Finite-Approximate Controllability for Fractional Composite Relaxation Equations with Different Nonlocal Conditions

Abstract

In this paper, the finite-approximate controllability for a class of fractional composite relaxation equations with different nonlocal conditions is discussed. Firstly, under the condition that the nonlocal term is compact, the existence of mild solutions to the equations is obtained by employing resolvent theory, the variational method, and Schauder’s fixed-point theorem. Moreover, under the assumption that the corresponding linear equation is approximately controllable, the fractional composite relaxation equation with the nonlocal condition is derived to be finite-approximately controllable. Furthermore, the existence of mild solutions and the finite-approximate controllability to the equations are considered for the weaker nonlocal problem. Finally, the example of nonlocal problem is provided to verify the feasibility of the results in this paper.

1. Introduction

In this paper, the finite-approximate controllability of the following fractional composite relaxation equations with nonlocal conditions is investigated:

$$\begin{array}{c}\left\{\begin{array}{c}{\eta }^{\prime }\left(t\right)=\mathcal{A}{}^{c}D_0^{\beta }\eta \left(t\right)-\eta \left(t\right)+h(t,\eta \left(t\right))+\mathcal{B}v\left(t\right),t\in [0,d],\beta \in (0,1),\hfill \\ \eta \left(0\right)={\eta }_0-w\left(\eta \right),\hfill \end{array}\right.\hfill \end{array}$$

(1)

where ${\eta }^{\prime }$ is the first derivative of $\eta$ with respect to t. The state variable $\eta (·)\in C\left(\right[0,d];E)$ and the control function $v(·)\in L^2([0,d];F)$. E and F are Hilbert spaces, respectively. ${}^{c}D_0^{\beta }$ denotes the fractional derivative in the Caputo sense. $\mathcal{A}:D\left(\mathcal{A}\right)\subseteq E\to E$ is a densely defined closed linear operator that generates a resolvent family $\{S_{1-\beta }\left(t\right):t\ge 0\}$ on E. $\mathcal{B}$ is the bounded linear operator from F to E. The nonlinear function $h:[0,d]\times E\to E$ and the continuous function $w:C\left(\right[0,d];E)\to E$ will be specified later. The initial condition ${\eta }_0\in E$.

In recent decades, the theory of fractional differential equations has attracted much attention from scholars. Because of its important role in medicine, finance, engineering, and so on, fractional differential equations can be used as models to describe many natural phenomena. For more information, the reader can refer to the literature [1,2] and references therein. In 1997, Gorenflo et al. [3] examined the system in which $\mathcal{A}$ is a positive constant in Equation (1) and contains no control terms. In particular, when $\beta ={\displaystyle \frac{1}{2}}$, Equation (1) is known as the Basset problem, which is generally used to characterize the problem of non-stationary acceleration under gravity in viscous fluids. Afterwards, some authors developed the fractional composite relaxation equations without control terms as shown in [4,5,6]. Under the influence of the above studies, the finite-approximate controllability of fractional composite relaxation equations with nonlocal problems is dealt with in this paper.

Since Kalman [7] put forward the concept of controllability in 1960, it has aroused great interest among scholars. The issue of controllability is of great significance for the development of control theory. It has a wide range of applications in circuit system, artificial intelligence, biopharmaceutical, and so on. It is well known that from a mathematical point of view, controllability is generally divided into exact controllability and approximate controllability. In recent years, significant progress has been made in the study of exact controllability and approximate controllability of systems. Readers are encouraged to read the literature [8,9,10] for more information.

Since the application of fractional calculus in the field of control in the 1960s, the research of fractional control theory has experienced a long period of slow development until the end of the 20th century, and there have been some remarkable results. See [11,12,13]. In recent decades, many scholars have conducted in-depth research on the controllability of fractional differential equations, which has been widely developed in both theory and application. For example, in 2020, Hernández et al. [14] proved the existence, uniqueness and approximate controllability of abstract differential equations with state-dependent delay in Lipschitz continuous function spaces by the method of iterative. In 2023, Liang et al. [9] discussed the process controllability of the semilinear evolution equations by using the idea of integral and the method of approximation. Shukla et al. [15] dealt with the approximate controllability results for Hilfer fractional stochastic differential inclusions of order $1<q<2$ with the help of cosine operators family theory as well as the fixed-point theorem in 2024.

Finite-approximate controllability means that the final state of the system satisfies both the approximate controllability condition and a finite number of constraints (exact controllability on a finite-dimensional subspace). Finite-approximate controllability is a stronger notion than approximate controllability and is a more robust form of approximate controllability. It has specific applications in the fields of engineering, chemistry, physics and so on. See [16,17,18]. In 1997, Lions and Zuazua [19] proposed the finite-approximate controllability problem as a consequence of approximate controllability in abstract linear control systems. However, Zuazua pointed out in [20] that the two are not equivalent in the case of semilinearity. In recent years, there have been some rich research results on finite-approximate controllability. In 2021, Mahmudov [21] provided a useful criterion for finite-approximate controllability of the Riemann–Liouville fractional linear evolution equations using resolvent-like operators. And a variational approach was introduced to discuss the approximate controllability and finite-approximate controllability of Riemann–Liouville fractional linear/semilinear evolution equations in Hilbert spaces. In 2020, Wang et al. [22] studied the finite approximate controllability of fractional semilinear differential equations with Hilfer derivatives under certain natural conditions. In 2008, Menezes [23] demonstrated finite-approximate controllability for a class of nonlocal parabolic problems. For more references on this type, see [24,25,26] and the references therein.

For the study of mathematical and physical problems such as the passage of fluids through fractured rocks, diffusion of gases in transparent tubes, heat transfer, and so on, the classical initial conditions, i.e., the nonlocal conditions, were popularized by the famous scholar Byszewski [27,28]. In fact, nonlocal problems are more common and have better results than classical problems in practical applications. Therefore, differential equations for nonlocal problems have been extensively studied. The main difficulty in dealing with nonlocal problems is to obtain the compactness of the solution operator at zero, especially when the nonlocal terms are Lipschitz continuous or continuous. Many authors have developed different techniques and methods to address this problem. For example, Byszewski and Akca [29] established the existence of solutions to functional differential equations when the semigroup is compact and the nonlocal term is convex and compact on a given ball. Liang, Liu and Xiao [30] assumed several cases in which the nonlocal term g is Lipschitz, compact, not Lipschitz and not compact, respectively, and studied the mild and classical solutions of the nonlocal impulse problem in Banach space X under corresponding conditions. Xue [31] discussed new existence theorems for semilinear and nonlinear nonlocal problems in terms of the non-compact measures under the condition that the nonlocal condition g is not compact. For more on nonlocal problems, the interested reader is referred to [32,33,34].

As far as we know, there is still a gap in the research on the finite-approximate controllability of a class of fractional composite relaxation equations with nonlocal problems. In fact, we have addressed the finite-approximate controllability of the abstract fractional composite relaxation equations in [35]. In contrast to [35], the major difficulty in this paper lies in the influence of nonlocal term on the existence of solutions to Equation (1) and on whether finite-approximate controllability can be achieved. And the nonlocal problem is essentially the problem of understanding the compactness of the operator at $t=0$. We consider this article from the following aspects:

• We directly consider the case where the nonlocal term w is compact. The existence of solutions and the finite-approximate controllability to Equation (1) are obtained with the help of the resolvent theory, the variational method, and Schauder’s fixed point theorem.
• We weaken the condition for the nonlocal term w. Under the assumption that $w\left(\overline{co}{\mathsf{\Omega }}_{\epsilon }{O}_r\right)$ is pre-compact, where $\overline{co}D$ is the bounded convex closed hull of $D\subseteq C\left(\right[0,d];E)$, the existence of solutions and the finite-approximate controllability to Equation (1) are derived.
• To verify the feasibility of the results in this paper, the specific application is developed, where the nonlocal term w is compact.

The organization of this article is divided into the following five sections. Section 2 reviews some basic symbols and important knowledge of this article. Section 3 investigates the existence of mild solutions to Equation (1) in the framework, where the nonlocal term w is assumed to be compact. Section 4 presents the finite-approximate controllability for Equation (1), where the nonlocal term is compact. Section 5 deduces the existence of mild solutions and the finite-approximate controllability to Equation (1) for the weaker nonlocal problem. Section 6 verifies the finite-approximate controllability of the Basset problem in fluid dynamics with a nonlocal term.

2. Preliminaries

First, we review some symbols throughout this text. Let $d>0$ be fixed. E and F are Hilbert spaces. Denote $\mathbb{N}$, $\mathbb{R}$, and ${\mathbb{R}}_{+}$ as positive integers, real numbers, and non-negative real numbers, respectively. The space of all E-valued continuous functions on $[0,d]$ is denoted $C\left(\right[0,d];E)$ and endowed with the norm ${\parallel \eta \parallel }_C=\underset{0\le t\le d}{sup}\parallel \eta \left(t\right)\parallel$. The space of all E-valued Bochner integrable functions on $[0,d]$ is defined by $L^p([0,d];E)$ with the norm ${\parallel h\parallel }_{L^p}=({\int }_0^d{\parallel h\left(t\right)\parallel }^p{\mathrm{d}t)}^{{\displaystyle \frac{1}{p}}}$, where $1\le p<\infty$. The space of all E-valued essentially bounded functions on $[0,d]$ is recorded as $L^{\infty }([0,d];E)$ and ${\parallel h\parallel }_{L^{\infty }}=\mathrm{ess}sup\{h\left(t\right):0\le t\le d\}$.

For the completeness of exposition, we now recall the knowledge related to this article.

Definition 1

([36]). The Riemann–Liouville fractional integral of order $0<\beta <1$ with the lower limit zero for a function $q\in L^1([0,d];\mathbb{R})$ is defined as

$$I_0^{\beta }q\left(t\right)={\int }_0^t{\displaystyle \frac{{(t-\mu )}^{\beta -1}q\left(\mu \right)}{\mathsf{\Gamma }\left(\beta \right)}}\mathrm{d}\mu ,t\in [0,d],$$

where Γ is the Gamma function.

Definition 2

([36]). The Riemann–Liouville fractional derivative of order $0<\beta <1$ with the lower limit zero for a function $q\in L^1([0,d];\mathbb{R})$ is defined as

$${}^{L}D_0^{\beta }q\left(t\right)={\displaystyle \frac{{\mathrm{d}}^k}{\mathrm{d}{t}^k}}{\int }_0^t{\displaystyle \frac{{(t-\mu )}^{k-\beta -1}q\left(\mu \right)}{\mathsf{\Gamma }(k-\beta )}}\mathrm{d}\mu ,k-1<\beta \le k,k\in \mathbb{N}.$$

Definition 3

([36]). The Caputo fractional fractional derivative of order $0<\beta <1$ with the lower limit zero for a function $q\in L^1([0,d];\mathbb{R})$ is defined as

$${}^{c}D_0^{\beta }q\left(t\right)={}^{L}D_0^{\beta }\left(q\left(t\right)-\sum _{m=0}^{k-1}{\displaystyle \frac{t^m}{m!}}{q}^{\left(m\right)}\left(0\right)\right),t>0,k-1<\beta \le k,k\in \mathbb{N}.$$

If $q\in C\left(\right[0,d];\mathbb{R})$, then

$${}^{c}D_0^{\beta }q\left(t\right)=I_0^{k-\beta }{q}^{\left(k\right)}\left(\mu \right)={\int }_0^t{\displaystyle \frac{{(t-\mu )}^{k-\beta -1}{q}^{\left(k\right)}\left(\mu \right)}{\mathsf{\Gamma }(k-\beta )}}\mathrm{d}\mu ,t>0,k-1<\beta \le k,k\in \mathbb{N}.$$

Since $q\left(t\right)$ takes values in Hilbert space E, the integrals mentioned above are all in the sense of Bochner.

