm{s},\mathrm{x}(\mathrm{s}),\mathrm{G}\mathrm{x}(\mathrm{s})\right)\right]\mathrm{d}\mathsf{\theta }\mathrm{d}\mathrm{s}\\ & ={\int }_0^{\mathrm{\infty }}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)\mathsf{\mu }\left({\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}\mathsf{\theta }\right){\mathrm{x}}_0\mathrm{d}\mathsf{\theta }\\ & +\mathrm{q}{\int }_0^{\mathrm{t}}{\int }_0^{\mathrm{\infty }}\mathsf{\theta }{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right){\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}\mathsf{\mu }\left({\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathrm{q}}\mathsf{\theta }\right)\\ & \times \left[\mathrm{B}\mathrm{u}(\mathrm{s}\mathrm{f}\left(\mathrm{s},\mathrm{x}(\mathrm{s}),\mathrm{G}\mathrm{x}(\mathrm{s})\right)\right]{\mathsf{\psi }}^{\prime }(\mathrm{s})\mathrm{d}\mathsf{\theta }\mathrm{d}\mathrm{s}.\end{array}$$

 □

Remark 1.

Here, ${\xi }_q\left(\theta \right)$ is a probability mass function on $\left(0,\infty \right)$, and ${\int }_0^{\infty }{\xi }_q(\theta )d\theta =1$.

For $x\in X$ and $0<q<1$, two families $\left\{U_{\psi }^q\left(t,s\right):0\le s\le t\le T\right\}$ and $\left\{V_{\psi }^q\left(t,s\right):0\le s\le t\le T\right\}$ of operators are defined by the following:

$${\mathrm{U}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{t},\mathrm{s}\right)\mathrm{x}={\int }_0^{\mathrm{\infty }}{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)\mathsf{\mu }\left({\left(\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(\mathrm{s})\right)}^{\mathrm{q}}\mathsf{\theta }\right)\mathrm{x}\mathrm{d}\mathsf{\theta }:\mathrm{X}\times \mathrm{X}\to {\mathrm{X}}_{\mathsf{\alpha }}$$

and

$${\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{t},\mathrm{s}\right)\mathrm{x}=\mathrm{q}{\int }_0^{\mathrm{\infty }}\mathsf{\theta }{\mathsf{\xi }}_{\mathrm{q}}\left(\mathsf{\theta }\right)\mathsf{\mu }\left({\left(\mathsf{\psi }(\mathrm{t})-\mathsf{\psi }(\mathrm{s})\right)}^{\mathrm{q}}\mathsf{\theta }\right)\mathrm{x}\mathrm{d}\mathsf{\theta }:\mathrm{X}\times \mathrm{X}\to {\mathrm{X}}_{\mathsf{\alpha }}$$

respectively.

Lemma 2.

[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] The operators $U_{\psi }^q\left(t,s\right)$ and $V_{\psi }^q\left(t,s\right)$ satisfy the following:

i. 

$\forall$ fixed $t\ge s\ge 0$ and $x\in X_{\alpha },$ the operators $U_{\psi }^q\left(t,s\right)$ and $V_{\psi }^q\left(t,s\right)$ are bounded linear operators, i.e., $\forall$ $x\in X_{\alpha },$

$${‖{\mathrm{U}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{t},\mathrm{s}\right)\mathrm{x}‖}_{\mathsf{\alpha }}\le \mathrm{M}{‖\mathrm{x}‖}_{\mathsf{\alpha }} \mathrm{and} {‖{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{t},\mathrm{s}\right)\mathrm{x}‖}_{\mathsf{\alpha }}\le \frac{\mathrm{M}}{\mathsf{\Gamma }\left(\mathrm{q}\right)}{‖\mathrm{x}‖}_{\mathsf{\alpha }}$$

ii. 

The operators $U_{\psi }^q\left(t,s\right)$ and $V_{\psi }^q\left(t,s\right)$ are strongly continuous $\forall$ $t\ge s\ge 0.$ That is, $\forall$ $x\in X_{\alpha }$ and $0\le s\le t_1\le t_2\le T$, we have the following:

$${‖{\mathrm{U}}_{\mathsf{\psi }}^{\mathrm{q}}\left({\mathrm{t}}_2,\mathrm{s}\right)\mathrm{x}-{\mathrm{U}}_{\mathsf{\psi }}^{\mathrm{q}}\left({\mathrm{t}}_1,\mathrm{s}\right)\mathrm{x}‖}_{\mathsf{\alpha }}\to 0$$

and

$${‖{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left({\mathrm{t}}_2,\mathrm{s}\right)\mathrm{x}-{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left({\mathrm{t}}_1,\mathrm{s}\right)\mathrm{x}‖}_{\mathsf{\alpha }}\to 0, \mathrm{as} {\mathrm{t}}_2\to {\mathrm{t}}_1$$

iii. 

$U_{\psi }^q\left(t,s\right)$ and $V_{\psi }^q\left(t,s\right)$ are compact operators $\forall t,s>0$.

iv. 

If $U_{\psi }^q\left(t,s\right)$ and $V_{\psi }^q\left(t,s\right)$ are strongly continuous compact semigroups of linear bounded operators $\forall$ $t,s>0,$ then $U_{\psi }^q\left(t,s\right)$ and $V_{\psi }^q\left(t,s\right)$ are continuous in the uniform operator topology.

v. 

If $0<q<1$, then

$${{}_0{}^{\mathrm{c}}\mathrm{D}}_{\mathsf{\psi }}^{\mathrm{q}}\left\{{\mathrm{U}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{t},0\right){\mathrm{x}}_0\right\}=\mathrm{A}\left\{{\mathrm{U}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{t},0\right){\mathrm{x}}_0\right\}$$

and

$${{}_0{}^{\mathrm{c}}\mathrm{D}}_{\mathsf{\psi }}^{\mathrm{q}}\left\{{\mathsf{{\rm Y}}}_0^{\mathrm{t}}\left\{\mathrm{g}\left(\mathrm{t}\right)\right\}\right\}+\mathrm{g}\left(\mathrm{t}\right)=\mathrm{A}\left\{{\mathsf{{\rm Y}}}_0^{\mathrm{t}}\left\{\mathrm{g}\left(\mathrm{t}\right)\right\}\right\}+\mathrm{g}\left(\mathrm{t}\right),$$

where

$${\mathsf{{\rm Y}}}_{{\mathrm{t}}_1}^{{\mathrm{t}}_2}\left\{\mathrm{x}\left(\mathrm{t}\right)\right\}:={\int }_{{\mathrm{t}}_1}^{{\mathrm{t}}_2}{\left(\mathsf{\psi }\left({\mathrm{t}}_2\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left({\mathrm{t}}_2,\mathrm{s}\right)\mathrm{x}\left(\mathrm{s}\right){\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s},$$

such that $t_1,t_2\in \left[0,T\right], x,g\in C\left(\left[0,T\right],X_{\alpha }\right).$

Definition 4.

[38,39] A solution $x\left(;x_0,u\right)\in C\left(\left[0,T\right],X_{\alpha }\right)$ is said to be a mild solution of system (1) if $\forall$ $u\in L_2\left(\left[0,T\right],U\right)$ the integral equation

$$\mathrm{x}\left(\mathrm{t}\right)={\mathrm{U}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{t},0\right){\mathrm{x}}_0+{\mathsf{{\rm Y}}}_0^{\mathrm{t}}\left\{\mathrm{B}\mathrm{u}\left(\mathrm{t}\right)+\mathrm{f}\left(\mathrm{t},\mathrm{x}\left(\mathrm{t}\right),\mathrm{G}\mathrm{x}\left(\mathrm{t}\right)\right)\right\}$$

(6)

 is satisfied.

Lemma 3.

[10] (Schauder’s fixed-point theorem). If $K$ is a closed bounded and convex subset of a Banach space $X$ and $F:K\to K$ is completely continuous, then $F$ has a fixed point in $K$.

3. Materials and Methods

In this section, we will introduce and demonstrate new conditions for the approximate controllability of semilinear fractional control differential systems. Our approach involves utilizing Schauder’s fixed-point theorem to prove the existence of a fixed point for the operator ${\mathit{F}}_{\mathit{\epsilon }}$. Additionally, Theorem 3 establishes that, under specific hypotheses, the approximate controllability of the considered fractional system (1) is inferred from the approximate controllability of the associated linear system.

Let $\mathbf{x}\left(\mathbf{T};{\mathbf{x}}_0,\mathbf{u}\right)$ represent the state of problem (1) at the final time $\mathbf{T},$ corresponding to the control $\mathbf{u}$ and the initial value ${\mathbf{x}}_0$. Given that the set $\mathfrak{R}\left(\mathbf{T},{\mathbf{x}}_0\right)=\left\{\mathbf{x}\left(\mathbf{T};{\mathbf{x}}_0,\mathbf{u}\right):\mathbf{u}\in {\mathbf{L}}_2\left(\left[0,\mathbf{T}\right],\mathbf{U}\right)\right\},$ the reachable set of system (1) at the final time $\mathbf{T}$ with its closure in ${\mathbf{X}}_{\mathsf{\alpha }}$ is denoted by $\overline{\mathfrak{R}\left(\mathbf{T},{\mathbf{x}}_0\right)}$.

Definition 5.

The system (1) is said to be A-controllable on the interval $\left[0,\mathit{T}\right]$ if the closure of the reachable set satisfies the relation $\overline{\mathfrak{R}\left(\mathit{T},{\mathit{x}}_0\right)}={\mathit{X}}_{\mathit{\alpha }}$. Namely, given an arbitrary $\mathit{\epsilon }>0,$ it is possible to steer from the initial point ${\mathit{x}}_0$ to a distance $\mathit{\epsilon }$ from all points in the state space ${\mathit{X}}_{\mathit{\alpha }}$ at time $\mathit{T}$.

Let us now consider the linear fractional differential system given by (7).

$$\left\{\begin{array}{l}{{}_0{}^{\mathbf{c}}\mathbf{D}}_{\mathsf{\psi }}^{\mathbf{q}}\mathbf{x}(\mathbf{t})=\mathbf{A}\mathbf{x}(\mathbf{t})+\mathbf{B}\mathbf{u}(\mathbf{t}),\mathbf{t}\in \left[0,\mathbf{T}\right]\\ \mathbf{x}(0)={\mathbf{x}}_0\end{array}\right.$$

(7)

At this stage, it is appropriate to introduce the resolvent and controllability operators belonging to Equation (7) as follows:

$$\mathbf{R}\left(\mathsf{\epsilon },{\mathsf{\Gamma }}_0^{\mathbf{T}}\right)={\left(\mathsf{\epsilon }\mathbf{I}+{\mathsf{\Gamma }}_0^{\mathbf{T}}\right)}^{-1}:{\mathbf{X}}_{\mathsf{\alpha }}\to {\mathbf{X}}_{\mathsf{\alpha }},\mathsf{\epsilon }>0,$$

and

$${\mathsf{\Gamma }}_0^{\mathbf{T}}={\int }_0^{\mathbf{T}}{\left(\mathsf{\psi }\left(\mathbf{T}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left(\mathbf{T},\mathbf{s}\right)\mathbf{B}{\mathbf{B}}^{\mathbf{*}}{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{*}\mathbf{q}}\left(\mathbf{T},\mathbf{s}\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}:{\mathbf{X}}_{\mathsf{\alpha }}\times {\mathbf{X}}_{\mathsf{\alpha }}\to {\mathbf{X}}_{\mathsf{\alpha }},$$

respectively, where ${\mathbf{B}}^{\mathbf{*}}$ stands for the adjoint of $\mathbf{B}$ and ${\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{*}\mathbf{q}}\left(\mathbf{T},\mathbf{s}\right)$ means the adjoint of ${\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left(\mathbf{T},\mathbf{s}\right)$. It is easy to see that the operator ${\mathsf{\Gamma }}_0^{\mathbf{T}}$ is a bounded linear operator.