The densely defined closed linear operator $\mathcal{A}:D\left(\mathcal{A}\right)\subseteq E\to E$ in this article generates a resolvent family $\{S_{1-\beta }\left(t\right):t\ge 0\}$. Therefore, we provide relevant knowledge of the resolvent family, and more detailed information can be found in reference [37].

Definition 4

([37]). Assume that a densely determined closed linear operator $\mathcal{A}:D\left(\mathcal{A}\right)\subseteq E\to E$ generates an operator family $\{S_{1-\beta }\left(t\right):t\ge 0\}$ from E to E. $\{S_{1-\beta }\left(t\right):t\ge 0\}$ is said to be resolvent if the following conditions are true:

(i) 

$S_{1-\beta }\left(0\right)=I$ and $S_{1-\beta }(·)x\in C([0,+\infty );E)$, for any $x\in E$, where I is the identity operator;

(ii) 

$S_{1-\beta }\left(t\right)D\left(\mathcal{A}\right)\subset D\left(\mathcal{A}\right)$ and $\mathcal{A}{S}_{1-\beta }\left(t\right)x=S_{1-\beta }\left(t\right)\mathcal{A}x$, for every $t\ge 0$ and $x\in D\left(\mathcal{A}\right)$;

(iii) 

For every $t\ge 0$ and $x\in D\left(\mathcal{A}\right)$, the resolvent equation holds

$$\begin{array}{c}\hfill S_{1-\beta }x=x+{\int }_0^t{g}_{1-\beta }(t-\mu )\mathcal{A}{S}_{1-\beta }\left(\mu \right)x\mathrm{d}\mu ,\end{array}$$

(2)

where $g_{1-\beta }\left(t\right)={\displaystyle \frac{t^{-\beta }}{\mathsf{\Gamma }(1-\beta )}}$, for all $\beta \in (0,1)$.

In fact, for every $x\in E$, the above resolvent Equation (2) is also valid.

Definition 5.

If $\{S_{1-\beta }\left(t\right):t>0\}$ is a compact operator, then the resolvent $\{S_{1-\beta }\left(t\right):t>0\}$ is also compact.

Next, using resolvent theory as well as the Laplace transform, the mild solution to Equation (1) is established.

Definition 6.

For any ${\eta }_0\in E$ and the control function $v\in L^2([0,d];F)$, the function $\eta \in C\left(\right[0,d];E)$ is referred to as a mild solution of Equation (1) provided that

$$\begin{array}{c}\hfill \eta \left(t\right)={\eta }_0-w\left(\eta \right)-{\int }_0^t{S}_{1-\beta }(t-\mu )\eta \left(\mu \right)\mathrm{d}\mu +{\int }_0^t{S}_{1-\beta }(t-\mu )[h(\mu ,\eta \left(\mu \right))+\mathcal{B}v(\mu ,\eta )]\mathrm{d}\mu ,t\in [0,d].\end{array}$$

Then, we give the definition of finite-approximate controllability.

Definition 7.

Equation (1) is said to be finite-approximately controllable on $[0,d]$ if for any $\epsilon >0$, every initial state ${\eta }_0\in E$ and desired final state ${\eta }_d\in E$, there exists a control function $v_{\epsilon }(·)\in L^2([0,d];F)$ such that

$$\parallel \eta (d,v_{\epsilon })-{\eta }_d\parallel <\epsilon$$

and

$${\pi }_Z\eta (d,v_{\epsilon })={\pi }_Z{\eta }_d,$$

where $Z\subset E$ is the finite-dimensional and ${\pi }_Z:E\to Z$ is the orthogonal projection.

Remark 1.

Let $0<\beta <1$. It follows from Lemma 3.8 in [38] that an analytic resolvent of the analyticity type $({\omega }_0,{\theta }_0)$ is uniformly continuous for all $t>0$.

The following Lemma is crucial to the existence of a mild solution to Equation (1).

Lemma 1

([38]). Let $\{S_{1-\beta }\left(t\right):t\ge 0\}$ be a compact analytic resolvent, then for any $t>0$, we have the following conclusions:

(i) 

$\underset{\xi \to 0}{lim}\parallel S_{1-\beta }\left(t\right)-S_{1-\beta }(\xi +t)\parallel =0$;

(ii) 

$\underset{\xi \to 0^{+}}{lim}\parallel S_{1-\beta }\left(t\right)S_{1-\beta }\left(\xi \right)-S_{1-\beta }(\xi +t)\parallel =0$;

(iii) 

$\underset{\xi \to 0^{+}}{lim}\parallel S_{1-\beta }(t-\xi )S_{1-\beta }\left(\xi \right)-S_{1-\beta }\left(t\right)\parallel =0$.

Finally, we present the fixed-point theorem required in this article.

Lemma 2

([39], Schauder). Let K be a bounded convex closed subset of the space X, and $T:K\to K$ be a compact and continuous mapping. Then, T has a fixed point in K.

3. The Nonlocal Term w Is Compact in $C\left(\right[0,d];E)$

It is well known that the existence of mild solutions of Equation (1) is a sufficient condition to obtain that Equation (1) is controllable. Therefore, this section focuses on proving the existence of mild solutions to Equation (1). Firstly, we provide the following assumptions.

($Q_S$) ${\left\{S_{1-\beta }\left(t\right)\right\}}_{t\ge 0}$ is a compact resolvent family generated by $\mathcal{A}$ with $M_S:=\underset{t\in [0,d]}{sup}\parallel S_{1-\beta }\left(t\right)\parallel$.

($Q_h$) (i) The nonlinear function $h:[0,d]\times E\to E$ satisfies the carathédory condition, i.e., $h(·,x)$ is strongly measurable for all $x\in E$, $h(t,·)$ is continuous almost everywhere for $t\in [0,d]$;

(ii) There is a positive integrable function $\varphi \in L^{\infty }([0,d];E)$ and a non-decreasing continuous function ${{\rm Y}}_h\in C({\mathbb{R}}_{+};{\mathbb{R}}_{+})$ such that

$$\parallel h(t,\eta )\parallel \le \varphi \left(t\right){{\rm Y}}_h(\parallel \eta \parallel ),\mathrm{for}\mathrm{all}(t,\eta )\in [0,d]\times E,\mathrm{and}\underset{r\to \infty }{lim\; inf}{\displaystyle \frac{{{\rm Y}}_h\left(r\right)}{r}}={\lambda }_h<\infty .$$

($Q_w$) $w:C\left(\right[0,d];E)\to E$ is continuous. w is the compact operator and let $\parallel w\left(\eta \right)\parallel \le K_w$, for any $\eta \in C\left(\right[0,d];E)$ and some $K_w>0$.

(L) The corresponding linear equation

$$\begin{array}{c}\left\{\begin{array}{c}{\eta }^{\prime }\left(t\right)=\mathcal{A}{}^{c}D_0^{\beta }\eta \left(t\right)-\eta \left(t\right)+\mathcal{B}v\left(t\right),t\in [0,d],\beta \in (0,1),\hfill \\ \eta \left(0\right)={\eta }_0,\hfill \end{array}\right.\hfill \end{array}$$

(3)

is approximately controllable.

Based on the above assumptions, we can derive an important conclusion of this section.

Theorem 1.

Let the assumptions $\left(Q_S\right)$, $\left(Q_h\right)$, $\left(Q_w\right)$ and $\left(L\right)$ hold. Then, there exists at least one mild solution to Equation (1) provided that

$$\begin{array}{c}\hfill \underset{r_{\epsilon }\to \infty }{lim}{\displaystyle \frac{C_{\epsilon }\left(r_{\epsilon }\right)}{r_{\epsilon }}}<{\displaystyle \frac{1-d{M}_S{(1+\parallel \varphi \parallel }_{L^{\infty }}·{\lambda }_h)}{d{M}_S^2{\parallel \mathcal{B}\parallel }^2}},\end{array}$$

(4)

where $C_{\epsilon }\left(r_{\epsilon }\right)$ is defined in Lemma 8.

Next, to prove Theorem 1, we introduce the functional as follows:

$$J_{\epsilon }(·;x):E\to \mathbb{R},$$

and for any $\epsilon >0$ and $x\in C\left(\right[0,d];E)$, one has

$$J_{\epsilon }(\zeta ;x)={\displaystyle \frac{1}{2}}{\int }_0^d\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}{(d-\mu )\zeta \parallel }^2\mathrm{d}\mu +\epsilon \parallel (I-{\pi }_Z)\zeta \parallel -⟨\zeta ,\rho \left(x\right)⟩,\zeta \in E,$$

(5)

where ${\mathcal{B}}^{*}$ and $S_{1-\beta }^{*}$ are the adjoint of operators $\mathcal{B}$ and $S_{1-\beta }$. The mapping $\rho :C\left(\right[0,d];E)\to E$ and for any ${\eta }_d\in E$, we know

$$\rho \left(x\right)={\eta }_d-{\eta }_0+w\left(x\right)+{\int }_0^d{S}_{1-\beta }(d-\mu )x\left(\mu \right)\mathrm{d}\mu -{\int }_0^d{S}_{1-\beta }(d-\mu )h(\mu ,x\left(\mu \right))\mathrm{d}\mu .$$

Before demonstrating Theorem 1, we discuss some properties of functional $J_{\epsilon }(·;x)$ and $\rho$.

3.1. Related Properties of the Mapping $\rho$

Lemma 3.

ρ is a continuous mapping from $C\left(\right[0,d];E)$ to E.

Proof. 

Let $\left\{x_k\right\}$ be a sequence in $C\left(\right[0,d];E)$ and $\underset{k\to \infty }{lim}{x}_k=x$. For any $x_k,x\in C([0,d];E)$, we estimate that

$$\begin{array}{cc}\hfill & \parallel \rho \left(x_k\right)-\rho \left(x\right)\parallel \hfill \\ \hfill & =∥{\eta }_d-{\eta }_0+w\left(x_k\right)+{\int }_0^d{S}_{1-\beta }(d-\mu )x_k\left(\mu \right)\mathrm{d}\mu -{\int }_0^d{S}_{1-\beta }(d-\mu )h(\mu ,x_k\left(\mu \right))\mathrm{d}\mu -{\eta }_d+{\eta }_0-w\left(x\right)\hfill \\ \hfill & -{\int }_0^d{S}_{1-\beta }(d-\mu )x\left(\mu \right)\mathrm{d}\mu +{\int }_0^d{S}_{1-\beta }(d-\mu )h(\mu ,x\left(\mu \right))\mathrm{d}\mu ∥\hfill \\ \hfill & \le \parallel w\left(x_k\right)-w\left(x\right)\parallel +{\int }_0^d∥S_{1-\beta }(d-\mu )[x_k\left(\mu \right)-x\left(\mu \right)]∥\mathrm{d}\mu +{\int }_0^d∥S_{1-\beta }(d-\mu )[h(\mu ,x_k\left(\mu \right))-h(\mu ,x\left(\mu \right))]∥\mathrm{d}\mu \hfill \\ \hfill & \le \parallel w\left(x_k\right)-w\left(x\right)\parallel +M_S{\int }_0^d∥x_k\left(\mu \right)-x\left(\mu \right)∥\mathrm{d}\mu +M_S{\int }_0^d∥h(\mu ,x_k\left(\mu \right))-h(\mu ,x\left(\mu \right))∥\mathrm{d}\mu .\hfill \end{array}$$

The assumption ($Q_w$) shows that w is continuous. Therefore, $\parallel w\left(x_k\right)-w\left(x\right)\parallel \to 0$ as $k\to \infty$. In addition, we know from $\underset{k\to \infty }{lim}{\parallel x_k-x\parallel }_C=0$ and the assumption ($Q_h$)(i) that $∥x_k\left(\mu \right)-x\left(\mu \right)∥\to 0$ and $∥h(\mu ,x_k\left(\mu \right))-h(\mu ,x\left(\mu \right))∥\to 0$, as $k\to \infty$, for any $\mu \in [0,d]$. It follows from the Lebesgue dominated convergence theorem that ${\int }_0^d∥x_k\left(\mu \right)-x\left(\mu \right)∥\mathrm{d}\mu \to 0$ and ${\int }_0^d∥h(\mu ,x_k\left(\mu \right))-h(\mu ,x\left(\mu \right))∥\mathrm{d}\mu \to 0$, as $k\to \infty$. Then, $\parallel \rho \left(x_k\right)-\rho \left(x\right)\parallel \to 0$, as $k\to \infty$. Hence, $\rho$ is a continuous mapping. □

Lemma 4.