Theorem 1.

[10] Let $\mathit{Z}$ denote a reflexive separable Banach space and ${\mathit{Z}}^{\mathit{*}}$ represent its dual space. Assume that $\mathit{\Gamma }:{\mathit{Z}}^{\mathit{*}}\to \mathit{Z}$ is symmetric. Then, the following expressions are equivalent:

i. 

$\mathit{\Gamma }:{\mathit{Z}}^{\mathit{*}}\to \mathit{Z}$ is positive, that is, $⟨{\mathit{z}}^{\mathit{*}},\mathit{\Gamma }{\mathit{z}}^{\mathit{*}}⟩>0$, $\forall$ non-zero ${\mathit{z}}^{\mathit{*}}\in {\mathit{Z}}^{\mathit{*}}$.

ii. 

$\forall$ $\mathit{h}\in \mathit{Z},$ ${\mathit{z}}_{\mathit{\epsilon }}\left(\mathit{h}\right)=\mathit{\epsilon }{\left(\mathit{\epsilon }\mathit{I}+\mathit{\Gamma }\mathit{J}\right)}^{-1}\left(\mathit{h}\right)$ strongly converges to zero as $\mathit{\epsilon }\to 0^{+},$ where $\mathit{J}$ is the duality mapping of $\mathit{Z}$ into ${\mathit{Z}}^{\mathit{*}}.$

Lemma 4.

[10] The fractional linear control system (7) is A-controllable on the interval $\left[0,\mathit{T}\right]$ if and only if $\mathit{\epsilon }\mathit{R}\left(\mathit{\epsilon },{\mathit{\Gamma }}_0^{\mathit{T}}\right)\to 0$ as $\mathit{\epsilon }\to 0^{+}$ in the strong operator topology.

Remark 2.

Note that the positivity of ${\mathit{\Gamma }}_0^{\mathit{T}}$ is equivalent to $⟨{\mathit{\Gamma }}_0^{\mathit{T}}\mathit{x},\mathit{x}⟩=0\Rightarrow \mathit{x}=0.$ That is, using the definition $⟨{\mathit{\Gamma }}_0^{\mathit{T}}\mathit{x},\mathit{x}⟩={\int }_0^{\mathit{T}}{\left(\mathit{\psi }\left(\mathit{T}\right)-\mathit{\psi }\left(\mathit{s}\right)\right)}^{\mathit{q}-1}{‖{\mathit{B}}^{\mathit{*}}{\mathit{V}}_{\mathit{\psi }}^{\mathit{*}\mathit{q}}\left(\mathit{T},\mathit{s}\right)\mathit{x}‖}^2\mathit{d}\mathit{s},$ one can easily see that the approximate controllability of the linear control system (7) is equivalent to ${\mathit{B}}^{\mathit{*}}{\mathit{V}}_{\mathit{\psi }}^{\mathit{*}\mathit{q}}\left(\mathit{T},\mathit{s}\right)\mathit{x}=0,0\le \mathit{s}<\mathit{T}\Rightarrow \mathit{x}=0$.

Before proceeding with the main result, it is necessary to present the underlying assumptions as follows:

(H1) $\mathsf{\mu }\left(\mathbf{t}\right),\mathbf{t}>0$ is a compact analytic semigroup and $\underset{\mathbf{t}\to 0^{+}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathsf{\mu }\left(\mathbf{t}\right)=\mathbf{I}$ (the identity operator).

(H2) $\exists$ $\mathsf{\beta }\in \left[\mathsf{\alpha },1\right]$ such that the function $\mathbf{f}:\left[0,\mathbf{T}\right]\times {\mathbf{X}}_{\mathsf{\alpha }}\times {\mathbf{X}}_{\mathsf{\alpha }}\to {\mathbf{X}}_{\mathsf{\beta }}$ satisfies the following properties:

(a)

$\forall$ $\left(\mathbf{x},\mathbf{y}\right)\in {\mathbf{X}}_{\mathsf{\alpha }}\times {\mathbf{X}}_{\mathsf{\alpha }},$ the function $\mathbf{f}\left(.,\mathbf{x},\mathbf{y}\right)$ is strongly measurable.

(b)

$\forall$ $\mathbf{t}\in \left[0,\mathbf{T}\right],$ the function $\mathbf{f}\left(\mathbf{t},,\right):{\mathbf{X}}_{\mathsf{\alpha }}\times {\mathbf{X}}_{\mathsf{\alpha }}\to {\mathbf{X}}_{\mathsf{\beta }}$ is continuous.

(c)

$\forall \mathbf{r}>0,$ $\exists$ a function ${\mathbf{g}}_{\mathbf{r}}\in {\mathbf{L}}^{\mathbf{\infty }}\left(\left[0,\mathbf{T}\right],\left.0,\mathbf{\infty }\right)\right)$ such that

$$\mathbf{sup}\left\{{‖\mathbf{f}\left(\mathbf{t},\mathbf{x},\mathbf{y}\right)‖}_{\mathsf{\beta }}:{‖\mathbf{x}‖}_{\mathsf{\alpha }}\le \mathbf{r},{‖\mathbf{y}‖}_{\mathsf{\alpha }}\le {\mathbf{k}}^{\mathbf{*}}\mathbf{T}\mathbf{r}\right\}\le {\mathbf{g}}_{\mathbf{r}}\left(\mathbf{t}\right),\mathbf{t}\in \left[0,\mathbf{T}\right],$$

and $\exists$ a constant $\mathsf{\omega }>0$ such that

$$\underset{\mathbf{r}\to +\mathbf{\infty }}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathbf{inf}\frac{1}{\mathbf{r}}{\int }_0^{\mathbf{t}}{\left(\mathsf{\psi }\left(\mathbf{t}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}{\mathbf{g}}_{\mathbf{r}}\left(\mathbf{s}\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\le \mathsf{\omega }<+\mathbf{\infty }.$$

where

$${\mathbf{k}}^{\mathbf{*}}:=\mathbf{max}\left\{\mathbf{K}\left(\mathbf{t},\mathbf{s}\right):\left(\mathbf{t},\mathbf{s}\right)\in \mathsf{\Delta }\right\}.$$

$\forall \mathsf{\epsilon }>0$ and $\in {\mathbf{X}}_{\mathsf{\alpha }}$ describe a control function $\mathbf{u}\left(\mathbf{t},\mathbf{x}\right)$ as follows:

$$\begin{array}{ll}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{t},\mathbf{x}\right)& ={\mathbf{B}}^{\mathbf{*}}{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{*}\mathbf{q}}\left(\mathbf{T},\mathbf{t}\right)\mathbf{R}\left(\mathsf{\epsilon },{\mathsf{\Gamma }}_0^{\mathbf{T}}\right)\\ & \times \left(\mathit{h}-{\mathit{U}}_{\mathit{\psi }}^{\mathit{q}}\left(\mathit{T},\mathit{s}\right){\mathit{x}}_0-{\int }_0^{\mathit{T}}{\left(\mathit{\psi }\left(\mathit{T}\right)-\mathit{\psi }\left(\mathit{s}\right)\right)}^{\mathit{q}-1}{\mathit{V}}_{\mathit{\psi }}^{\mathit{q}}\left(\mathit{T},\mathit{s}\right)\mathit{f}\left(\mathit{s},\mathit{x}\left(\mathit{s}\right),\mathit{G}\mathit{x}\left(\mathit{s}\right)\right){\mathit{\psi }}^{\mathbf{\prime }}\left(\mathit{s}\right)\mathit{d}\mathit{s}\right).\end{array}$$

Lemma 5.

Under the hypotheses (H1) and (H2), the following inequalities hold:

$${‖\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{t},\mathbf{x}\right)‖}_{\mathsf{\alpha }}\le \frac{1}{\mathsf{\epsilon }}{\mathbf{L}}_{\mathbf{u}},$$

and

$${\int }_0^{\mathbf{t}}{\left(\mathsf{\psi }\left(\mathbf{t}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}‖{\mathbf{A}}^{\mathsf{\alpha }}\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right)‖{\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\le \frac{1}{\mathsf{\epsilon }}\frac{{\left(\mathsf{\psi }\left(\mathbf{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathbf{q}}}{\mathbf{q}}{\mathbf{L}}_{\mathbf{u}},{‖\mathbf{x}‖}_{\mathsf{\alpha }}\le \mathbf{r}.$$

where

$$\begin{array}{l}{\mathrm{L}}_{\mathrm{u}}:={\mathrm{L}}_{\mathrm{B}}\mathrm{M}\left({\mathrm{C}}_{\mathsf{\alpha }}‖{\mathrm{A}}^{\mathsf{\alpha }}\mathrm{h}‖+\mathrm{M}{\mathrm{C}}_{\mathsf{\alpha }}{‖{\mathrm{x}}_0‖}_{\mathsf{\alpha }}+\frac{\mathrm{M}{\mathrm{C}}_{\mathsf{\beta }}}{\mathsf{\Gamma }\left(\mathrm{q}\right)}{\int }_0^{\mathrm{T}}{\left(\mathsf{\psi }\left(\mathrm{T}\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}{\mathrm{g}}_{\mathrm{r}}\left(\mathrm{s}\right){\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\right),\\ {\mathrm{L}}_{\mathrm{B}}:={‖\mathrm{B}‖}_{\mathsf{\alpha }}\underset{0\le \mathrm{t}\le \mathrm{T}}{\mathrm{s}\mathrm{u}\mathrm{p}}‖{\mathrm{B}}^{\mathrm{*}}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{*}\mathrm{q}}\left(\mathrm{T},\mathrm{t}\right)‖.\end{array}$$

Proof. 