Let $O_r=\{\eta \in C([0,d];E):\parallel \eta \parallel \le r,r>0$ is a real number}. Then, for any $\left\{x_k\right\}\subset O_r$, $\rho \left(x_k\right)$ has a convergent subsequence in E.

Proof. 

Define

$$\begin{array}{c}\hfill {\rho }_{\epsilon }\left(x\right)={\eta }_d-{\eta }_0+w\left(x\right)+S_{1-\beta }\left(\epsilon \right){\int }_0^{d-\epsilon }{S}_{1-\beta }(d-\epsilon -\mu )x\left(\mu \right)\mathrm{d}\mu -S_{1-\beta }\left(\epsilon \right){\int }_0^{d-\epsilon }{S}_{1-\beta }(d-\epsilon -\mu )h(\mu ,x\left(\mu \right))\mathrm{d}\mu ,\end{array}$$

for any $0<\epsilon <d$ and $x\in O_r$. We know from the assumption ($Q_w$) that w is a compact operator, so $w\left(O_r\right)$ is relatively compact. Since

$$\begin{array}{cc}\hfill & ∥{\int }_0^{d-\epsilon }{S}_{1-\beta }(d-\epsilon -\mu )x\left(\mu \right)\mathrm{d}\mu -{\int }_0^{d-\epsilon }{S}_{1-\beta }(d-\epsilon -\mu )h(\mu ,x\left(\mu \right))\mathrm{d}\mu ∥\hfill \\ \hfill & \le M_S{\int }_0^{d-\epsilon }∥x\left(\mu \right)∥\mathrm{d}\mu +M_S{\int }_0^{d-\epsilon }∥h(\mu ,x(\mu \left)\right)∥\mathrm{d}\mu \hfill \\ \hfill & <M_{S}d\left(r+{\parallel \varphi \parallel }_{L^{\infty }}{{\rm Y}}_h(\parallel r\parallel )\right),\hfill \end{array}$$

it follows from the boundedness of ${\int }_0^{d-\epsilon }{S}_{1-\beta }(d-\epsilon -\mu )[x\left(\mu \right)-h(\mu ,x\left(\mu \right))]\mathrm{d}\mu$ and the compactness of $S_{1-\beta }\left(\epsilon \right)$ that $\{{\rho }_{\epsilon }\left(x\right):x\in O_r\}$ is relatively compact in E.

Moreover, one obtains

$$\begin{array}{cc}\hfill & \parallel \rho \left(x\right)-{\rho }_{\epsilon }\left(x\right)\parallel \hfill \\ \hfill & =∥{\int }_0^d{S}_{1-\beta }(d-\mu )[x\left(\mu \right)-h(\mu ,x\left(\mu \right))]\mathrm{d}\mu -S_{1-\beta }\left(\epsilon \right){\int }_0^{d-\epsilon }{S}_{1-\beta }(d-\epsilon -\mu )[x\left(\mu \right)-h(\mu ,x\left(\mu \right))]\mathrm{d}\mu ∥\hfill \\ \hfill & =∥{\int }_0^{d-\epsilon }{S}_{1-\beta }(d-\mu )[x\left(\mu \right)-h(\mu ,x\left(\mu \right))]\mathrm{d}\mu +{\int }_{d-\epsilon }^d{S}_{1-\beta }(d-\mu )[x\left(\mu \right)-h(\mu ,x\left(\mu \right))]\mathrm{d}\mu \hfill \\ \hfill & -S_{1-\beta }\left(\epsilon \right){\int }_0^{d-\epsilon }{S}_{1-\beta }(d-\epsilon -\mu )[x\left(\mu \right)-h(\mu ,x\left(\mu \right))]\mathrm{d}\mu ∥\hfill \\ \hfill & \le ∥{\int }_0^{d-\epsilon }{S}_{1-\beta }(d-\mu )[x\left(\mu \right)-h(\mu ,x\left(\mu \right))]\mathrm{d}\mu -S_{1-\beta }\left(\epsilon \right){\int }_0^{d-\epsilon }{S}_{1-\beta }(d-\epsilon -\mu )[x\left(\mu \right)-h(\mu ,x\left(\mu \right))]\mathrm{d}\mu ∥\hfill \\ \hfill & +M_S\epsilon \left(r+{\parallel \varphi \parallel }_{L^{\infty }}{{\rm Y}}_h(\parallel r\parallel )\right)\hfill \\ \hfill & \le {\int }_0^{d-\epsilon }∥S_{1-\beta }(d-\mu )-S_{1-\beta }\left(\epsilon \right)S_{1-\beta }(d-\epsilon -\mu )∥·∥x\left(\mu \right)-h(\mu ,x(\mu \left)\right)∥\mathrm{d}\mu +M_S\epsilon \left(r+{\parallel \varphi \parallel }_{L^{\infty }}{{\rm Y}}_h(\parallel r\parallel )\right).\hfill \end{array}$$

According to Lemma 1, for any $0\le \mu \le d-\epsilon$, we have

$$\begin{array}{c}\hfill \underset{\epsilon \to 0^{+}}{lim}∥S_{1-\beta }(d-\mu )-S_{1-\beta }\left(\epsilon \right)S_{1-\beta }(d-\epsilon -\mu )∥=0.\end{array}$$

Therefore, by the arbitrariness of $\epsilon$, we draw the conclusion that $\parallel \rho \left(x\right)-{\rho }_{\epsilon }\left(x\right)\parallel \to 0$. □

3.2. Related Properties of the Functional $J_{\epsilon }$

Before discussing the properties of $J_{\epsilon }$, we give the following conclusion.

Lemma 5

([40]). The assumption $\left(L\right)$ is satisfied if and only if ${\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\phi =0$, for any $\mu \in [0,d]$. This means that $\phi =0$.

More sufficient and necessary conditions for the approximate controllability of the linear system can be found in reference [40].

Lemma 6.

For any $\epsilon >0$ and $x\in C\left(\right[0,d];E)$, the functional $J_{\epsilon }(·;x):E\to \mathbb{R}$ is strictly convex and continuous.

Proof. 

From (5) and the Lebesgue dominated convergence theorem, we know that if $\underset{k\to \infty }{lim}{\zeta }_k=\zeta$, for any ${\zeta }_k,\zeta \in E$, then $J_{\epsilon }(·;x)$ is continuous.

Moreover, for any $0<\gamma <1$, ${\zeta }^{\prime },{\zeta }^{\prime \prime }\in E$, it is straightforward to verify that

$$\begin{array}{cc}\hfill & J_{\epsilon }(\gamma {\zeta }^{\prime }+(1-\gamma ){\zeta }^{\prime \prime };x)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\int }_0^d{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )[\gamma {\zeta }^{\prime }+(1-\gamma ){\zeta }^{\prime \prime }]∥}^2\mathrm{d}\mu +\epsilon ∥(I-{\pi }_Z)[\gamma {\zeta }^{\prime }+(1-\gamma ){\zeta }^{\prime \prime }]∥-⟨\gamma {\zeta }^{\prime }+(1-\gamma ){\zeta }^{\prime \prime },\rho \left(x\right)⟩\hfill \\ \hfill & \le {\displaystyle \frac{{\gamma }^2}{2}}{\int }_0^d{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime }∥}^2\mathrm{d}\mu +\epsilon \gamma ∥(I-{\pi }_Z){\zeta }^{\prime }∥+{\displaystyle \frac{{(1-\gamma )}^2}{2}}{\int }_0^d{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime \prime }∥}^2\mathrm{d}\mu +\epsilon (1-\gamma )∥(I-{\pi }_Z){\zeta }^{\prime \prime }∥\hfill \\ \hfill & +\gamma (1-\gamma ){\int }_0^d∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime }∥∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime \prime }∥\mathrm{d}\mu -\gamma ⟨{\zeta }^{\prime },\rho \left(x\right)⟩-(1-\gamma )⟨{\zeta }^{\prime \prime },\rho \left(x\right)⟩.\hfill \end{array}$$

Since

$$\begin{array}{c}\hfill 2∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime }∥∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime \prime }∥⩽{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime }∥}^2+{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime \prime }∥}^2.\end{array}$$

If

$$\begin{array}{c}\hfill ∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime }∥∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime \prime }∥={∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime }∥}^2+{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime \prime }∥}^2,\end{array}$$

then it follows from Lemma 5 that ${\zeta }^{\prime }={\zeta }^{\prime \prime }$. Thus,

$$\begin{array}{c}\hfill 2{\int }_0^d∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime }∥∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime \prime }∥\mathrm{d}\mu <{\int }_0^d{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime }∥}^2\mathrm{d}\mu +{\int }_0^d{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime \prime }∥}^2\mathrm{d}\mu .\end{array}$$

Then

$$\begin{array}{cc}\hfill & J_{\epsilon }(\gamma {\zeta }^{\prime }+(1-\gamma ){\zeta }^{\prime \prime };x)\hfill \\ \hfill & <{\displaystyle \frac{{\gamma }^2}{2}}{\int }_0^d{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime }∥}^2\mathrm{d}\mu +\epsilon \gamma ∥(I-{\pi }_Z){\zeta }^{\prime }∥+{\displaystyle \frac{{(1-\gamma )}^2}{2}}{\int }_0^d{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime \prime }∥}^2\mathrm{d}\mu +\epsilon (1-\gamma )∥(I-{\pi }_Z){\zeta }^{\prime \prime }∥\hfill \\ \hfill & +{\displaystyle \frac{\gamma (1-\gamma )}{2}}{\int }_0^d{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime }∥}^2\mathrm{d}\mu -\gamma ⟨{\zeta }^{\prime },\rho \left(x\right)⟩+{\displaystyle \frac{\gamma (1-\gamma )}{2}}{\int }_0^d{∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }^{\prime \prime }∥}^2\mathrm{d}\mu -(1-\gamma )⟨{\zeta }^{\prime \prime },\rho \left(x\right)⟩\hfill \\ \hfill & =\gamma J_{\epsilon }({\zeta }^{\prime };x)+(1-\gamma )J_{\epsilon }({\zeta }^{\prime \prime };x).\hfill \end{array}$$

In summary, by the definition of strict convexity, the functional $J_{\epsilon }(·;x)$ is strictly convex and continuous. □

Lemma 7.

If the assumption $\left(L\right)$ is valid, then for any $\epsilon >0$ and $\zeta \in E$, the following inequality holds:

$$\begin{array}{c}\hfill \underset{\zeta \to \infty }{\underline{lim}}\underset{x\in O_r}{inf}{\displaystyle \frac{J_{\epsilon }(\zeta ;x)}{\parallel \zeta \parallel }}\ge \epsilon .\end{array}$$

(6)

Proof. 