$$\begin{array}{ll}{‖\mathrm{B}{\mathrm{u}}_{\mathsf{\epsilon }}\left(\mathrm{t},\mathrm{x}\right)‖}_{\mathsf{\alpha }}& \le \frac{1}{\mathsf{\epsilon }}‖{\mathrm{A}}^{\mathsf{\alpha }}\mathrm{B}{\mathrm{B}}^{\mathrm{*}}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{*}\mathrm{q}}\left(\mathrm{T},\mathrm{t}\right)‖\\ & \times \left[‖{\mathrm{A}}^{-\mathsf{\alpha }}{\mathrm{A}}^{\mathsf{\alpha }}\mathrm{h}‖+‖{\mathrm{A}}^{-\mathsf{\alpha }}{\mathrm{U}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{T},\mathrm{s}\right){\mathrm{A}}^{\mathsf{\alpha }}{\mathrm{x}}_0‖+‖{\int }_0^{\mathrm{T}}{\left(\mathsf{\psi }\left(\mathrm{T}\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}{\mathrm{A}}^{-\mathsf{\beta }}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{T},\mathrm{s}\right){\mathrm{A}}^{\mathsf{\beta }}\mathrm{f}\left(\mathrm{s},\mathrm{x}\left(\mathrm{s}\right),\mathrm{G}\mathrm{x}\left(\mathrm{s}\right)\right){\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}‖\right]\\ & \le \frac{1}{\mathsf{\epsilon }}‖{\mathrm{A}}^{\mathsf{\alpha }}\mathrm{B}‖\underset{0\le \mathrm{t}\le \mathrm{T}}{\mathrm{s}\mathrm{u}\mathrm{p}}‖{\mathrm{B}}^{\mathrm{*}}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{*}\mathrm{q}}\left(\mathrm{T},\mathrm{t}\right)‖\left[{\mathrm{C}}_{\mathsf{\alpha }}‖{\mathrm{A}}^{\mathsf{\alpha }}\mathrm{h}‖+\mathrm{M}{\mathrm{C}}_{\mathsf{\alpha }}{‖{\mathrm{x}}_0‖}_{\mathsf{\alpha }}+\frac{\mathrm{M}{\mathrm{C}}_{\mathsf{\beta }}}{\mathsf{\Gamma }\left(\mathrm{q}\right)}{\int }_0^{\mathrm{T}}{\left(\mathsf{\psi }\left(\mathrm{T}\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}{\mathrm{g}}_{\mathrm{r}}\left(\mathrm{s}\right){\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\right]\\ & \le \frac{1}{\mathsf{\epsilon }}{\mathrm{L}}_{\mathrm{u}},\end{array}$$

and

$${\int }_0^{\mathbf{t}}{\left(\mathsf{\psi }\left(\mathbf{t}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}‖{\mathbf{A}}^{\mathsf{\alpha }}\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right)‖{\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\le \frac{1}{\mathsf{\epsilon }}\frac{{\left(\mathsf{\psi }\left(\mathbf{t}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathbf{q}}}{\mathbf{q}}{\mathbf{L}}_{\mathbf{u}}.$$

 □

Theorem 2.

Given that hypotheses (H1) and (H2) are fulfilled and

$$\frac{\mathbf{M}\mathsf{\omega }}{\mathsf{\Gamma }\left(\mathbf{q}\right)}\left({\mathbf{C}}_{\mathsf{\beta }-\mathsf{\alpha }}+\frac{1}{\mathsf{\epsilon }}\frac{{\left(\mathsf{\psi }\left(\mathbf{T}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathbf{q}}}{\mathbf{q}}\frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\alpha }}{\mathbf{C}}_{\mathsf{\beta }}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}{\mathbf{L}}_{\mathbf{B}}\right)<1$$

(8)

then the fractional semilinear control system (1) has a mild solution on the interval $\left[0,\mathit{T}\right]$.

Proof. 

$\forall$ $\mathbf{r}>0,$ let ${\mathbf{B}}_{\mathbf{r}}:=\left\{\mathbf{x}\in \mathbf{C}\left(\left[0,\mathbf{T}\right],{\mathbf{X}}_{\mathsf{\alpha }}\right):\mathbf{x}\left(0\right)={\mathbf{x}}_0,{‖\mathbf{x}‖}_{\mathsf{\alpha }}\le \mathbf{r}\right\}.$ Then, ${\mathbf{B}}_{\mathbf{r}}$ is a closed, convex, and bounded subset of the Banach space $\mathbf{C}\left(\left[0,\mathbf{T}\right],{\mathbf{X}}_{\mathsf{\alpha }}\right).$ For $\mathsf{\epsilon }>0,$ we describe the operator ${\mathbf{F}}_{\mathsf{\epsilon }}:\mathbf{C}\left(\left[0,\mathbf{T}\right],{\mathbf{X}}_{\mathsf{\alpha }}\right)\to \mathbf{C}\left(\left[0,\mathbf{T}\right],{\mathbf{X}}_{\mathsf{\alpha }}\right)$ as follows:

$$\left({\mathbf{F}}_{\mathsf{\epsilon }}\mathbf{x}\right)\left(\mathbf{t}\right)={\mathbf{z}}_{\mathsf{\epsilon }}\left(\mathbf{t}\right)$$

where

$${\mathbf{z}}_{\mathsf{\epsilon }}\left(\mathbf{t}\right)={\mathbf{U}}_{\mathsf{\psi }}^{\mathbf{q}}\left(\mathbf{t},0\right){\mathbf{x}}_0+{\mathsf{{\rm Y}}}_0^{\mathbf{t}}\left\{\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{t},\mathbf{x}\right)+\mathbf{f}\left(\mathbf{t},\mathbf{x}\left(\mathbf{t}\right),\mathbf{G}\mathbf{x}\left(\mathbf{t}\right)\right)\right\}.$$

(9)

It is required to prove that $\forall \mathsf{\epsilon }>0,$ the operator ${\mathbf{F}}_{\mathsf{\epsilon }}:\mathbf{C}\left(\left[0,\mathbf{T}\right],{\mathbf{X}}_{\mathsf{\alpha }}\right)\to \mathbf{C}\left(\left[0,\mathbf{T}\right],{\mathbf{X}}_{\mathsf{\alpha }}\right)$ has a fixed point. The proof of the statement is long and technical; therefore, it is appropriate to give it in three steps.

Step 1: For an arbitrary $\mathsf{\epsilon }>0,$ $\exists$ a positive constant $\mathbf{r}:=\mathbf{r}\left(\mathsf{\epsilon }\right)$ such that ${\mathbf{F}}_{\mathsf{\epsilon }}\left({\mathbf{B}}_{\mathbf{r}\left(\mathsf{\epsilon }\right)}\right)\subset {\mathbf{B}}_{\mathbf{r}\left(\mathsf{\epsilon }\right)}$.

We assume that $\forall$ $\mathbf{r}>0,$ $\exists \mathbf{x}\in {\mathbf{B}}_{\mathbf{r}\left(\mathsf{\epsilon }\right)}$ and ${\mathbf{t}}_{\mathbf{r}}\in \left[0,\mathbf{T}\right]$ such that ${‖{\mathbf{F}}_{\mathsf{\epsilon }}\left({\mathbf{B}}_{\mathbf{r}\left(\mathsf{\epsilon }\right)}\right)‖}_{\mathsf{\alpha }}>\mathbf{r}.$ According to Lemma 2 and assumption (H2), we have the following:

$$\begin{array}{ll}\mathbf{r}& <{‖{\mathbf{z}}_{\mathsf{\epsilon }}\left({\mathbf{t}}_{\mathbf{r}}\right)‖}_{\mathsf{\alpha }}\le {‖{\mathbf{U}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_{\mathbf{r}},0\right){\mathbf{x}}_0‖}_{\mathsf{\alpha }}+{‖{\mathsf{{\rm Y}}}_0^{{\mathbf{t}}_{\mathbf{r}}}\left\{\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{t},\mathbf{x}\right)\right\}‖}_{\mathsf{\alpha }}+{‖{\mathsf{{\rm Y}}}_0^{{\mathbf{t}}_{\mathbf{r}}}\left\{\mathbf{f}\left(\mathbf{t},\mathbf{x}\left(\mathbf{t}\right),\mathbf{G}\mathbf{x}\left(\mathbf{t}\right)\right)\right\}‖}_{\mathsf{\alpha }}\\ & \le \mathbf{M}{‖{\mathbf{x}}_0‖}_{\mathsf{\alpha }}+{\int }_0^{{\mathbf{t}}_{\mathbf{r}}}{\left(\mathsf{\psi }\left({\mathbf{t}}_{\mathbf{r}}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}‖{\mathbf{A}}^{-\mathsf{\alpha }}{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_{\mathbf{r}},\mathbf{s}\right){\mathbf{A}}^{\mathsf{\alpha }}\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right)‖{\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\\ & +{\int }_0^{{\mathbf{t}}_{\mathbf{r}}}{\left(\mathsf{\psi }\left({\mathbf{t}}_{\mathbf{r}}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}‖{\mathbf{A}}^{\mathsf{\alpha }-\mathsf{\beta }}{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_{\mathbf{r}},\mathbf{s}\right){\mathbf{A}}^{\mathsf{\beta }}\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right)‖{\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\\ & \le \mathbf{M}{‖{\mathbf{x}}_0‖}_{\mathsf{\alpha }}+\frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\beta }-\mathsf{\alpha }}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}{\int }_0^{{\mathbf{t}}_{\mathbf{r}}}{\left(\mathsf{\psi }\left({\mathbf{t}}_{\mathbf{r}}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}{\mathbf{g}}_{\mathbf{r}}\left(\mathbf{s}\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\\ & +\frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}{\int }_0^{{\mathbf{t}}_{\mathbf{r}}}{\left(\mathsf{\psi }\left({\mathbf{t}}_{\mathbf{r}}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}‖{\mathbf{A}}^{\mathsf{\alpha }}\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right)‖{\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\\ & \le \mathbf{M}{‖{\mathbf{x}}_0‖}_{\mathsf{\alpha }}+\frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\beta }-\mathsf{\alpha }}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}{\int }_0^{{\mathbf{t}}_{\mathbf{r}}}{\left(\mathsf{\psi }\left({\mathbf{t}}_{\mathbf{r}}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}{\mathbf{g}}_{\mathbf{r}}\left(\mathbf{s}\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\\ & +\frac{1}{\mathsf{\epsilon }}\frac{{\left(\mathsf{\psi }\left(\mathbf{T}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathbf{q}}}{\mathbf{q}}\frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}{‖\mathbf{B}‖}_{\mathsf{\alpha }}‖{\mathbf{B}}^{\mathbf{*}}{{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{*}}}^{\mathbf{q}}\left(\mathbf{T},\mathbf{s}\right)‖\\ & \times \left({\mathbf{C}}_{\mathsf{\alpha }}‖{\mathbf{A}}^{\mathsf{\alpha }}\mathbf{h}‖+\mathbf{M}{\mathbf{C}}_{\mathsf{\alpha }}{‖{\mathbf{x}}_0‖}_{\mathsf{\alpha }}+\frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\beta }}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}{\int }_0^{\mathbf{T}}{\left(\mathsf{\psi }\left(\mathbf{T}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}{\mathbf{g}}_{\mathbf{r}}\left(\mathbf{s}\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\right)\end{array}$$

Multiplying both sides by $\frac{1}{\mathbf{r}}$ and taking the lower limit as $\mathbf{r}\to \mathbf{\infty },$ we obtain the following:

$$1\le \frac{\mathbf{M}\mathsf{\omega }}{\mathsf{\Gamma }\left(\mathbf{q}\right)}\left({\mathbf{C}}_{\mathsf{\beta }-\mathsf{\alpha }}+\frac{1}{\mathsf{\epsilon }}\frac{{\left(\mathsf{\psi }\left(\mathbf{T}\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathbf{q}}}{\mathbf{q}}\frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\alpha }}{\mathbf{C}}_{\mathsf{\beta }}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}{\mathbf{L}}_{\mathbf{B}}\right)<1,$$

which is a contradiction. Hence, ${\mathbf{F}}_{\mathsf{\epsilon }}\left({\mathbf{B}}_{\mathbf{r}\left(\mathsf{\epsilon }\right)}\right)\subset {\mathbf{B}}_{\mathbf{r}\left(\mathsf{\epsilon }\right)}$ for some $\mathbf{r}\left(\mathsf{\epsilon }\right)>0$.