Suppose, towards a contradiction, that there are subsequences $\left\{{\zeta }_k\right\}\subset E$ and $\left\{x_k\right\}\subset O_r$ with $\parallel {\zeta }_k\parallel \to \infty$ such that

$$\begin{array}{c}\hfill \underset{{\zeta }_k\to \infty }{\underline{lim}}\underset{x_k\in O_r}{inf}{\displaystyle \frac{J_{\epsilon }({\zeta }_k;x_k)}{\parallel {\zeta }_k\parallel }}<\epsilon .\end{array}$$

(7)

According to the expression (5) for the functional $J_{\epsilon }$, we can see that

$$\begin{array}{c}\hfill J_{\epsilon }({\zeta }_k;x_k)={\displaystyle \frac{1}{2}}{\int }_0^d\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }_k{\parallel }^2\mathrm{d}\mu +\epsilon \parallel (I-{\pi }_Z){\zeta }_k\parallel -⟨{\zeta }_k,\rho \left(x_k\right)⟩.\end{array}$$

The normalization of ${\zeta }_k$ yields

$$\begin{array}{c}\hfill {\displaystyle \frac{J_{\epsilon }({\zeta }_k;x_k)}{\parallel {\zeta }_k\parallel }}={\displaystyle \frac{\parallel {\zeta }_k\parallel }{2}}{\int }_0^d\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\overline{{\zeta }_k}{\parallel }^2\mathrm{d}\mu +\epsilon \parallel (I-{\pi }_Z)\overline{{\zeta }_k}\parallel -⟨\overline{{\zeta }_k},\rho \left(x_k\right)⟩,\end{array}$$

where $\overline{{\zeta }_k}={\displaystyle \frac{{\zeta }_k}{\parallel {\zeta }_k\parallel }}$ and $\parallel \overline{{\zeta }_k}\parallel =1$. Thus, a subsequence (still denoted $\overline{{\zeta }_k}$) can be extracted such that $\overline{{\zeta }_k}\rightharpoonup \overline{\zeta }$. Furthermore, since $\{S_{1-\beta }\left(t\right):t\ge 0\}$ is a compact resolvent and $\mathcal{B}$ is continuous, we have

$$\begin{array}{c}\hfill \underset{\mu \in [0,d]}{sup}∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\overline{{\zeta }_k}-{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\overline{\zeta }∥\to 0.\end{array}$$

Then

$$\begin{array}{c}\hfill \underset{k\to \infty }{\underline{lim}}{\int }_0^d{\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\overline{{\zeta }_k}\parallel }^2\mathrm{d}\mu =0,\end{array}$$

and

$$\begin{array}{cc}\hfill {\int }_0^d{\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\overline{\zeta }\parallel }^2\mathrm{d}\mu & ={\int }_0^d\underset{k\to \infty }{\underline{lim}}{\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\overline{{\zeta }_k}\parallel }^2\mathrm{d}\mu \hfill \\ \hfill & \le \underset{k\to \infty }{\underline{lim}}{\int }_0^d{\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\overline{{\zeta }_k}\parallel }^2\mathrm{d}\mu =0,\hfill \end{array}$$

this is obtained from the Fatou lemma. Therefore, based on the hypothesis (L) and Lemma 5, we can deduce that $\overline{\zeta }=0$ and $\overline{{\zeta }_k}\rightharpoonup 0$. Additionally, it follows from the compactness of ${\pi }_Z$ that ${\pi }_Z\overline{{\zeta }_k}\to 0$ in E. Then,

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel (I-{\pi }_Z)\overline{{\zeta }_k}\parallel =\underset{k\to \infty }{lim}\sqrt{⟨(I-{\pi }_Z)\overline{{\zeta }_k},(I-{\pi }_Z)\overline{{\zeta }_k}⟩}=\underset{k\to \infty }{lim}\sqrt{\parallel \overline{{\zeta }_k}{\parallel }^2+{\parallel {\pi }_Z\overline{{\zeta }_k}\parallel }^2}=1.\end{array}$$

On the other hand, we know from Lemma 4 that $\rho \left(x_k\right)$ is relatively compact. Thus, $⟨\overline{{\zeta }_k},\rho \left(x_k\right)⟩\to 0.$ Then,

$$\begin{array}{cc}\hfill \underset{{\zeta }_k\to \infty }{\underline{lim}}{\displaystyle \frac{J_{\epsilon }({\zeta }_k;x_k)}{\parallel {\zeta }_k\parallel }}& =\underset{{\zeta }_k\to \infty }{\underline{lim}}({\displaystyle \frac{\parallel {\zeta }_k\parallel }{2}}{\int }_0^d\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\overline{{\zeta }_k}{\parallel }^2\mathrm{d}\mu +\epsilon \parallel (I-{\pi }_Z)\overline{{\zeta }_k}\parallel -⟨\overline{{\zeta }_k},\rho \left(x_k\right)⟩)\hfill \\ \hfill & \ge \underset{{\zeta }_k\to \infty }{\underline{lim}}(\epsilon \parallel (I-{\pi }_Z)\overline{{\zeta }_k}\parallel -⟨\overline{{\zeta }_k},\rho \left(x_k\right)⟩)=\epsilon .\hfill \end{array}$$

We arrive at a contradiction. □

Based on Lemmas 5–7, define the mapping

$$\begin{array}{cc}\hfill {\mathfrak{P}}_{\epsilon }:C([0,d];E)& \to E\hfill \\ \hfill x& \mapsto \widetilde{{\zeta }_{\epsilon }},\mathrm{for}\mathrm{any}x\in C([0,d];E),\hfill \end{array}$$

where $\widetilde{{\zeta }_{\epsilon }}$ is a unique minimum of $J_{\epsilon }(·;x)$. Next, the properties of ${\mathfrak{P}}_{\epsilon }$ are addressed.

3.3. Related Properties of the Mapping ${\mathfrak{P}}_{\epsilon }$

Lemma 8.

There exists a constant $C_{\epsilon }\left(r\right)>0$ such that $\parallel {\mathfrak{P}}_{\epsilon }\left(x\right)\parallel \le C_{\epsilon }\left(r\right)$, for any $x\in O_r$. Here $\left\{{\mathfrak{P}}_{\epsilon }\left(x\right)\right\}$ is the minimum point set.

Proof. 

By Lemma 7, for any $x\in O_r$, there exists a constant $C_{\epsilon }\left(r\right)>0$ such that $\parallel \zeta \parallel >C_{\epsilon }\left(r\right)$, and

$$\begin{array}{c}\hfill \underset{x\in O_r}{inf}{\displaystyle \frac{J_{\epsilon }(\zeta ;x)}{\parallel \zeta \parallel }}\ge {\displaystyle \frac{\epsilon }{2}}.\end{array}$$

If ${\mathfrak{P}}_{\epsilon }$ is not bounded, we may assume that there is $x^{\prime }\in O_r$ such that $\parallel {\mathfrak{P}}_{\epsilon }\left(x^{\prime }\right)\parallel >C_{\epsilon }\left(r\right)$. Then, one has

$$\begin{array}{c}\hfill \underset{x^{\prime }\in O_r}{inf}{\displaystyle \frac{J_{\epsilon }({\mathfrak{P}}_{\epsilon }\left(x^{\prime }\right);x^{\prime })}{\parallel {\mathfrak{P}}_{\epsilon }\left(x^{\prime }\right)\parallel }}\ge {\displaystyle \frac{\epsilon }{2}}.\end{array}$$

However, we infer from the definition of ${\mathfrak{P}}_{\epsilon }$ that

$$\begin{array}{c}\hfill J_{\epsilon }({\mathfrak{P}}_{\epsilon }\left(x\right);x)\le J_{\epsilon }(0;x)=0,\mathrm{for}\mathrm{any}x\in O_r.\end{array}$$

This is contradictory to

$$\begin{array}{c}\hfill \underset{x^{\prime }\in O_r}{inf}{\displaystyle \frac{J_{\epsilon }({\mathfrak{P}}_{\epsilon }\left(x^{\prime }\right);x^{\prime })}{\parallel {\mathfrak{P}}_{\epsilon }\left(x^{\prime }\right)\parallel }}\ge {\displaystyle \frac{\epsilon }{2}}.\end{array}$$

Thus, we obtain

$$\parallel {\mathfrak{P}}_{\epsilon }\left(x\right)\parallel \le C_{\epsilon }\left(r\right),\mathrm{for}\mathrm{any}x\in O_r.$$

□

Lemma 9.

${\mathfrak{P}}_{\epsilon }\left(x_k\right)\rightharpoonup {\mathfrak{P}}_{\epsilon }\left(x\right)$ and $\underset{k\to \infty }{lim}\parallel {\mathfrak{P}}_{\epsilon }\left(x_k\right)\parallel =\parallel {\mathfrak{P}}_{\epsilon }\left(x\right)\parallel$ provided that $x_k\to x$ in $O_r$.

Proof. 

On one hand, because of the definition of ${\mathfrak{P}}_{\epsilon }$, we get $\left\{\widetilde{{\zeta }_{\epsilon ,k}}\right\}=\left\{{\mathfrak{P}}_{\epsilon }\left(x_k\right)\right\}$, for any $x_k\in O_r$. Let $\widetilde{{\zeta }_{\epsilon }}$ be the minimum of $J_{\epsilon }(·;x)$ and $\widetilde{{\zeta }_{\epsilon }}={\mathfrak{P}}_{\epsilon }\left(x\right)$. Next verify that $\widetilde{{\zeta }_{\epsilon ,k}}\rightharpoonup \widetilde{{\zeta }_{\epsilon }}$. By Lemma 8, $\left\{\widetilde{{\zeta }_{\epsilon ,k}}\right\}=\left\{{\mathfrak{P}}_{\epsilon }\left(x_k\right)\right\}$ is bounded. Thus, extracting a subsequence is still denoted as $\widetilde{{\zeta }_{\epsilon ,k}}$ such that $\widetilde{{\zeta }_{\epsilon ,k}}\rightharpoonup \widehat{{\zeta }_{\epsilon }}$, as $k\to \infty$. Using the optimality of $\left\{\widetilde{{\zeta }_{\epsilon ,k}}\right\}=\left\{{\mathfrak{P}}_{\epsilon }\left(x_k\right)\right\}$ and $\widetilde{{\zeta }_{\epsilon }}={\mathfrak{P}}_{\epsilon }\left(x\right)$, we can obtain

$$\begin{array}{c}\hfill J_{\epsilon }(\widetilde{{\zeta }_{\epsilon }};x)\le J_{\epsilon }(\widehat{{\zeta }_{\epsilon }};x)\le \underset{k\to \infty }{\underline{lim}}{J}_{\epsilon }(\widetilde{{\zeta }_{\epsilon ,k}};x_k)\le \underset{k\to \infty }{\overline{lim}}{J}_{\epsilon }(\widetilde{{\zeta }_{\epsilon ,k}};x_k)\le \underset{k\to \infty }{lim}{J}_{\epsilon }(\widetilde{{\zeta }_{\epsilon }};x_k)=J_{\epsilon }(\widetilde{{\zeta }_{\epsilon }};x).\end{array}$$

Then, $J_{\epsilon }(\widetilde{{\zeta }_{\epsilon }};x)=J_{\epsilon }(\widehat{{\zeta }_{\epsilon }};x).$ The uniqueness of the minimum yields $\widetilde{{\zeta }_{\epsilon }}=\widehat{{\zeta }_{\epsilon }}$. Hence $\underset{k\to \infty }{lim}{J}_{\epsilon }(\widetilde{{\zeta }_{\epsilon ,k}};x_k)=J_{\epsilon }(\widetilde{{\zeta }_{\epsilon }};x).$

On the other hand, according to the expression (5) of the functional $J_{\epsilon }$, one has

$$\begin{array}{c}\hfill J_{\epsilon }({\zeta }_k;x_k)={\displaystyle \frac{1}{2}}{\int }_0^d\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\zeta }_k{\parallel }^2\mathrm{d}\mu +\epsilon \parallel (I-{\pi }_Z){\zeta }_k\parallel -⟨{\zeta }_k,\rho \left(x_k\right)⟩.\end{array}$$

Based on the fact that $\{S_{1-\beta }\left(t\right):t\ge 0\}$ is the compact resolvent, $\rho$ is continuous and $\widetilde{{\zeta }_{\epsilon ,k}}\rightharpoonup \widetilde{{\zeta }_{\epsilon }}$, one has

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\int }_0^d\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\widetilde{{\zeta }_{\epsilon ,k}}{\parallel }^2\mathrm{d}\mu ={\int }_0^d{\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\widetilde{{\zeta }_{\epsilon }}\parallel }^2\mathrm{d}\mu ,\end{array}$$

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}⟨\widetilde{{\zeta }_{\epsilon ,k}},\rho \left(x\right)⟩=⟨\widetilde{{\zeta }_{\epsilon }},\rho \left(x\right)⟩,\end{array}$$

and

$$\begin{array}{c}\hfill \parallel (I-{\pi }_Z)\widetilde{{\zeta }_{\epsilon }}\parallel \le \underset{k\to \infty }{lim}\parallel (I-{\pi }_Z)\widetilde{{\zeta }_{\epsilon ,k}}\parallel .\end{array}$$

Using the weak convergence and weak lower semicontinuity of the norm in E, we get

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel (I-{\pi }_Z)\widetilde{{\zeta }_{\epsilon ,k}}\parallel =\parallel (I-{\pi }_Z)\widetilde{{\zeta }_{\epsilon }}\parallel .\end{array}$$

Furthermore, since ${\pi }_Z$ is compact, then

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel \widetilde{{\zeta }_{\epsilon ,k}}{\parallel }^2=\underset{k\to \infty }{lim}\parallel (I-{\pi }_Z)\widetilde{{\zeta }_{\epsilon ,k}}{\parallel }^2+\underset{k\to \infty }{lim}\parallel {\pi }_Z\widetilde{{\zeta }_{\epsilon ,k}}{\parallel }^2=\parallel (I-{\pi }_Z)\widetilde{{\zeta }_{\epsilon }}{\parallel }^2+\parallel {\pi }_Z\widetilde{{\zeta }_{\epsilon }}{\parallel }^2={\parallel \widetilde{{\zeta }_{\epsilon }}\parallel }^2.\end{array}$$

□

Now, we directly prove Theorem 1.

Proof of Theorem 1. 

Construct the operator ${\mathsf{\Omega }}_{\epsilon }$ from $C\left(\right[0,d];E)$ to $C\left(\right[0,d];E)$ as follows

$$\begin{array}{cc}\hfill \left({\mathsf{\Omega }}_{\epsilon }\eta \right)\left(t\right)& ={\eta }_0-w\left(\eta \right)-{\int }_0^t{S}_{1-\beta }(t-\mu )\eta \left(\mu \right)\mathrm{d}\mu +{\int }_0^t{S}_{1-\beta }(t-\mu )[h(\mu ,\eta \left(\mu \right))+\mathcal{B}{v}_{\epsilon }(\mu ,\eta )]\mathrm{d}\mu ,\hfill \end{array}$$

for any ${\eta }_0\in E$ and $t\in [0,d]$, where

$$\begin{array}{c}\hfill v_{\epsilon }(\mu ,\eta ):={\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\mathfrak{P}}_{\epsilon }\left(\eta \right).\end{array}$$

(8)

We divide into the following four steps to derive that the operator ${\mathsf{\Omega }}_{\epsilon }$ exists at least one fixed point.

Step 1. Demonstrate that the operator ${\mathsf{\Omega }}_{\epsilon }$ maps from itself to itself. By the assumptions ($Q_h$)(ii), ($Q_w$), (8) and Lemma 8, for any $\epsilon >0$ and $\eta \in O_{r_{\epsilon }}$, one obtains

$$\begin{array}{cc}\hfill \parallel \left({\mathsf{\Omega }}_{\epsilon }\eta \right)\left(t\right)\parallel & =∥{\eta }_0-w\left(\eta \right)-{\int }_0^t{S}_{1-\beta }(t-\mu )\eta \left(\mu \right)\mathrm{d}\mu +{\int }_0^t{S}_{1-\beta }(t-\mu )[h(\mu ,\eta \left(\mu \right))+\mathcal{B}{v}_{\epsilon }(\mu ,\eta )]\mathrm{d}\mu ∥\hfill \\ \hfill & \le \parallel {\eta }_0\parallel +K_w+M_S{\int }_0^t∥\eta \left(\mu \right)∥\mathrm{d}\mu +M_S{\int }_0^t∥h(\mu ,\eta (\mu \left)\right)∥\mathrm{d}\mu +M_S{\int }_0^t∥\mathcal{B}{v}_{\epsilon }(\mu ,\eta )∥\mathrm{d}\mu \hfill \\ \hfill & \le \parallel {\eta }_0\parallel +K_w+M_{S}d\left(r_{\epsilon }+{\parallel \varphi \parallel }_{L^{\infty }}{{\rm Y}}_h(\parallel r_{\epsilon }\parallel )+M_S{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right)\right).\hfill \end{array}$$

From (4), we know that $\parallel {\mathsf{\Omega }}_{\epsilon }\eta \parallel \le r_{\epsilon }$, where $r_{\epsilon }>0$ is sufficiently large.

Step 2. Discuss that the operator ${\mathsf{\Omega }}_{\epsilon }$ is continuous. Let ${\left\{{\eta }_k\right\}}_{k\in N}$ be a subsequence in $C\left(\right[0,d];E)$ and $\underset{k\to \infty }{lim}{\eta }_k=\eta$. Then, for any $t\in [0,d]$, one has

$$\begin{array}{cc}\hfill & \parallel \left({\mathsf{\Omega }}_{\epsilon }{\eta }_k\right)\left(t\right)-\left({\mathsf{\Omega }}_{\epsilon }\eta \right)\left(t\right)\parallel \hfill \\ \hfill & \le \parallel w\left({\eta }_k\right)-w\left(\eta \right)\parallel +{\int }_0^t∥S_{1-\beta }(t-\mu )[{\eta }_k\left(\mu \right)-\eta \left(\mu \right)]∥\mathrm{d}\mu +{\int }_0^t∥S_{1-\beta }(t-\mu )[h(\mu ,{\eta }_k\left(\mu \right))-h(\mu ,\eta \left(\mu \right))]∥\mathrm{d}\mu \hfill \\ \hfill & +{\int }_0^t∥S_{1-\beta }(t-\mu )[\mathcal{B}{v}_{\epsilon }(\mu ,{\eta }_k)-\mathcal{B}{v}_{\epsilon }(\mu ,\eta )]∥\mathrm{d}\mu \hfill \\ \hfill & \le \parallel w\left({\eta }_k\right)-w\left(\eta \right)\parallel +M_S{\int }_0^t∥{\eta }_k\left(\mu \right)-\eta \left(\mu \right)∥\mathrm{d}\mu +M_S{\int }_0^t∥h(\mu ,{\eta }_k\left(\mu \right))-h(\mu ,\eta \left(\mu \right))∥\mathrm{d}\mu \hfill \\ \hfill & +M_S\parallel \mathcal{B}\parallel {\int }_0^t∥v_{\epsilon }(\mu ,{\eta }_k)-v_{\epsilon }(\mu ,\eta )∥\mathrm{d}\mu .\hfill \end{array}$$

Based on the assumptions ($Q_h$), ($Q_w$) and the Lebesgue dominated convergence theorem, we know that $\parallel w\left({\eta }_k\right)-w\left(\eta \right)\parallel \to 0$, ${\int }_0^t∥{\eta }_k\left(\mu \right)-\eta \left(\mu \right)∥\mathrm{d}\mu \to 0$ and ${\int }_0^t∥h(\mu ,{\eta }_k\left(\mu \right))-h(\mu ,\eta \left(\mu \right))∥\mathrm{d}\mu \to 0$, as $k\to \infty$.

In addition, by (8) and Lemma 9, one finds

$$\begin{array}{cc}\hfill \parallel v_{\epsilon }(\mu ,{\eta }_k)-v_{\epsilon }(\mu ,\eta )\parallel & =∥{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )[{\mathfrak{P}}_{\epsilon }\left({\eta }_k\right)-{\mathfrak{P}}_{\epsilon }\left(\eta \right)]∥\hfill \\ \hfill & \le M_S\parallel \mathcal{B}\parallel ·\parallel {\mathfrak{P}}_{\epsilon }\left({\eta }_k\right)-{\mathfrak{P}}_{\epsilon }\left(\eta \right)\parallel \to 0,\mathrm{as}k\to \infty .\hfill \end{array}$$

Therefore, ${\mathsf{\Omega }}_{\epsilon }$ is continuous.

Step 3. Consider the set $\{{\mathsf{\Omega }}_{\epsilon }\eta :\eta \in O_{r_{\epsilon }}\}$ is equicontinuous in $C\left(\right[0,d];E)$. For any $0\le s<t\le d$ and $\sigma >0$ that is sufficiently small, we estimate

$$\begin{array}{cc}\hfill & \parallel \left({\mathsf{\Omega }}_{\epsilon }\eta \right)\left(t\right)-\left({\mathsf{\Omega }}_{\epsilon }\eta \right)\left(s\right)\parallel \hfill \\ \hfill & =∥{\int }_0^t{S}_{1-\beta }(t-\mu )\eta \left(\mu \right)\mathrm{d}\mu -{\int }_0^s{S}_{1-\beta }(s-\mu )\eta \left(\mu \right)\mathrm{d}\mu ∥+∥{\int }_0^t{S}_{1-\beta }(t-\mu )h(\mu ,\eta \left(\mu \right))\mathrm{d}\mu -{\int }_0^s{S}_{1-\beta }(s-\mu )h(\mu ,\eta \left(\mu \right))\mathrm{d}\mu ∥\hfill \\ \hfill & +∥{\int }_0^t{S}_{1-\beta }(t-\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu -{\int }_0^s{S}_{1-\beta }(s-\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu ∥\hfill \\ \hfill & \le {\int }_0^{s-\sigma }∥S_{1-\beta }(t-\mu )-S_{1-\beta }(s-\mu )∥∥\eta \left(\mu \right)∥\mathrm{d}\mu +{\int }_{s-\sigma }^s∥S_{1-\beta }(t-\mu )-S_{1-\beta }(s-\mu )∥∥\eta \left(\mu \right)∥\mathrm{d}\mu \hfill \\ \hfill & +{\int }_s^t∥S_{1-\beta }(t-\mu )∥∥\eta \left(\mu \right)∥\mathrm{d}\mu +{\int }_0^{s-\sigma }∥S_{1-\beta }(t-\mu )-S_{1-\beta }(s-\mu )∥∥h(\mu ,\eta (\mu \left)\right)∥\mathrm{d}\mu \hfill \\ \hfill & +{\int }_{s-\sigma }^s∥S_{1-\beta }(t-\mu )-S_{1-\beta }(s-\mu )∥∥h(\mu ,\eta (\mu \left)\right)∥\mathrm{d}\mu +{\int }_s^t∥S_{1-\beta }(t-\mu )∥∥h(\mu ,\eta (\mu \left)\right)∥\mathrm{d}\mu \hfill \\ \hfill & +{\int }_0^{s-\sigma }∥S_{1-\beta }(t-\mu )-S_{1-\beta }(s-\mu )∥∥\mathcal{B}{v}_{\epsilon }(\mu ,\eta )∥\mathrm{d}\mu +{\int }_{s-\sigma }^s∥S_{1-\beta }(t-\mu )-S_{1-\beta }(s-\mu )∥∥\mathcal{B}{v}_{\epsilon }(\mu ,\eta )∥\mathrm{d}\mu \hfill \\ \hfill & +{\int }_s^t∥S_{1-\beta }(t-\mu )-S_{1-\beta }(s-\mu )∥∥\mathcal{B}{v}_{\epsilon }(\mu ,\eta )∥\mathrm{d}\mu \hfill \\ \hfill & \le \left(r_{\epsilon }+{\parallel \varphi \parallel }_{L^{\infty }}{{\rm Y}}_h(\parallel r_{\epsilon }\parallel )+M_S{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right)\right){\int }_0^{s-\sigma }∥S_{1-\beta }(t-\mu )-S_{1-\beta }(s-\mu )∥\mathrm{d}\mu \hfill \\ \hfill & +2M_S\sigma \left(r_{\epsilon }+{\parallel \varphi \parallel }_{L^{\infty }}{{\rm Y}}_h(\parallel r_{\epsilon }\parallel )+M_S{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right)\right)+M_S\left(t-s\right)\left(r_{\epsilon }+{\parallel \varphi \parallel }_{L^{\infty }}{{\rm Y}}_h(\parallel r_{\epsilon }\parallel )+M_S{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right)\right).\hfill \end{array}$$