Step 2: We will now show the continuity of ${\mathbf{F}}_{\mathsf{\epsilon }}:{\mathbf{B}}_{\mathbf{r}}\to {\mathbf{B}}_{\mathbf{r}}$.

Let $\left\{{\mathbf{x}}_{\mathbf{n}}\right\}\subset {\mathbf{B}}_{\mathbf{r}}$ with ${\mathbf{x}}_{\mathbf{n}}\to \mathbf{x}\in {\mathbf{B}}_{\mathbf{r}}$ as $\mathbf{n}\to \mathbf{\infty }.$ From Hypothesis (H2)(b), $\forall \mathbf{s}\in \left[0,\mathbf{T}\right],$ we reach to the following:

$$\mathbf{f}\left(\mathbf{s},{\mathbf{x}}_{\mathbf{n}}\left(\mathbf{s}\right),\mathbf{G}{\mathbf{x}}_{\mathbf{n}}\left(\mathbf{s}\right)\right)\to \mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right) \mathrm{and} \mathbf{u}\left(\mathbf{s},{\mathbf{x}}_{\mathbf{n}}\right)\to \mathbf{u}\left(\mathbf{s},\mathbf{x}\right) \mathrm{as} \mathbf{n}\to \mathbf{\infty },\forall \mathbf{n}\in \mathbb{N}.$$

Using the following estimations:

$${‖\mathbf{f}\left(\mathbf{s},{\mathbf{x}}_{\mathbf{n}}\left(\mathbf{s}\right),\mathbf{G}{\mathbf{x}}_{\mathbf{n}}\left(\mathbf{s}\right)\right)-\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right)‖}_{\mathsf{\beta }}\le 2{\mathbf{g}}_{\mathbf{r}}\left(\mathbf{s}\right)$$

and

$${‖\mathbf{B}\left[{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},{\mathbf{x}}_{\mathbf{n}}\right)-{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right)\right]‖}_{\mathsf{\alpha }}\le \frac{2}{\mathsf{\epsilon }}{\mathbf{L}}_{\mathbf{u}}$$

and the Lebesgue’s dominated convergence theorem, $\forall \mathbf{s}\in \left[0,\mathbf{T}\right],$ we obtain the following:

$$\begin{array}{l}{‖\left({\mathbf{F}}_{\mathsf{\epsilon }}\mathbf{x}\right)\left(\mathbf{t}\right)-\left({\mathbf{F}}_{\mathsf{\epsilon }}{\mathbf{x}}_{\mathbf{n}}\right)\left(\mathbf{t}\right)‖}_{\mathsf{\alpha }}\\ \le {\int }_0^{\mathbf{t}}{\left(\mathsf{\psi }\left(\mathbf{t}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}‖{\mathbf{A}}^{\mathsf{\alpha }-\mathsf{\beta }}{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left(\mathbf{t},\mathbf{s}\right){\mathbf{A}}^{\mathsf{\beta }}\left[\mathbf{f}\left(\mathbf{s},{\mathbf{x}}_{\mathbf{n}}\left(\mathbf{s}\right),\mathbf{G}{\mathbf{x}}_{\mathbf{n}}\left(\mathbf{s}\right)\right)-\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right)\right]‖{\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\\ +{\int }_0^{\mathbf{t}}{\left(\mathsf{\psi }\left(\mathbf{t}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}‖{\mathbf{A}}^{\mathsf{\alpha }-\mathsf{\beta }}{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left(\mathbf{t},\mathbf{s}\right){\mathbf{A}}^{\mathsf{\beta }}\mathbf{B}\left[{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},{\mathbf{x}}_{\mathbf{n}}\right)-{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right)\right]‖{\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\\ \le \frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\beta }-\mathsf{\alpha }}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}{\int }_0^{\mathbf{t}}{\left(\mathsf{\psi }\left(\mathbf{t}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}{‖\mathbf{f}\left(\mathbf{s},{\mathbf{x}}_{\mathbf{n}}\left(\mathbf{s}\right),\mathbf{G}{\mathbf{x}}_{\mathbf{n}}\left(\mathbf{s}\right)\right)-\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right)‖}_{\mathsf{\beta }}{\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\\ +\frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}{\int }_0^{\mathbf{t}}{\left(\mathsf{\psi }\left(\mathbf{t}\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}{‖\mathbf{B}\left[{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},{\mathbf{x}}_{\mathbf{n}}\right)-{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right)\right]‖}_{\mathsf{\alpha }}{\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}\to 0\end{array}$$

which implies that ${\mathbf{F}}_{\mathsf{\epsilon }}:{\mathbf{B}}_{\mathbf{r}}\to {\mathbf{B}}_{\mathbf{r}}$ is continuous.

Step 3: $\forall \mathsf{\epsilon }>0,$ the set $\mathsf{\Phi }\left(\mathbf{t}\right)=\left\{\left({\mathbf{F}}_{\mathsf{\epsilon }}\mathbf{x}\right)\left(\mathbf{t}\right):\mathbf{x}\in {\mathbf{B}}_{\mathbf{r}}\right\},\mathbf{r}=\mathbf{r}\left(\mathsf{\epsilon }\right)$ is relatively compact in ${\mathbf{X}}_{\mathsf{\alpha }}$.

Apparently, $\mathsf{\Phi }\left(0\right)=\left\{{\mathbf{x}}_0\right\}$ is relatively compact in ${\mathbf{X}}_{\mathsf{\alpha }}.$ Let $\mathsf{\tau }$ be a given real number such that $0<\mathsf{\tau }<\mathbf{t}$, where $\mathbf{t}\in [0,\mathbf{T}]$ is a fixed real number. $\forall \mathsf{\delta }>0$, we define the following:

$${\mathsf{\Phi }}_{\mathsf{\tau }}\left(\mathbf{t}\right)=\left\{\left({\mathbf{F}}_{\mathsf{\epsilon }}^{\mathsf{\tau },\mathsf{\delta }}\mathbf{x}\right)\left(\mathbf{t}\right):\mathbf{x}\in {\mathbf{B}}_{\mathbf{r}}\right\},$$

where

$$\begin{array}{ll}\left({\mathbf{F}}_{\mathsf{\epsilon }}^{\mathsf{\tau },\mathsf{\delta }}\mathbf{x}\right)\left(\mathbf{t}\right)& ={\int }_{\mathsf{\delta }}^{\mathbf{\infty }}{\mathsf{\xi }}_{\mathbf{q}}(\mathsf{\theta })\mathsf{\mu }((\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(0){)}^{\mathbf{q}}\mathsf{\theta }){\mathbf{x}}_0\mathbf{d}\mathsf{\theta }\\ & +\mathbf{q}{\int }_0^{\mathbf{t}-\mathsf{\delta }}{\int }_{\mathsf{\delta }}^{\mathbf{\infty }}\mathsf{\theta }(\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(\mathbf{s}){)}^{\mathbf{q}-1}{\mathsf{\xi }}_{\mathbf{q}}(\mathsf{\theta })\\ & \times \mathsf{\mu }((\mathsf{\psi }\left(\mathbf{t}\right)-\mathsf{\psi }(0){)}^{\mathbf{q}}\mathsf{\theta })\left[\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}(\mathbf{s},\mathbf{x})+\mathbf{f}(\mathbf{s},\mathbf{x}(\mathbf{s}),\mathbf{G}\mathbf{x}(\mathbf{s}))\right]{\mathsf{\psi }}^{\mathbf{\prime }}(\mathbf{s})\mathbf{d}\mathsf{\theta }\mathbf{d}\mathbf{s}\\ & ={\int }_{\mathsf{\delta }}^{\mathbf{\infty }}{\mathsf{\xi }}_{\mathbf{q}}(\mathsf{\theta })\mathsf{\mu }((\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(0){)}^{\mathbf{q}}\mathsf{\theta }+{\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta }-{\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta }){\mathbf{x}}_0\mathbf{d}\mathsf{\theta }\\ & +\mathbf{q}{\int }_0^{\mathbf{t}-\mathsf{\delta }}{\int }_{\mathsf{\delta }}^{\mathbf{\infty }}\mathsf{\theta }(\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(\mathbf{s}){)}^{\mathbf{q}-1}{\mathsf{\xi }}_{\mathbf{q}}(\mathsf{\theta })\mathsf{\mu }((\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(0){)}^{\mathbf{q}}\mathsf{\theta }+{\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta }-{\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta })\left[\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right)+\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right)\right]{\mathsf{\psi }}^{\mathbf{\prime }}(\mathbf{s})\mathbf{d}\mathsf{\theta }\mathbf{d}\mathbf{s}\\ & ={\int }_{\mathsf{\delta }}^{\mathbf{\infty }}{\mathsf{\xi }}_{\mathbf{q}}\left(\mathsf{\theta }\right)\left[\mathsf{\mu }({\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta })\mathsf{\mu }((\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(0){)}^{\mathbf{q}}\mathsf{\theta }-{\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta })\right]{\mathbf{x}}_0\mathbf{d}\mathsf{\theta }\\ & +\mathbf{q}{\int }_0^{\mathbf{t}-\mathsf{\delta }}{\int }_{\mathsf{\delta }}^{\mathbf{\infty }}\mathsf{\theta }(\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(\mathbf{s}){)}^{\mathbf{q}-1}{\mathsf{\xi }}_{\mathbf{q}}\left(\mathsf{\theta }\right)\left[\mathsf{\mu }({\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta })\mathsf{\mu }((\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(0){)}^{\mathbf{q}}\mathsf{\theta }-{\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta })\right]\\ & \times \left[\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right)+\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right)\right]{\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathsf{\theta }\mathbf{d}\mathbf{s}\\ & =\mathsf{\mu }\left({\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta }\right){\int }_{\mathsf{\delta }}^{\mathbf{\infty }}{\mathsf{\xi }}_{\mathbf{q}}\left(\mathsf{\theta }\right)\left[\mathsf{\mu }((\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(0){)}^{\mathbf{q}}\mathsf{\theta }-{\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta })\right]{\mathbf{x}}_0\mathbf{d}\mathsf{\theta }\\ & +\mathsf{\mu }\left({\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta }\right)\mathbf{q}{\int }_0^{\mathbf{t}-\mathsf{\delta }}{\int }_{\mathsf{\delta }}^{\mathbf{\infty }}\mathsf{\theta }(\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(\mathbf{s}){)}^{\mathbf{q}-1}{\mathsf{\xi }}_{\mathbf{q}}\left(\mathsf{\theta }\right)\left[\mathsf{\mu }((\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(0){)}^{\mathbf{q}}\mathsf{\theta }-{\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta })\right]\\ & \times \left[\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}(\mathbf{s},\mathbf{x})+\mathbf{f}(\mathbf{s},\mathbf{x}(\mathbf{s}),\mathbf{G}\mathbf{x}(\mathbf{s}))\right]{\mathsf{\psi }}^{\mathbf{\prime }}(\mathbf{s})\mathbf{d}\mathsf{\theta }\mathbf{d}\mathbf{s}\\ & :=\mathsf{\mu }({\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta })\mathbf{y}\left(\mathbf{t},\mathsf{\tau }\right).\end{array}$$

Then, by the compactness of $\mathsf{\mu }({\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta })$ for ${\mathsf{\tau }}^{\mathbf{q}}\mathsf{\delta }>0$ and the boundedness of $\mathbf{y}\left(\mathbf{t},\mathsf{\tau }\right)$ on ${\mathbf{B}}_{\mathbf{r}},$ one can easily see that the set ${\mathsf{\Phi }}_{\mathsf{\tau }}\left(\mathbf{t}\right)$ is a relatively compact set in ${\mathbf{X}}_{\mathsf{\alpha }}.$ Moreover,

$$\begin{array}{l}{‖\left({\mathbf{F}}_{\mathsf{\epsilon }}\mathbf{x}\right)\left(\mathbf{t}\right)-\left({\mathbf{F}}_{\mathsf{\epsilon }}^{\mathsf{\tau },\mathsf{\delta }}{\mathbf{x}}_{\mathbf{n}}\right)\left(\mathbf{t}\right)‖}_{\mathsf{\alpha }}\\ \le {‖{\int }_0^{\mathsf{\delta }}{\mathsf{\xi }}_{\mathbf{q}}(\mathsf{\theta })\mathsf{\mu }((\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(0){)}^{\mathbf{q}}\mathsf{\theta }){\mathbf{x}}_0\mathbf{d}\mathsf{\theta }‖}_{\mathsf{\alpha }}\\ +\mathbf{q}{‖{\int }_0^{\mathbf{t}}{\int }_{\mathsf{\delta }}^{\mathsf{\delta }}\mathsf{\theta }(\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(\mathbf{s}){)}^{\mathbf{q}-1}{\mathsf{\xi }}_{\mathbf{q}}\left(\mathsf{\theta }\right)\mathsf{\mu }\left(\left(\mathsf{\psi }\left(\mathbf{t}\right)-\mathsf{\psi }\left(0\right){)}^{\mathbf{q}}\mathsf{\theta }\right)-\mathsf{\psi }\left(0\right){)}^{\mathbf{q}}\mathsf{\theta }\right)\left[\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right)+\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right)\right]{\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathsf{\theta }\mathbf{d}\mathbf{s}‖}_{\mathsf{\alpha }}\\ +\mathbf{q}{‖{\int }_{\mathbf{t}-\mathsf{\tau }}^{\mathbf{t}}{\int }_{\mathsf{\delta }}^{\mathbf{\infty }}\mathsf{\theta }(\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(\mathbf{s}){)}^{\mathbf{q}-1}{\mathsf{\xi }}_{\mathbf{q}}(\mathsf{\theta })\mathsf{\mu }((\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(0){)}^{\mathbf{q}}\mathsf{\theta })\left[\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right)+\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right)\right]{\mathsf{\psi }}^{\mathbf{\prime }}(\mathbf{s})\mathbf{d}\mathsf{\theta }\mathbf{d}\mathbf{s}‖}_{\mathsf{\alpha }}\\ \le \mathbf{M}{‖{\mathbf{x}}_0‖}_{\mathsf{\alpha }}{\int }_0^{\mathsf{\delta }}{\mathsf{\xi }}_{\mathbf{q}}\left(\mathsf{\theta }\right)\mathbf{d}\mathsf{\theta }+\mathbf{q}\mathbf{M}\left({\mathbf{C}}_{\mathsf{\beta }-\mathsf{\alpha }}{‖{\mathbf{g}}_{\mathbf{r}}‖}_{{\mathbf{L}}^{\mathbf{\infty }}}+\frac{1}{\mathsf{\epsilon }}{\mathbf{C}}_{\mathsf{\alpha }}{\mathbf{L}}_{\mathbf{u}}\right){\int }_{\mathbf{t}-\mathsf{\tau }}^{\mathbf{t}}(\mathsf{\psi }(\mathbf{t})-\mathsf{\psi }(\mathbf{s}){)}^{\mathbf{q}-1}{\mathsf{\psi }}^{\mathbf{\prime }}(\mathbf{s}){\int }_0^{\mathsf{\delta }}\mathsf{\theta }{\mathsf{\xi }}_{\mathbf{q}}\left(\mathsf{\theta }\right)\mathbf{d}\mathsf{\theta }\mathbf{d}\mathbf{s}.\end{array}$$

This implies that there are relatively compact sets arbitrarily close to the set $\mathsf{\Phi }\left(\mathbf{t}\right)$, $\forall$ $\mathbf{t}\in [0,\mathbf{T}].$ Consequently, $\mathsf{\Phi }\left(\mathbf{t}\right),\mathbf{t}\in [0,\mathbf{T}]$ is relatively compact in ${\mathbf{X}}_{\mathsf{\alpha }}.$ Since it is compact at $\mathbf{t}=0,$ we obtain the relative compactness of $\mathsf{\Phi }\left(\mathbf{t}\right)$ in ${\mathbf{X}}_{\mathsf{\alpha }},$ $\forall$ $\mathbf{t}\in \left[0,\mathbf{T}\right]$.

Step 4: $\mathsf{\Phi }:=\left\{{\mathbf{F}}_{\mathsf{\epsilon }}\mathbf{x}\in \mathbf{C}\left(\left[0,\mathbf{T}\right],{\mathbf{X}}_{\mathsf{\alpha }}\right):\mathbf{x}\in {\mathbf{B}}_{\mathbf{r}}\right\}$ is an equi-continuous family of functions on $\left[0,\mathbf{T}\right].$ For $0<{\mathbf{t}}_1<{\mathbf{t}}_2<\mathbf{T},$

$$\begin{array}{l}{‖\mathbf{z}\left({\mathbf{t}}_2\right)-\mathbf{z}\left({\mathbf{t}}_1\right)‖}_{\mathsf{\alpha }}\\ \le {‖{\mathbf{U}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,0\right){\mathbf{x}}_0-{\mathbf{U}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_1,0\right){\mathbf{x}}_0‖}_{\mathsf{\alpha }}+{‖{\mathsf{{\rm Y}}}_0^{{\mathbf{t}}_2}\left\{\mathbf{f}\left(\mathbf{t},\mathbf{x}\left(\mathbf{t}\right),\mathbf{G}\mathbf{x}\left(\mathbf{t}\right)\right)\right\}-{\mathsf{{\rm Y}}}_0^{{\mathbf{t}}_1}\left\{\mathbf{f}\left(\mathbf{t},\mathbf{x}\left(\mathbf{t}\right),\mathbf{G}\mathbf{x}\left(\mathbf{t}\right)\right)\right\}‖}_{\mathsf{\alpha }}\\ +{‖{\mathsf{{\rm Y}}}_0^{{\mathbf{t}}_2}\left\{\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{t},\mathbf{x}\right)\right\}-{\mathsf{{\rm Y}}}_0^{{\mathbf{t}}_1}\left\{\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{t},\mathbf{x}\right)\right\}‖}_{\mathsf{\alpha }}\\ \le {‖{\mathbf{U}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,0\right){\mathbf{x}}_0-{\mathbf{U}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_1,0\right){\mathbf{x}}_0‖}_{\mathsf{\alpha }}+{‖{\mathsf{{\rm Y}}}_{{\mathbf{t}}_1}^{{\mathbf{t}}_2}\left\{\mathbf{f}\left(\mathbf{t},\mathbf{x}\left(\mathbf{t}\right),\mathbf{G}\mathbf{x}\left(\mathbf{t}\right)\right)\right\}‖}_{\mathsf{\alpha }}\\ +{‖{\int }_0^{{\mathbf{t}}_1}\left[{\left(\mathsf{\psi }\left({\mathbf{t}}_2\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}-{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}\right]{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,\mathbf{s}\right)\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}‖}_{\mathsf{\alpha }}\\ +{‖{\int }_0^{{\mathbf{t}}_1}{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}\left[{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,\mathbf{s}\right)-{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_1,\mathbf{s}\right)\right]\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}‖}_{\mathsf{\alpha }}\\ +{‖{\mathsf{{\rm Y}}}_{{\mathbf{t}}_1}^{{\mathbf{t}}_2}\left\{\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{t},\mathbf{x}\right)\right\}‖}_{\mathsf{\alpha }}\\ +{‖{\int }_0^{{\mathbf{t}}_1}\left[{\left(\mathsf{\psi }\left({\mathbf{t}}_2\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}-{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}\right]{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,\mathbf{s}\right)\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}‖}_{\mathsf{\alpha }}\\ +{‖{\int }_0^{{\mathbf{t}}_1}{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}\left[{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,\mathbf{s}\right)-{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_1,\mathbf{s}\right)\right]\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}‖}_{\mathsf{\alpha }}\\ ={\mathbf{I}}_1+{\mathbf{I}}_2+{\mathbf{I}}_3+{\mathbf{I}}_4+{\mathbf{I}}_5+{\mathbf{I}}_6+{\mathbf{I}}_7\end{array}$$

(10)

By using Hypothesis (H2) and Hölder’s inequality, one can obtain the following:

$$\begin{array}{l}{\mathbf{I}}_2\le \frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\beta }-\mathsf{\alpha }}{‖\mathbf{g}‖}_{{\mathbf{L}}^{\mathbf{\infty }}}{\left(\mathsf{\psi }\left({\mathbf{t}}_2\right)-\mathsf{\psi }\left({\mathbf{t}}_1\right)\right)}^{\mathbf{q}}}{\mathsf{\Gamma }\left(\mathbf{q}\right)},\\ {\mathbf{I}}_3\le \frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\beta }-\mathsf{\alpha }}{‖\mathbf{g}‖}_{{\mathbf{L}}^{\mathbf{\infty }}}\left[{\left(\mathsf{\psi }\left({\mathbf{t}}_2\right)\right)}^{\mathbf{q}}-{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)\right)}^{\mathbf{q}}-{\left(\mathsf{\psi }\left({\mathbf{t}}_2\right)-\mathsf{\psi }\left({\mathbf{t}}_1\right)\right)}^{\mathbf{q}}\right]}{\mathsf{\Gamma }\left(\mathbf{q}\right)},\\ {\mathbf{I}}_5\le \frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\alpha }}{\left(\mathsf{\psi }\left({\mathbf{t}}_2\right)-\mathsf{\psi }\left({\mathbf{t}}_1\right)\right)}^{\mathbf{q}}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}\frac{1}{\mathsf{\epsilon }}{\mathbf{L}}_{\mathbf{u}},\end{array}$$

and

$${\mathbf{I}}_6\le \frac{\mathbf{M}{\mathbf{C}}_{\mathsf{\alpha }}\left[{\left(\mathsf{\psi }\left({\mathbf{t}}_2\right)\right)}^{\mathbf{q}}-{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)\right)}^{\mathbf{q}}-{\left(\mathsf{\psi }\left({\mathbf{t}}_2\right)-\mathsf{\psi }\left({\mathbf{t}}_1\right)\right)}^{\mathbf{q}}\right]}{\mathsf{\Gamma }\left(\mathbf{q}\right)}\frac{1}{\mathsf{\epsilon }}{\mathbf{L}}_{\mathbf{u}}.$$