It follows from the arbitrariness of $\sigma$ and Lemma 1 that

$$\underset{s\to t}{lim}∥S_{1-\beta }(t-\mu )-S_{1-\beta }(s-\mu )∥=0.$$

Hence, $\parallel \left({\mathsf{\Omega }}_{\epsilon }\eta \right)\left(t\right)-\left({\mathsf{\Omega }}_{\epsilon }\eta \right)\left(s\right)\parallel \to 0$, as $s\to t$.

Step 4. Verify that the set $\{{\mathsf{\Omega }}_{\epsilon }\eta :\eta \in O_{r_{\epsilon }}\}$ is relatively compact in $C\left(\right[0,d];E)$. For any $t\in [0,d]$, define

$$\left({\mathsf{\Omega }}_{\epsilon }\eta \right)\left(t\right):=\left({\mathsf{\Omega }}_{\epsilon }^1\eta \right)\left(t\right)+\left({\mathsf{\Omega }}_{\epsilon }^2\eta \right)\left(t\right),$$

where

$$\begin{array}{c}\hfill \left({\mathsf{\Omega }}_{\epsilon }^1\eta \right)\left(t\right)={\eta }_0-w\left(\eta \right)-{\int }_0^t{S}_{1-\beta }(t-\mu )\eta \left(\mu \right)\mathrm{d}\mu +{\int }_0^t{S}_{1-\beta }(t-\mu )h(\mu ,\eta \left(\mu \right))\mathrm{d}\mu \end{array}$$

and

$$\begin{array}{c}\hfill \left({\mathsf{\Omega }}_{\epsilon }^2\eta \right)\left(t\right)={\int }_0^t{S}_{1-\beta }(t-\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu .\end{array}$$

We consider the following two situations.

(i) If $t=0$, according to the boundedness, one concludes that

$$\left({\mathsf{\Omega }}_{\epsilon }\eta \right)\left(0\right)=\left({\mathsf{\Omega }}_{\epsilon }^1\eta \right)\left(0\right)+\left({\mathsf{\Omega }}_{\epsilon }^2\eta \right)\left(0\right)={\eta }_0-w\left(\eta \right).$$

$\left({\mathsf{\Omega }}_{\epsilon }\eta \right)\left(0\right)$ is obviously relatively compact by the assumption ($Q_w$).

(ii) For any $t\in (0,d]$, we consider the relative compactness of $\left({\mathsf{\Omega }}_{\epsilon }\eta \right)\left(t\right)$. In view of Lemma 4, we observe that $\left({\mathsf{\Omega }}_{\epsilon }^1\eta \right)\left(t\right)$ is relatively compact. Now, for every $0<\tau <t$, define

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_{\epsilon ,\tau }^2\eta =S_{1-\beta }\left(\tau \right){\int }_0^{t-\tau }{S}_{1-\beta }(t-\tau -\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu .\end{array}$$

Since $v_{\epsilon }(\mu ,\eta )={\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\mathfrak{P}}_{\epsilon }\left(\eta \right)$,

$$\begin{array}{cc}\hfill ∥{\int }_0^{t-\tau }{S}_{1-\beta }(t-\tau -\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu ∥& =∥{\int }_0^{t-\tau }{S}_{1-\beta }(t-\tau -\mu )\mathcal{B}{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\mathfrak{P}}_{\epsilon }\left(\eta \right)\mathrm{d}\mu ∥\hfill \\ \hfill & \le M_S^2{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right)d.\hfill \end{array}$$

Then, it is easy to derive that $\left({\mathsf{\Omega }}_{\epsilon ,\tau }^2\eta \right)\left(t\right)$ is relatively compact via the assumption ($Q_S$).

Moreover, for any $\tau <\sigma <d$ and $\eta \in O_{r_{\epsilon }}$, one observes that

$$\begin{array}{cc}\hfill & \parallel \left({\mathsf{\Omega }}_{\epsilon ,\tau }^2\eta \right)\left(t\right)-\left({\mathsf{\Omega }}_{\epsilon }^2\eta \right)\left(t\right)\parallel \hfill \\ \hfill & =∥S_{1-\beta }\left(\tau \right){\int }_0^{t-\tau }{S}_{1-\beta }(t-\tau -\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu -{\int }_0^t{S}_{1-\beta }(t-\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu ∥\hfill \\ \hfill & \le ∥S_{1-\beta }\left(\tau \right){\int }_0^{t-\tau }{S}_{1-\beta }(t-\tau -\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu -{\int }_0^{t-\tau }{S}_{1-\beta }(t-\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu ∥+∥{\int }_{t-\tau }^t{S}_{1-\beta }(t-\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu ∥\hfill \\ \hfill & \le ∥S_{1-\beta }\left(\tau \right){\int }_0^{t-\sigma }{S}_{1-\beta }(t-\tau -\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu -{\int }_0^{t-\sigma }{S}_{1-\beta }(t-\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu ∥\hfill \\ \hfill & +∥S_{1-\beta }\left(\tau \right){\int }_{t-\sigma }^{t-\tau }{S}_{1-\beta }(t-\tau -\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu -{\int }_{t-\sigma }^{t-\tau }{S}_{1-\beta }(t-\mu )\mathcal{B}{v}_{\epsilon }(\mu ,\eta )\mathrm{d}\mu ∥+M_S^2{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right)\tau \hfill \\ \hfill & \le M_S{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right){\int }_0^{t-\sigma }∥S_{1-\beta }\left(\tau \right)S_{1-\beta }(t-\tau -\mu )-S_{1-\beta }(t-\mu )∥\mathrm{d}\mu \hfill \\ \hfill & +M_S{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right){\int }_{t-\sigma }^{t-\tau }∥S_{1-\beta }\left(\tau \right)S_{1-\beta }(t-\tau -\mu )-S_{1-\beta }(t-\mu )∥\mathrm{d}\mu +M_S^2{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right)\tau \hfill \\ \hfill & \le M_S{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right){\int }_0^{t-\sigma }∥S_{1-\beta }\left(\tau \right)S_{1-\beta }(t-\tau -\mu )-S_{1-\beta }(t-\mu )∥\mathrm{d}\mu +(\sigma -\tau )M_S{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right)(M_S^2+M_S)+M_S^2{\parallel \mathcal{B}\parallel }^2{C}_{\epsilon }\left(r_{\epsilon }\right)\tau .\hfill \end{array}$$

In view of Lemma 1, for every $0\le \mu \le t-\sigma$, we obtain that

$$\underset{\tau \to 0}{lim}∥S_{1-\beta }\left(\tau \right)S_{1-\beta }(t-\tau -\mu )-S_{1-\beta }(t-\mu )∥=0.$$

Thus, $\parallel \left({\mathsf{\Omega }}_{\epsilon ,\tau }^2\eta \right)\left(t\right)-\left({\mathsf{\Omega }}_{\epsilon }^2\eta \right)\left(t\right)\parallel \to 0$, as $\tau \to 0$. In conclusion, the set $\{{\mathsf{\Omega }}_{\epsilon }\eta :\eta \in O_{r_{\epsilon }}\}$ is relatively compact in $C\left(\right[0,d];E)$.

Owing to the theorem of Arzelà–Ascoli, the operator ${\mathsf{\Omega }}_{\epsilon }$ is compact. Application of Schauder’s fixed-point trick yields that there exists at least one fixed point of the operator ${\mathsf{\Omega }}_{\epsilon }$. □

4. Result of the Controllability When the Nonlocal Term w Is Compact

This section focuses on establishing the result related to the controllability of Equation (1). The finite-approximate controllability of Equation (1) is derived on the basis of the existence of the solutions to Equation (1).

Theorem 2.

Suppose that the conditions in Theorem 1 are satisfied, then Equation (1) can achieve finite-approximate controllability.

Proof. 

Let $\widetilde{{\zeta }_{\epsilon }}$ be the critical point of $J_{\epsilon }(·;\eta )$. Then,

$$J_{\epsilon }(\widetilde{{\zeta }_{\epsilon }};{\eta }_{\epsilon })=\underset{\zeta \in E}{min}{J}_{\epsilon }(\zeta ;{\eta }_{\epsilon }),\widetilde{{\zeta }_{\epsilon }}\in E.$$

From Lemma 6, we know that for any $\psi \in E$ and $\varpi >0$,

$$J_{\epsilon }(\widetilde{{\zeta }_{\epsilon }}+\psi \varpi ;{\eta }_{\epsilon })\ge J_{\epsilon }(\widetilde{{\zeta }_{\epsilon }};{\eta }_{\epsilon }).$$

Consequently, it is clear from the expression (5) that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\varpi }^2}{2}}{\int }_0^d{\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\psi \parallel }^2\mathrm{d}\mu +\varpi {\int }_0^d⟨{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\widetilde{{\zeta }_{\epsilon }},{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\psi ⟩\mathrm{d}\mu +\epsilon \left|\varpi \right|∥(I-{\pi }_E)\psi ∥-\varpi ⟨\psi ,\rho \left({\eta }_{\epsilon }\right)⟩\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

The collation yields

$$\begin{array}{c}\hfill \varpi ⟨\psi ,\rho \left({\eta }_{\epsilon }\right)⟩-{\displaystyle \frac{{\varpi }^2}{2}}{\int }_0^d{\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\psi \parallel }^2\mathrm{d}\mu -\varpi {\int }_0^d⟨{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\widetilde{{\zeta }_{\epsilon }},{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\psi ⟩\mathrm{d}\mu \le \epsilon \left|\varpi \right|∥(I-{\pi }_E)\psi ∥.\end{array}$$

Both sides of the inequality are divided by $\varpi >0$ to get

$$\begin{array}{c}\hfill ⟨\psi ,\rho \left({\eta }_{\epsilon }\right)⟩-{\displaystyle \frac{\varpi }{2}}{\int }_0^d{\parallel {\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\psi \parallel }^2\mathrm{d}\mu -{\int }_0^d⟨{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\widetilde{{\zeta }_{\epsilon }},{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\psi ⟩\mathrm{d}\mu \le {\displaystyle \frac{\epsilon \left|\varpi \right|∥(I-{\pi }_E)\psi ∥}{\varpi }}.\end{array}$$

Taking the limit on each side of the inequality, as $\varpi \to 0^{+}$ and $\varpi \to 0^{-}$, we give

$$\begin{array}{c}\hfill \left|⟨\psi ,\rho \left({\eta }_{\epsilon }\right)⟩-{\int }_0^d⟨{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\widetilde{{\zeta }_{\epsilon }},{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\psi ⟩\mathrm{d}\mu \right|\le \epsilon ∥(I-{\pi }_E)\psi ∥.\end{array}$$

(9)

As we know from the preceding,

$$\begin{array}{c}\hfill \rho \left({\eta }_{\epsilon }\right)={\eta }_d-{\eta }_0+w\left({\eta }_{\epsilon }\right)+{\int }_0^d{S}_{1-\beta }(d-\mu ){\eta }_{\epsilon }\left(\mu \right)\mathrm{d}\mu -{\int }_0^d{S}_{1-\beta }(d-\mu )h(\mu ,{\eta }_{\epsilon }\left(\mu \right))\mathrm{d}\mu ,{\eta }_d\in E\end{array}$$

(10)

and

$$v_{\epsilon }={\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu ){\mathfrak{P}}_{\epsilon }\left(\eta \right).$$

Then

$$\begin{array}{cc}\hfill {\int }_0^d⟨{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\widetilde{{\zeta }_{\epsilon }},{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\psi ⟩\mathrm{d}\mu & ={\int }_0^d⟨S_{1-\beta }(d-\mu )\mathcal{B}{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\widetilde{{\zeta }_{\epsilon }},\psi ⟩\mathrm{d}\mu \hfill \\ \hfill & ={\int }_0^d⟨S_{1-\beta }(d-\mu )\mathcal{B}{v}_{\epsilon },\psi ⟩\mathrm{d}\mu .\hfill \end{array}$$

Combining (9) and (10) estimates that

$$\begin{array}{cc}\hfill & \left|⟨\psi ,\rho \left({\eta }_{\epsilon }\right)⟩-{\int }_0^d⟨{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\widetilde{{\zeta }_{\epsilon }},{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\psi ⟩\mathrm{d}\mu \right|\hfill \\ \hfill & =\left|⟨\psi ,\rho \left({\eta }_{\epsilon }\right)⟩-{\int }_0^d⟨S_{1-\beta }(d-\mu )\mathcal{B}{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\widetilde{{\zeta }_{\epsilon }},\psi ⟩\mathrm{d}\mu \right|\hfill \\ \hfill & =\left|⟨{\int }_0^d{S}_{1-\beta }(d-\mu )\left[\mathcal{B}{\mathcal{B}}^{*}{S}_{1-\beta }^{*}(d-\mu )\widetilde{{\zeta }_{\epsilon }}-{\eta }_{\epsilon }\left(\mu \right)+h(\mu ,{\eta }_{\epsilon }\left(\mu \right))-w\left({\eta }_{\epsilon }\right)+{\eta }_0-{\eta }_d\right]\mathrm{d}\mu ,\psi ⟩\right|\hfill \\ \hfill & \le \epsilon ∥(I-{\pi }_E)\psi ∥\hfill \\ \hfill & \le \epsilon ∥\psi ∥,\psi \in E,\hfill \end{array}$$

which implies

$$\parallel \eta (d,v_{\epsilon })-{\eta }_d\parallel \le \epsilon .$$

Taking $\psi \in Z\subset E$, it follows from the arbitrariness of $\psi$ that

$$⟨\eta \left(d\right)-{\eta }_d,{\pi }_Z\eta \left(d\right)-{\pi }_Z{\eta }_d⟩=0.$$

This means that ${\pi }_Z\eta \left(d\right)={\pi }_Z{\eta }_d$. Therefore, Equation (1) is finite-approximately controllable. □

5. Results of Existence for Solutions and Controllability of Equation (1) Under the Weak Nonlocal Condition

In this section, we investigate the existence of solutions to Equation (1) by providing the following assumption.

${\left(Q_w\right)}^{\prime }w:C\left(\right[0,d];E)\to E$ is continuous and the set $w\left(\overline{co}{\mathsf{\Omega }}_{\epsilon }{O}_r\right)$ is pre-compact, where $\overline{co}D$ is the bounded convex closed hull of $D\subseteq C\left(\right[0,d];E)$.

Remark 2.

In reference [41,42], the authors discussed the existence of solutions to semilinear nonlocal problems under the above assumption when the strongly continuous semigroup generated by a densely defined closed linear operator $\mathcal{A}$. Clearly, the compactness and convexity of w are stronger than the assumption ${\left(Q_w\right)}^{\prime }$.

Next, before establishing the existence of solutions to Equation (1), we need to verify the correctness of Lemmas 3–9.

Remark 3.

In fact, a crucial step in all the preceding lemmas is to verify that for any $\left\{x_k\right\}\subset O_r$, $\rho \left(x_k\right)$ is relatively compact in Lemma 4. The relative compactness of $\rho \left(x_k\right)$ in Lemma 4 ensures that the inequality of functional $J_{\epsilon }$ in Lemma 7 holds, as well as the properties of mapping ${\mathfrak{P}}_{\epsilon }$ in Lemmas 8 and 9 being satisfied. So according to the hypothesis ${\left(Q_w\right)}^{\prime }$, we just need to modify the lemmas as follows.

Lemma 10.

Let the assumption ${\left(Q_w\right)}^{\prime }$ holds, for any $\left\{x_k\right\}\subset {\mathsf{\Omega }}_{\epsilon }{O}_r$, $\rho \left(x_k\right)$ has a convergent subsequence.

Lemma 11.

If the assumption $\left(L\right)$ is valid, then for any $\epsilon >0$ and $\zeta \in E$, the following inequality holds:

$$\begin{array}{c}\hfill \underset{\zeta \to \infty }{\underline{lim}}\underset{x\in {\mathsf{\Omega }}_{\epsilon }{O}_r}{inf}{\displaystyle \frac{J_{\epsilon }(\zeta ;x)}{\parallel \zeta \parallel }}\ge \epsilon .\end{array}$$

Define the mapping

$$\begin{array}{cc}\hfill {\mathfrak{P}}_{\epsilon }:{\mathsf{\Omega }}_{\epsilon }{O}_r\subseteq O_r& \to E\hfill \\ \hfill x& \mapsto \widetilde{{\zeta }_{\epsilon }},\mathrm{for}\mathrm{any}x\in {\mathsf{\Omega }}_{\epsilon }{O}_r,\hfill \end{array}$$

where $\widetilde{{\zeta }_{\epsilon }}$ is a unique minimum of $J_{\epsilon }(·;x)$.

Lemma 12.

There exists a constant $C_{\epsilon }\left(r\right)>0$ such that $\parallel {\mathfrak{P}}_{\epsilon }\left(x\right)\parallel \le C_{\epsilon }\left(r\right)$, for any $x\in {\mathsf{\Omega }}_{\epsilon }{O}_r$. Here, $\left\{{\mathfrak{P}}_{\epsilon }\left(x\right)\right\}$ is the minimum point set.

Lemma 13.

${\mathfrak{P}}_{\epsilon }\left(x_k\right)\rightharpoonup {\mathfrak{P}}_{\epsilon }\left(x\right)$ and $\underset{k\to \infty }{lim}\parallel {\mathfrak{P}}_{\epsilon }\left(x_k\right)\parallel =\parallel {\mathfrak{P}}_{\epsilon }\left(x\right)\parallel$ provided that $x_k\to x$ in ${\mathsf{\Omega }}_{\epsilon }{O}_r$.

Lemmas 10–13 correspond to Lemma 4 and Lemmas 7–9, respectively. The proof processes are similar. The details are omitted. Next, we introduce the following theorem.

Theorem 3.

Suppose that the conditions $\left(Q_S\right)$, $\left(Q_h\right)$, ${\left(Q_w\right)}^{\prime }$ and $\left(L\right)$ are satisfied, and

$$\begin{array}{c}\hfill \underset{r_{\epsilon }\to \infty }{lim}{\displaystyle \frac{C_{\epsilon }\left(r_{\epsilon }\right)}{r_{\epsilon }}}<{\displaystyle \frac{1-d{M}_S{(1+\parallel \varphi \parallel }_{L^{\infty }}·{\lambda }_h)}{d{M}_S^2{\parallel \mathcal{B}\parallel }^2}}.\end{array}$$

(11)

Then, there exists at least one mild solution to Equation (1).

Proof. 

The steps of the proof are the same as in Theorem 1. However, it remains to prove the relative compactness at $t=0$. Let $O=\overline{co}{\mathsf{\Omega }}_{\epsilon }{O}_{r_{\epsilon }}$. Obviously, O is a bounded closed and convex subset of $C\left(\right[0,d];E)$ and ${\mathsf{\Omega }}_{\epsilon }O\subseteq O$. According to the proof process in steps 3 and 4 of Theorem 1, ${\mathsf{\Omega }}_{\epsilon }O\left(t\right)$ is relatively compact for any $t\in (0,d]$, and ${\mathsf{\Omega }}_{\epsilon }O$ is equicontinuous for any $t\in [0,d]$. Furthermore, based on the assumption ${\left(Q_w\right)}^{\prime }$, we have that $w\left(O\right)=w\left(\overline{co}{\mathsf{\Omega }}_{\epsilon }{O}_{r_{\epsilon }}\right)$ is pre-compact.

Therefore,

$$\left({\mathsf{\Omega }}_{\epsilon }O\right)\left(0\right)=\left({\mathsf{\Omega }}_{\epsilon }^{1}O\right)\left(0\right)+\left({\mathsf{\Omega }}_{\epsilon }^{2}O\right)\left(0\right)={\eta }_0-w\left(O\right)={\eta }_0-w\left(\overline{co}{\mathsf{\Omega }}_{\epsilon }{O}_{r_{\epsilon }}\right)$$

is relatively compact. Using the Arzelà–Ascoli theorem, we obtain that ${\mathsf{\Omega }}_{\epsilon }:O\to O$ is a compact mapping. From the Schauder’s fixed-point theorem, Equation (1) has at least one mild solution. □

Remark 4.

Due to the specificity of the model in this paper, the nonlocal term is only related to the solutions and not to the resolvent family generated by $\mathcal{A}$. Therefore, it is difficult to find concrete examples to verify the condition ${\left(Q_w\right)}^{\prime }$. The compactness problem of w at 0 cannot be guaranteed. In fact, the nonlocal term is related to the semigroup $\left\{T\right(t);t\ge 0\}$ generated by $\mathcal{A}$ in the general model. Specific information can be found in [41,42]. In general, the problem of nonlocal term at 0 can be solved using the properties of the semigroup $\left\{T\right(t);t\ge 0\}$. In this case, we can find concrete examples that are neither Lipschitz nor compact to verify that that the condition ${\left(Q_w\right)}^{\prime }$ is true.

Theorem 4.

Assume that the conditions of Theorem 3 are met, then Equation (1) is finite-approximately controllable.

In fact, Theorem 4 is justified by the same procedure as Theorem 2. Here, we omit the proof processes.

6. Applications

In this section, we consider the example of fractional composite relaxation equations under a nonlocal condition. The fractional composite relaxation equations can be used to characterize the classical problem in the fluid dynamics of the non-constant motion of accelerated particles in viscous fluids under the action of gravity.

Example 1.