For ${\mathbf{t}}_1=0, 0<{\mathbf{t}}_2\le \mathbf{T},$ it can be easily seen that ${\mathbf{I}}_4={\mathbf{I}}_7=0.$ For ${\mathbf{t}}_1>0$ and $\mathsf{\eta }>0$ small enough, we obtain the following:

$$\begin{array}{ll}{\mathbf{I}}_4& \le {‖{\int }_0^{{\mathbf{t}}_1-\mathsf{\eta }}{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}\left[{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,\mathbf{s}\right)-{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_1,\mathbf{s}\right)\right]\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}‖}_{\mathsf{\alpha }}\\ & +{‖{\int }_{{\mathbf{t}}_1-\mathsf{\eta }}^{{\mathbf{t}}_1}{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}\left[{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,\mathbf{s}\right)-{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_1,\mathbf{s}\right)\right]\mathbf{f}\left(\mathbf{s},\mathbf{x}\left(\mathbf{s}\right),\mathbf{G}\mathbf{x}\left(\mathbf{s}\right)\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}‖}_{\mathsf{\alpha }}\\ & \le \frac{1}{\mathbf{q}}{\mathbf{C}}_{\mathsf{\beta }-\mathsf{\alpha }}{‖\mathbf{g}‖}_{{\mathbf{L}}^{\mathbf{\infty }}}\left[{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathbf{q}}-{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left({\mathbf{t}}_1-\mathsf{\eta }\right)\right)}^{\mathbf{q}}\right]\underset{\mathbf{s}\in \left[0,{\mathbf{t}}_1-\mathsf{\eta }\right]}{\mathbf{s}\mathbf{u}\mathbf{p}}‖{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,\mathbf{s}\right)-{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_1,\mathbf{s}\right)‖\\ & +\frac{2\mathbf{M}{\mathbf{C}}_{\mathsf{\beta }-\mathsf{\alpha }}{‖\mathbf{g}‖}_{{\mathbf{L}}^{\mathbf{\infty }}}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}\left[{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left({\mathbf{t}}_1-\mathsf{\eta }\right)\right)}^{\mathbf{q}}\right],\end{array}$$

and

$$\begin{array}{ll}{\mathbf{I}}_7& \le {‖{\int }_0^{{\mathbf{t}}_1-\mathsf{\eta }}{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}\left[{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,\mathbf{s}\right)-{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_1,\mathbf{s}\right)\right]\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}‖}_{\mathsf{\alpha }}\\ & +{‖{\int }_{{\mathbf{t}}_1-\mathsf{\eta }}^{{\mathbf{t}}_1}{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left(\mathbf{s}\right)\right)}^{\mathbf{q}-1}\left[{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,\mathbf{s}\right)-{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_1,\mathbf{s}\right)\right]\mathbf{B}{\mathbf{u}}_{\mathsf{\epsilon }}\left(\mathbf{s},\mathbf{x}\right){\mathsf{\psi }}^{\mathbf{\prime }}\left(\mathbf{s}\right)\mathbf{d}\mathbf{s}‖}_{\mathsf{\alpha }}\\ & \le \frac{1}{\mathbf{q}}{\mathbf{C}}_{\mathsf{\alpha }}\frac{1}{\mathsf{\epsilon }}{\mathbf{L}}_{\mathbf{u}}\left[{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left(0\right)\right)}^{\mathbf{q}}-{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left({\mathbf{t}}_1-\mathsf{\eta }\right)\right)}^{\mathbf{q}}\right]\underset{\mathbf{s}\in \left[0,{\mathbf{t}}_1-\mathsf{\eta }\right]}{\mathbf{s}\mathbf{u}\mathbf{p}}‖{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_2,\mathbf{s}\right)-{\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left({\mathbf{t}}_1,\mathbf{s}\right)‖\\ & +\frac{2\mathbf{M}{\mathbf{C}}_{\mathsf{\alpha }}}{\mathsf{\Gamma }\left(\mathbf{q}\right)}\frac{1}{\mathsf{\epsilon }}{\mathbf{L}}_{\mathbf{u}}\left[{\left(\mathsf{\psi }\left({\mathbf{t}}_1\right)-\mathsf{\psi }\left({\mathbf{t}}_1-\mathsf{\eta }\right)\right)}^{\mathbf{q}}\right].\end{array}$$

Hypothesis (H1) and Lemma 2 imply the continuity of ${\mathbf{V}}_{\mathsf{\psi }}^{\mathbf{q}}\left(\mathbf{t},\mathbf{s}\right),\mathbf{t}>0$ in the uniform operator topology on $\mathbf{t}$. One can clearly notice that ${\mathbf{I}}_4$ and ${\mathbf{I}}_7$ tend to zero independently of $\mathbf{x}\in {\mathbf{B}}_{\mathbf{r}}$ as ${\mathbf{t}}_2-{\mathbf{t}}_1\to 0,\mathsf{\eta }\to 0.$ It is also obvious that ${\mathbf{I}}_1,{\mathbf{I}}_2,{\mathbf{I}}_3,{\mathbf{I}}_5,$ and ${\mathbf{I}}_6\to 0$ as ${\mathbf{t}}_2-{\mathbf{t}}_1\to 0.$ Therefore, the right-hand side of inequality (10) does not depend on the particular choices of $\mathbf{x}$, and it tends to zero as ${\mathbf{t}}_2-{\mathbf{t}}_1\to 0$. This enables us to conclude that $\left\{\left({\mathbf{F}}_{\mathsf{\epsilon }}\mathbf{x}\right),\mathbf{x}\in {\mathbf{B}}_{\mathbf{r}}\right\}$ is equi-continuous.

Since ${\mathbf{F}}_{\mathsf{\epsilon }}\left[{\mathbf{B}}_{\mathbf{r}}\right]$ is equi-continuous and bounded by the Ascoli–Arzela theorem, ${\mathbf{F}}_{\mathsf{\epsilon }}\left[{\mathbf{B}}_{\mathbf{r}}\right]$ is relatively compact in $\mathbf{C}\left(\left[0,\mathbf{T}\right],{\mathbf{X}}_{\mathsf{\alpha }}\right).$ Additionally, it is easy to conclude that $\forall \mathsf{\epsilon }>0,{\mathbf{F}}_{\mathsf{\epsilon }}$ is continuous on $\mathbf{C}\left(\left[0,\mathbf{T}\right],{\mathbf{X}}_{\mathsf{\alpha }}\right).$ Hence, $\forall \mathsf{\epsilon }>0,{\mathbf{F}}_{\mathsf{\epsilon }}$ is a completely continuous operator on $\mathbf{C}\left(\left[0,\mathbf{T}\right],{\mathbf{X}}_{\mathsf{\alpha }}\right).$ Therefore, ${\mathbf{F}}_{\mathsf{\epsilon }}$ has a fixed point as a result of the Schauder’s fixed-point theorem, which implies the desired result that the fractional control system (1) has a mild solution on $\left[0,\mathbf{T}\right].$ Hence, the proof is complete.  □

Remark 3.

From Theorem 2, we notice that if $\mathit{\psi }$ is a bijection function, then the control system (1) has at least one mild solution, provided that

$$\mathrm{T}<{\mathsf{\psi }}^{-1}\left[{\left(\frac{\mathsf{\epsilon }\mathrm{q}\mathsf{\Gamma }\left(\mathrm{q}\right)}{\mathrm{M}{\mathrm{C}}_{\mathsf{\alpha }}{\mathrm{C}}_{\mathsf{\beta }}}\frac{1}{{\mathrm{L}}_{\mathrm{B}}}\left(\frac{\mathsf{\Gamma }\left(\mathrm{q}\right)}{\mathrm{M}\mathsf{\omega }}-{\mathrm{C}}_{\mathsf{\beta }-\mathsf{\alpha }}\right)\right)}^{\frac{1}{\mathrm{q}}}+\mathsf{\psi }\left(0\right)\right].$$

Theorem 3.

Assume that hypotheses (H1) and (H2) are fulfilled. In addition, the function $f:\left[0,T\right]\times X_{\alpha }\times X_{\alpha }\to X_{\beta }$ is bounded and the linear control system (7) is A-controllable on $\left[0,T\right].$ Then, the semilinear control system (1) is A-controllable on $\left[0,T\right]$.

Proof. 

Note that the conditions of Theorem 2 are satisfied with $\mathsf{\omega }=0.$ Let ${\mathrm{x}}_{\mathsf{\epsilon }}$ be a fixed point of ${\mathrm{F}}_{\mathsf{\epsilon }}$ in ${\mathrm{B}}_{\mathrm{r}}.$ Using Theorem 2, we deduce that any fixed point of ${\mathrm{F}}_{\mathsf{\epsilon }}$ is a mild solution of (1) under the control

$$\begin{array}{ll}{\mathrm{u}}_{\mathsf{\epsilon }}\left(\mathrm{t},{\mathrm{x}}_{\mathsf{\epsilon }}\right)& ={\mathrm{B}}^{\mathrm{*}}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{*}\mathrm{q}}\left(\mathrm{T},\mathrm{t}\right)\mathrm{R}\left(\mathsf{\epsilon },{\mathsf{\Gamma }}_0^{\mathrm{T}}\right)\\ & \times \left(\mathrm{h}-{\mathrm{U}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{T},\mathrm{s}\right){\mathrm{x}}_0-{\int }_0^{\mathrm{T}}{\left(\mathsf{\psi }\left(\mathrm{T}\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{T},\mathrm{s}\right)\mathrm{f}\left(\mathrm{s},{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right),\mathrm{G}{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right)\right){\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\right)\end{array}$$

and satisfies the inequality

$$\begin{array}{c}{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{T}\right)=\mathrm{h}-\mathsf{\epsilon }\mathrm{R}\left(\mathsf{\epsilon },{\mathsf{\Gamma }}_0^{\mathrm{T}}\right)\mathrm{p}\left({\mathrm{x}}_{\mathsf{\epsilon }}\right),\\ \mathrm{p}\left({\mathrm{x}}_{\mathsf{\epsilon }}\right)=\left(\mathrm{h}-{\mathrm{U}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{T},\mathrm{s}\right){\mathrm{x}}_0-{\int }_0^{\mathrm{T}}{\left(\mathsf{\psi }\left(\mathrm{T}\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{T},\mathrm{s}\right)\mathrm{f}\left(\mathrm{s},{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right),\mathrm{G}{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right)\right){\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\right).\end{array}$$

(11)