The following fractional composite relaxation equation is given:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\displaystyle \frac{\partial y(t,\zeta )}{\partial t}}& ={\displaystyle \frac{{\partial }^2}{\partial {\zeta }^2}}{}^{c}D_t^{{\displaystyle \frac{1}{2}}}y(t,\zeta )+t^{2}y(t,\zeta )-y(t,\zeta )+v(t,\zeta ),t\in (0,1],\zeta \in [0,\pi ],\hfill \\ \hfill y(t,0)& =y(t,\pi )=0,t\in [0,1],\hfill \\ \hfill y(0,\zeta )& =y_0\left(\zeta \right)-{\int }_0^{\pi }{\int }_0^{1}g\left(t,\zeta ,z,y(t,z)\right)\mathrm{d}t\mathrm{d}z,\zeta \in [0,\pi ],\hfill \end{array}\right.\end{array}$$

(12)

where ${}^{c}D_t^{{\displaystyle \frac{1}{2}}}$ is the derivative operator of Caputo with the order ${\displaystyle \frac{1}{2}}$.

Let $E=F=L^2([0,\pi ];\mathbb{R})$. The operator $\mathcal{A}\subseteq E\to E$ denoted as

$$\mathcal{A}u=u^{\prime \prime }$$

with

$$D\left(\mathcal{A}\right):=\{u\in E:,u^{\prime }\mathit{are}\mathit{absolutely}\mathit{continuous},u\left(0\right)=u\left(\pi \right)=0,u^{\prime \prime }\in E\}.$$

Then, $\mathcal{A}$ is a closed, linear and densely defined operator, and

$$\mathcal{A}u=-\sum _{j=1}^{\infty }{\displaystyle \frac{j^2}{1+j^2}}⟨u,e_j⟩e_j,u\in D\left(\mathcal{A}\right),$$

where $e_j\left(\zeta \right)=\sqrt{{\displaystyle \frac{2}{\pi }}}sinj\zeta$, $j=1,2,\cdots$ is an orthonormal basis of E. Then, we obtain that the self-adjoint compact and analytic semigroup ${\left\{S\left(t\right)\right\}}_{t>0}$ generated by $\mathcal{A}$, and

$$S\left(t\right)u=\sum _{j=1}^{\infty }\exp \left({\displaystyle \frac{-j^{2}t}{1+j^2}}\right)⟨u,e_j⟩e_j,u\in E,$$

with $\parallel S\left(t\right)\parallel \le 1$. Moreover, it follows from the subordination principle [38] that $\mathcal{A}$ can also generate a compact ${\displaystyle \frac{1}{2}}$-order fractional analytic resolvent $\{S_{{\displaystyle \frac{1}{2}}}\left(t\right),t\ge 0\}$ of analytic type $({\omega }_0,{\theta }_0)$, and

$$S_{{\displaystyle \frac{1}{2}}}\left(t\right)={\int }_0^{\infty }{\mathsf{\Omega }}_{{\displaystyle \frac{1}{2}}}\left(\mu \right)S(\mu t^{{\displaystyle \frac{1}{2}}})\mathrm{d}\mu ,t\in [0,\infty ),$$

where

$${\mathsf{\Omega }}_{{\displaystyle \frac{1}{2}}}\left(\tau \right):=\sum _{j=0}^{\infty }{\displaystyle \frac{{(-\tau )}^j}{j!\Gamma (\frac{1}{2}-\frac{1}{2}j)}},\tau >0$$

is a probability density function and ${\int }_0^{\infty }{\mathsf{\Omega }}_{{\displaystyle \frac{1}{2}}}\left(\tau \right)\mathrm{d}\tau =1$. Hence, the assumption $\left(Q_S\right)$ is satisfied and

$$\parallel S_{{\displaystyle \frac{1}{2}}}\left(t\right)\parallel =\parallel {\int }_0^{\infty }{\mathsf{\Omega }}_{{\displaystyle \frac{1}{2}}}\left(\mu \right)S(\mu t^{{\displaystyle \frac{1}{2}}})\mathrm{d}\mu \parallel \le 1.$$

Then, $M_S=\underset{0\le t\le 1}{sup}\parallel S_{{\displaystyle \frac{1}{2}}}\left(t\right)\parallel =1$.

Furthermore, let $\mathcal{B}=I$, where I is an identity operator. Then, $\parallel \mathcal{B}\parallel =1$. Now, let

$$\eta \left(t\right)\left(\zeta \right)=y(t,\zeta ),$$

$$h(t,\eta \left(t\right))\left(\zeta \right)=t^{2}y(t,\zeta )=t^2\eta \left(t\right)\left(\zeta \right),$$

and

$$w\left(\eta \right)\left(\zeta \right)={\int }_0^{\pi }{\int }_0^{1}g\left(t,\zeta ,z,\eta (t,z)\right)\mathrm{d}t\mathrm{d}z,\zeta \in [0,\pi ].$$

Thus, Equation (12) can be converted into the abstract fractional composite relaxation Equation (1). In addition,

$$\begin{array}{c}\hfill {∥h(t,\eta (t\left)\right)(·)∥}_E={\left({\int }_0^{\pi }{∥t^2\eta \left(t\right)\left(\zeta \right)∥}^2\mathrm{d}\zeta \right)}^{{\displaystyle \frac{1}{2}}}\le t^2{\parallel \eta \left(t\right)(·)\parallel }_E,\end{array}$$

where $\varphi \left(t\right)=t^2$, ${{\rm Y}}_h(\parallel \eta \left(t\right)(·)\parallel )={\parallel \eta \left(t\right)(·)\parallel }_E$ and $\underset{\parallel \eta \left(t\right)(·)\parallel \to \infty }{lim\; inf}{\displaystyle \frac{{{\rm Y}}_h(\parallel \eta \left(t\right)(·)\parallel )}{{\parallel \eta \left(t\right)(·)\parallel }_E}}=1={\lambda }_h<\infty$. Thus, the hypothesis $\left(Q_h\right)$ holds.

Now, let the function $g:[0,1]\times [0,\pi ]\times [0,\pi ]\times \mathbb{R}\to \mathbb{R}$ meet the following conditions:

(1) $g\left(t,\zeta ,z,x\right)$ is a continuous function with respect to x, for almost everywhere $(t,\zeta ,z)\in [0,1]\times [0,\pi ]\times [0,\pi ]$;

(2) $g\left(t,\zeta ,z,x\right)$ is measurable with respect to $(t,\zeta ,z)$, for any fixed $x\in \mathbb{R}$.

In addition, we also assume that the following conditions of g are true:

(i) 

for any $\left(t,\zeta ,z,x\right),\left(t,{\zeta }^{\prime },z,x\right)\in [0,1]\times [0,\pi ]\times [0,\pi ]\times \mathbb{R}$ with $\left|x\right|\le n$, one has $\left|g\left(t,\zeta ,z,x\right)-g\left(t,{\zeta }^{\prime },z,x\right)\right|\le P_n\left(t,\zeta ,{\zeta }^{\prime },z\right)$, where $P_n\in L^1\left([0,1]\times [0,\pi ]\times [0,\pi ]\times \mathbb{R};{\mathbb{R}}^{+}\right)$ satisfies ${\int }_0^{\pi }{\int }_0^1{P}_n\left(t,\zeta ,{\zeta }^{\prime },z\right)\mathrm{d}t\mathrm{d}z\to 0$, as $\zeta \to {\zeta }^{\prime }$, and uniformly in $0\le {\zeta }^{\prime }\le \pi$;

(ii) 

for any $z\in \mathbb{R}$, $\left|g\left(t,\zeta ,z,x\right)\right|\le \varpi \left|x\right|+\phi \left(t,\zeta ,z\right)$, where $\phi \in L^2([0,1]\times [0,\pi ]\times [0,\pi ];{\mathbb{R}}^{+})$ and $\varpi >0$.

Therefore, it is known from the literature [43] that the mapping w is continuous and compact. Moreover, by (ii), for any $\eta \in C\left(\right[0,d];E)$, one obtains

$$\begin{array}{cc}\hfill \left|w\left(\eta \right)\left(\zeta \right)\right|& \le {\int }_0^{\pi }{\int }_0^{1}g\left(t,\zeta ,z,\eta (t,z)\right)\mathrm{d}t\mathrm{d}z\hfill \\ \hfill & \le \varpi {\int }_0^{\pi }{\int }_0^1\left|\eta (t,z)\right|\mathrm{d}t\mathrm{d}z+{\int }_0^{\pi }{\int }_0^1\phi \left(t,\zeta ,z\right)\mathrm{d}t\mathrm{d}z\hfill \\ \hfill & \le \varpi {\int }_0^1{\left({\int }_0^{\pi }\left|\eta (t,z)\right|\mathrm{d}z\right)}^{{\displaystyle \frac{1}{2}}}\mathrm{d}t+{\int }_0^{\pi }{\int }_0^1\phi \left(t,\zeta ,z\right)\mathrm{d}t\mathrm{d}z\hfill \\ \hfill & \le {\varpi \parallel \eta \parallel }_{C([0,1];L^2\left([0,\pi ]\right))}+{\parallel \phi \parallel }_{L^2([0,1]\times [0,\pi ]\times [0,\pi ];{\mathbb{R}}^{+})}.\hfill \end{array}$$

Then, the mapping w is the one which satisfies the assumption $\left(Q_w\right)$. Therefore, the assumptions $\left(Q_S\right)$, $\left(Q_h\right)$ and $\left(Q_w\right)$ are satisfied. If the following inequality,

$$\underset{r_{\epsilon }\to \infty }{lim\; inf}{\displaystyle \frac{C_{\epsilon }\left(r_{\epsilon }\right)}{r_{\epsilon }}}<1$$

is true, for some $r_{\epsilon }>0$, then Equation (12) has at least one fixed point by Theorem 1.

On the other hand,

$$\begin{array}{cc}\hfill \mathcal{B}{S}_{{\displaystyle \frac{1}{2}}}(1-\mu )u& =S_{{\displaystyle \frac{1}{2}}}(1-\mu )u\hfill \\ \hfill & =\sum _{j=1}^{\infty }{\int }_0^{\infty }{\mathsf{\Omega }}_{{\displaystyle \frac{1}{2}}}\left(q\right)\exp {\displaystyle \frac{-j^2{(1-\mu )}^{\frac{1}{2}}q}{1+j^2}}\mathrm{d}q⟨u,e_j⟩e_j.\hfill \end{array}$$

(13)

In view of Lemma 5, we know that $\mathcal{B}{S}_{{\displaystyle \frac{1}{2}}}(1-\mu )u=0,\mu \in [0,1]$ implies that $u=0$. Since ${\mathsf{\Omega }}_{{\displaystyle \frac{1}{2}}}\left(q\right)\ge 0$ and $\exp {\displaystyle \frac{-j^2{(1-\mu )}^{\frac{1}{2}}q}{1+j^2}}>0$ in (13), $u=0$. So the assumption $\left(L\right)$ holds. To conclude, the finite-approximate controllability of Equation (1) is derived.

7. Conclusions

In this paper, we study the existence and finite-approximate controllability of fractional composite relaxation equations under nonlocal conditions. We discuss two cases where the nonlocal term is compact and the nonlocal term is precompact on a bounded convex closed set. By using the resolvent operators family theory and the variational method, we respectively provide the sufficient conditions for the existence of the solution of and the finite-approximate controllability of fractional composite relaxation equations. Finally, the feasibility of the results in this paper is verified by an example. In fact, the finite-approximate controllability of fractional composite relaxation equations for which the nonlocal term is non-compact via variational methods has some limitations. In the future, we will consider the finite-approximate controllability of fractional equations with nonlocal terms in the non-compact case in conjunction with resolvent conditions.