Since $\mathrm{f}$ is bounded,

$${\int }_0^{\mathrm{T}}{‖\mathrm{f}\left(\mathrm{s},{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right),\mathrm{G}{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right)\right)‖}_{\mathsf{\alpha }}^2\mathrm{d}\mathrm{s}\le {\mathrm{N}}^2\mathrm{T},$$

for some $\mathrm{N}>0.$ Consequently, the sequence $\left\{\mathrm{f}\left(\mathrm{s},{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right),\mathrm{G}{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right)\right)\right\}$ is bounded in ${\mathrm{L}}_2\left(\left[0,\mathrm{T}\right],{\mathrm{X}}_{\mathsf{\alpha }}\right).$ Then $\exists$ is a subsequence denoted by $\left\{\mathrm{f}\left(\mathrm{s},{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right),\mathrm{G}{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right)\right)\right\}$ that converges weakly to say $\mathrm{f}\left(\mathrm{s}\right)$ in ${\mathrm{L}}_2\left(\left[0,\mathrm{T}\right],{\mathrm{X}}_{\mathsf{\alpha }}\right).$ Define

$$\mathrm{w}=\mathrm{h}-{\mathrm{U}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{T},\mathrm{s}\right){\mathrm{x}}_0-{\int }_0^{\mathrm{T}}{\left(\mathsf{\psi }\left(\mathrm{T}\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{T},\mathrm{s}\right)\mathrm{f}\left(\mathrm{s}\right){\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}.$$

It follows that

$$\begin{array}{ll}{‖\mathrm{p}\left({\mathrm{x}}_{\mathsf{\epsilon }}\right)-\mathrm{w}‖}_{\mathsf{\alpha }}& ={‖{\int }_0^{\mathrm{T}}{\left(\mathsf{\psi }\left(\mathrm{T}\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{T},\mathrm{s}\right)‖\mathrm{f}\left(\mathrm{s},{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right),\mathrm{G}{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right)\right)-\mathrm{f}\left(\mathrm{s}\right)‖{\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}‖}_{\mathsf{\alpha }}\\ & \le \underset{0\le \mathrm{t}\le \mathrm{T}}{\mathrm{s}\mathrm{u}\mathrm{p}}{‖{\int }_0^{\mathrm{T}}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left(\mathrm{T},\mathrm{s}\right)\left[\mathrm{f}\left(\mathrm{s},{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right),\mathrm{G}{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{s}\right)\right)-\mathrm{f}\left(\mathrm{s}\right)\right]{\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}‖}_{\mathsf{\alpha }}\end{array}$$

(12)

Next, by the compactness of the operator

$$\mathrm{l}\left(.\right)\to {\int }_0^{.}{\left(-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{q}}\left(-\mathsf{\psi }\left(\mathrm{s}\right)\right)\mathrm{l}\left(\mathrm{s}\right){\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}:{\mathrm{L}}_2\left(\left[0,\mathrm{T}\right],{\mathrm{X}}_{\mathsf{\alpha }}\right)\to \mathrm{C}\left(\left[0,\mathrm{T}\right],{\mathrm{X}}_{\mathsf{\alpha }}\right),$$

the right-hand side of (11) tends to zero as $\mathsf{\epsilon }\to 0^{+}.$ Then, from (10), we obtain the following:

$$\begin{array}{ll}{‖{\mathrm{x}}_{\mathsf{\epsilon }}\left(\mathrm{T}\right)-\mathrm{h}‖}_{\mathsf{\alpha }}& \le {‖\mathsf{\epsilon }\mathrm{R}\left(\mathsf{\epsilon },{\mathsf{\Gamma }}_0^{\mathrm{T}}\right)\left(\mathrm{w}\right)‖}_{\mathsf{\alpha }}+‖\mathsf{\epsilon }\mathrm{R}\left(\mathsf{\epsilon },{\mathsf{\Gamma }}_0^{\mathrm{T}}\right)‖{‖\mathrm{p}\left({\mathrm{x}}_{\mathsf{\epsilon }}\right)-\mathrm{w}‖}_{\mathsf{\alpha }}\\ & \le {‖\mathsf{\epsilon }\mathrm{R}\left(\mathsf{\epsilon },{\mathsf{\Gamma }}_0^{\mathrm{T}}\right)\left(\mathrm{w}\right)‖}_{\mathsf{\alpha }}+{‖\mathrm{p}\left({\mathrm{x}}_{\mathsf{\epsilon }}\right)-\mathrm{w}‖}_{\mathsf{\alpha }}\to 0.\end{array}$$

(13)

Furthermore, (13) tends to zero as $\mathsf{\epsilon }\to 0^{+}$ by Lemma 4 and the estimation (12). Hence, the A-controllability of (1) is proved.  □

4. Application

In the last section, we consider the following example to support the obtained results of Theorem 2.

Consider a control system driven by a fractional partial differential equation of the following form:

$$\left\{\begin{array}{l}{{}_0{}^{\mathrm{c}}\mathrm{D}}_{\mathsf{\psi }}^{\mathrm{q}}\mathrm{x}\left(\mathrm{t},\mathrm{z}\right)={\partial }_{\mathrm{z}}^2\mathrm{x}\left(\mathrm{t},\mathrm{z}\right)+\mathrm{u}\left(\mathrm{t},\mathrm{z}\right)+\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\cos \left(\mathrm{x}\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathrm{x}\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)\\ \mathrm{x}\left(\mathrm{t},0\right)=\mathrm{x}\left(\mathrm{t},\mathsf{\pi }\right)=0,\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{t}\in \left[0,\mathrm{T}\right], \\ \mathrm{x}\left(0,\mathrm{z}\right)={\mathrm{x}}_0\left(\mathrm{z}\right),\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{z}\in \left[0,\mathsf{\pi }\right], \end{array}\right.$$

(14)

where

$$\mathrm{q}=\frac{3}{4},\mathrm{A}\mathrm{x}={\partial }_{\mathrm{z}}^2\mathrm{x},\mathrm{B}=\mathrm{I},\mathrm{x}\left(\mathrm{t}\right)=\mathrm{x}\left(\mathrm{t},\mathrm{z}\right),\mathrm{u}\left(\mathrm{t}\right)=\mathrm{u}\left(\mathrm{t},\mathrm{z}\right),$$

$\mathrm{f}\left(\mathrm{t},\mathrm{x}\left(\mathrm{t}\right),\mathrm{G}\mathrm{x}\left(\mathrm{t}\right)\right)=\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\cos \left(\mathrm{x}\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathrm{x}\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)$, and $\mathsf{\psi }\left(\mathrm{t}\right)=\mathrm{t}$.

Let $\mathrm{X}$ be defined as $\mathrm{X}={\mathrm{L}}^2\left[0,\mathsf{\pi }\right]$ and $\mathrm{A}$ by $\mathrm{A}\mathrm{w}=-{\mathrm{w}}^{″}$ on the domain $\mathrm{D}\left(\mathrm{A}\right)=\left\{\mathrm{w}\left(.\right)\in {\mathrm{L}}^2\left[0,\mathsf{\pi }\right],\mathrm{w},{\mathrm{w}}^{\prime } \mathrm{are} \mathrm{absolutely} \mathrm{continuous}, {\mathrm{w}}^{″}\in {\mathrm{L}}^2\left[0,\mathsf{\pi }\right],\mathrm{w}\left(0\right)=\mathrm{w}\left(\mathsf{\pi }\right)=0\right\}$.

It is prominent that $\mathrm{A}$ has a discrete spectrum and the eigenvalues are $\left\{-{\mathrm{n}}^2:\mathrm{n}\in \mathbb{N}\right\}$ with the corresponding normalized eigenvectors ${\mathrm{e}}_{\mathrm{n}}\left(\mathrm{z}\right)=\sqrt{\frac{2}{\mathsf{\pi }}}\sin \mathrm{n}\mathrm{z}$. Therefore,

$$\mathrm{A}\mathrm{w}=-{\sum }_{\mathrm{n}=1}^{\mathrm{\infty }}{\mathrm{n}}^2⟨\mathrm{w},{\mathrm{e}}_{\mathrm{n}}⟩{\mathrm{e}}_{\mathrm{n}},\hspace{1em}\hspace{1em}\mathrm{w}\in \mathrm{D}\left(\mathrm{A}\right).$$

Moreover, $\mathrm{A}$ is the infinitesimal generator of a uniformly bounded analytic semigroup ${\left\{\mathsf{\mu }\left(\mathrm{t}\right)\right\}}_{\mathrm{t}\ge 0},$ where

$$\mathsf{\mu }\left(\mathrm{t}\right)\mathrm{w}={\sum }_{\mathrm{n}=1}^{\mathrm{\infty }}{\mathrm{e}}^{-{\mathrm{n}}^2\mathrm{t}}⟨\mathrm{w},{\mathrm{e}}_{\mathrm{n}}⟩{\mathrm{e}}_{\mathrm{n}},\hspace{1em}\hspace{1em}\mathrm{w}\in \mathrm{X}.$$

Surely, $‖\mathsf{\mu }\left(\mathrm{t}\right)‖\le {\mathrm{e}}^{-\mathrm{t}}$ $\forall \mathrm{t}\ge 0.$ Hence, we take $\mathrm{M}=1,$ which implies that $\underset{\mathrm{t}\in \left.0,\mathrm{\infty }\right)}{\mathrm{s}\mathrm{u}\mathrm{p}}‖\mathsf{\mu }\left(\mathrm{t}\right)‖=1$ and (H1) are satisfied.

For q $= \frac{1}{2}$, the operator ${\mathrm{A}}^{\frac{1}{2}}$ is given by the following:

$${\mathrm{A}}^{\frac{1}{2}}\mathrm{w}=-{\sum }_{\mathrm{n}=1}^{\mathrm{\infty }}\mathrm{n}⟨\mathrm{w},{\mathrm{e}}_{\mathrm{n}}⟩{\mathrm{e}}_{\mathrm{n}},\hspace{1em}\hspace{1em}\mathrm{w}\in \mathrm{D}\left({\mathrm{A}}^{\frac{1}{2}}\right),$$

where $\mathrm{D}\left({\mathrm{A}}^{\frac{1}{2}}\right)=\left\{\mathrm{w}\in \mathrm{X}:{\sum }_{\mathrm{n}=1}^{\mathrm{\infty }}\mathrm{n}⟨\mathrm{w},{\mathrm{e}}_{\mathrm{n}}⟩{\mathrm{e}}_{\mathrm{n}}\in \mathrm{X}\right\}$ and $‖{\mathrm{A}}^{-\frac{1}{2}}‖=1.$ let ${\mathrm{X}}_{\frac{1}{2}}=\left(\mathrm{D}\left({\mathrm{A}}^{\frac{1}{2}}\right),{‖ ‖}_{\frac{1}{2}}\right)$, $\mathrm{B}=\mathrm{I}$, and $\mathrm{U}={\mathrm{X}}_{\frac{1}{2}},$ where ${‖\mathrm{x}‖}_{\frac{1}{2}}={‖{\mathrm{A}}^{\frac{1}{2}}\mathrm{x}‖}_{\mathrm{X}}$ for $\mathrm{x}\in \mathrm{D}\left({\mathrm{A}}^{\frac{1}{2}}\right)$. It is now clear that the function $\mathrm{f}:\left[0,\mathrm{T}\right]\times \left[0,\mathsf{\pi }\right]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ fulfills the following conditions:

1.

$\forall$ $\left(\mathrm{t},\mathrm{x}\right)\in \left[0,\mathrm{T}\right]\times \left[0,\mathsf{\pi }\right],$ the function $\mathrm{f}\left(\mathrm{t},\mathrm{x},.,.\right)$ is continuous.

2.

$\forall$ $\left(\mathsf{\varsigma },\mathsf{\nu }\right)\in \mathbb{R}\times \mathbb{R}$ the function $\mathrm{f}\left(.,.,\mathsf{\varsigma },\mathsf{\nu }\right)$ is measurable.

3.

$\forall$ $\mathrm{t}\in \left[0,\mathrm{T}\right]$ and $\left(\mathsf{\varsigma },\mathsf{\nu }\right)\in \mathbb{R}\times \mathbb{R}$, $\mathrm{f}\left(\mathrm{t},,\mathsf{\varsigma },\mathsf{\nu }\right)$ is differentiable and $\frac{\partial }{\partial \mathrm{x}}\mathrm{f}\left(\mathrm{t},\mathrm{x},\mathsf{\varsigma },\mathsf{\nu }\right)\in \mathrm{X}$.

4.

$\mathrm{f}\left(0,.,.,.\right)=\mathrm{f}\left(\mathsf{\pi },.,.,.\right)=0$.

5.

$\exists \mathrm{C}>0$ such that $\left|\frac{\partial }{\partial \mathrm{x}}\mathrm{f}\left(\mathrm{t},\mathrm{x},\mathsf{\varsigma },\mathsf{\nu }\right)\right|\le \mathrm{C}$ $\forall$ $\left(\mathrm{t},\mathrm{x},\mathsf{\varsigma },\mathsf{\nu }\right)\in \left[0,\mathrm{T}\right]\times \left[0,\mathsf{\pi }\right]\times \mathbb{R}\times \mathbb{R}$.

Now, $\forall$ $\mathsf{\varphi }\in {\mathrm{X}}_{\frac{1}{2}}$, we have the following:

$$\begin{array}{ll}⟨\mathrm{f}\left(\mathrm{t},\mathsf{\varphi },\mathrm{G}\mathsf{\varphi }\right),{\mathrm{e}}_{\mathrm{n}}⟩& ={\int }_0^{\mathsf{\pi }}\left(\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\cos \left(\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)\right).\left(\sqrt{\frac{2}{\mathsf{\pi }}}\sin \mathrm{n}\mathrm{z}\right)\mathrm{d}\mathrm{z}\\ & =\frac{1}{\mathrm{n}}{\int }_0^{\mathsf{\pi }}\frac{\partial }{\partial \mathrm{z}}\left(\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\cos \left(\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)\right).\left(\sqrt{\frac{2}{\mathsf{\pi }}}\cos \mathrm{n}\mathrm{z}\right)\mathrm{d}\mathrm{z}\end{array}$$

This implies that $\mathrm{f}:\left[0,\mathrm{T}\right]\times {\mathrm{X}}_{\frac{1}{2}}\times {\mathrm{X}}_{\frac{1}{2}}\to {\mathrm{X}}_{\frac{1}{2}}.$ Moreover, $\forall$ $\mathrm{r}>0$ by the Minkowski inequality, we have the following:

$$\begin{array}{ll}\underset{{‖\mathsf{\varphi }‖}_{\frac{1}{2}}\le \mathrm{r}}{\mathrm{s}\mathrm{u}\mathrm{p}}{‖\mathrm{f}\left(\mathrm{t},\mathsf{\varphi },\mathrm{G}\mathsf{\varphi }\right)‖}_{\frac{1}{2}}& =\underset{{‖\mathsf{\varphi }‖}_{\frac{1}{2}}\le \mathrm{r}}{\mathrm{s}\mathrm{u}\mathrm{p}}{‖\frac{\partial }{\partial \mathrm{z}}\left(\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\cos \left(\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)\right)‖}_{\mathrm{X}}\\ & =\underset{{‖\mathsf{\varphi }‖}_{\frac{1}{2}}\le \mathrm{r}}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left({\int }_0^{\mathsf{\pi }}{\left|\frac{\partial }{\partial \mathrm{z}}\left(\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\cos \left(\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)\right)\right|}^2\mathrm{d}\mathrm{z}\right)}^{\frac{1}{2}}\\ & \le \underset{{‖\mathsf{\varphi }‖}_{1/2}\le \mathrm{r}}{\mathrm{s}\mathrm{u}\mathrm{p}}{\left({\int }_0^{\mathsf{\pi }}{\left[\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\sin \left(\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)\left(\left|\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)\right|+\left|\frac{\partial }{\partial \mathrm{z}}{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right|\right)\right]}^2\mathrm{d}\mathrm{z}\right)}^{1/2}\\ & \le \underset{{‖\mathsf{\varphi }‖}_{1/2}\le \mathrm{r}}{\mathrm{s}\mathrm{u}\mathrm{p}}\left[\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\sin \left(\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)\left({‖\mathsf{\varphi }‖}_{\mathrm{X}}+{\mathrm{K}}^{\mathrm{*}}\mathrm{T}{‖\mathsf{\varphi }‖}_{\mathrm{X}}\right)\right]\\ & =\underset{{‖\mathsf{\varphi }‖}_{1/2}\le \mathrm{r}}{\mathrm{s}\mathrm{u}\mathrm{p}}\left[\left(+{\mathrm{K}}^{\mathrm{*}}\mathrm{T}\right)\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\sin \left(\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)\right.\left.{\times ‖{\mathrm{A}}^{\left(-\frac{1}{2}\right)}\cdot {\mathrm{A}}^{\left(\frac{1}{2}\right)}\mathsf{\varphi }‖}_{\mathrm{X}}\right]\\ & \le \left(1+{\mathrm{K}}^{\mathrm{*}}\mathrm{T}\right)\mathrm{r}\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\sin \left(\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)\\ & \le {\mathrm{g}}_{\mathrm{r}}\left(\mathrm{t}\right)\end{array}$$

Therefore, $\mathrm{f}$ satisfies the condition (H2)(c) with the following:

$$\begin{array}{l}\underset{\mathrm{r}\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\inf \frac{1}{\mathrm{r}}{\int }_0^{\mathrm{t}}{\left(\mathsf{\psi }\left(\mathrm{t}\right)-\mathsf{\psi }\left(\mathrm{s}\right)\right)}^{\mathrm{q}-1}{\mathrm{g}}_{\mathrm{r}}\left(\mathrm{s}\right){\mathsf{\psi }}^{\prime }\left(\mathrm{s}\right)\mathrm{d}\mathrm{s}\\ =\underset{\mathrm{r}\to +\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\inf \frac{1}{\mathrm{r}}{\int }_0^{\mathrm{t}}{\left(\mathrm{t}-\mathrm{s}\right)}^{-\frac{1}{4}}\left(1+{\mathrm{K}}^{\mathrm{*}}\mathrm{T}\right)\mathrm{r}\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\sin \left(\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)\mathrm{d}\mathrm{s}\\ \le \left(1+{\mathrm{K}}^{\mathrm{*}}\mathrm{T}\right){‖\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\sin \left(\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)‖}_{{\mathrm{L}}^{\mathrm{\infty }}}{\int }_0^{\mathrm{t}}{\left(\mathrm{t}-\mathrm{s}\right)}^{-\frac{1}{4}}\mathrm{d}\mathrm{s}\\ \le \frac{4}{3}{\mathrm{T}}^{3/4}\left(1+{\mathrm{K}}^{\mathrm{*}}\mathrm{T}\right){‖\frac{\sin \mathrm{t}}{{\mathrm{e}}^{\mathrm{t}}+1}\sin \left(\mathsf{\varphi }\left(\mathrm{t},\mathrm{z}\right)+{\int }_0^{\mathrm{t}}\cos \left(\mathrm{t}\mathrm{s}\right)\mathsf{\varphi }\left(\mathrm{s},\mathrm{z}\right)\mathrm{d}\mathrm{s}\right)‖}_{{\mathrm{L}}^{\mathrm{\infty }}}=\mathsf{\omega }<+\mathrm{\infty }.\end{array}$$

We now need to prove the approximate controllability of the linear system corresponding to (14) on $\left[0,\mathrm{T}\right].$ It is obvious that

$${\mathrm{V}}_{\mathsf{\psi }}^{3/4}\left(\mathrm{t},\mathrm{s}\right)\mathrm{x}=\frac{3}{4}{\sum }_{\mathrm{n}=1}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\mathsf{\theta }{\mathsf{\xi }}_{3/4}\left(\mathsf{\theta }\right){\mathrm{e}}^{\left(-{\mathrm{n}}^2{\left(\mathrm{t}-\mathrm{s}\right)}^{3/4}\mathsf{\theta }\right)}\mathrm{d}\mathsf{\theta }⟨\mathrm{x},{\mathrm{e}}_{\mathrm{n}}⟩{\mathrm{e}}_{\mathrm{n}},\hspace{1em}\mathrm{x}\in {\mathrm{X}}_{\frac{1}{2}}$$

and

$${\mathrm{B}}^{\mathrm{*}}{\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{*}3/4}\left(\mathrm{T},\mathrm{t}\right)\mathrm{x}=\frac{3}{4}{\sum }_{\mathrm{n}=1}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\mathsf{\theta }{\mathsf{\xi }}_{3/4}\left(\mathsf{\theta }\right){\mathrm{e}}^{\left(-{\mathrm{n}}^2{\left(\mathrm{T}-\mathrm{t}\right)}^{3/4}\mathsf{\theta }\right)}\mathrm{d}\mathsf{\theta }⟨\mathrm{x},{\mathrm{e}}_{\mathrm{n}}⟩{\mathrm{e}}_{\mathrm{n}},\hspace{1em}\mathrm{x}\in {\mathrm{X}}_{\frac{1}{2}},\hspace{1em}0\le \mathrm{t}\le \mathrm{T}.$$

Recall that Remark 3 implies that the linear system corresponding to (14) is A-controllable on $\left[0,\mathrm{T}\right]$ if and only if ${\mathrm{V}}_{\mathsf{\psi }}^{\mathrm{*}3/4}\left(\mathrm{T},\mathrm{t}\right)\mathrm{x}=0,0\le \mathrm{t}\le \mathrm{T}$ implies that $\mathrm{x}=0$.

Hence, by Theorem 3, we deduce that the control system (14) is approximately controllable on $\left[0,\mathrm{T}\right]$. Whence, the proof.

5. Conclusions

In conclusion, the article highlights the importance of studying the notion of approximate controllability in fractional evolution equations with the ψ-Caputo derivative. The research focuses on cases where the semigroup is considered compact and analytic and employs the theory of fractional calculus, semigroup theory, and the fixed-point method, specifically Schauder’s fixed-point theorem, to obtain results. The study also utilizes the hypothesis that the analogous linear system is approximately controllable. The article’s findings demonstrate that under certain conditions, approximate controllability can be achieved, emphasizing the practical applications of this research in fields such as engineering, physics, and biology. Moving forward, there are several potential avenues for further research in this area. For example, it would be interesting to investigate the notion of approximate controllability in more general classes of fractional evolution equations beyond those considered in this study. Additionally, it may be worthwhile to explore the use of alternative fixed-point theorems to obtain results for non-compact semigroups. Finally, it would be valuable to investigate the applicability of our results in the context of numerical simulations and control system designs. Overall, this research contributes to the growing body of knowledge in the field of fractional calculus and highlights the importance of understanding approximate controllability in fractional evolution equations with the ψ-Caputo derivative.