Existence and Controls for Fractional Evolution Equations

Abstract

In this paper, we investigate the existence and uniqueness of mild solutions for non-autonomous fractional evolution equations (NFEEs) using the technique of non-compactness measure, focusing on scenarios where the semigroup is non-compact. Furthermore, the optimal control of nonlinear NFEEs with integral index functionals is studied, and the existence of optimal control pairs is proven. Finally, by constructing a corresponding Gramian controllability operator using the solution operator, a sufficient condition is provided for the existence of approximate controllability of the corresponding problem.

1. Introduction

The fractional evolution equation deals with the process by which complex systems change over time. It has important applications in physics, chemistry, economics, biological sciences, and so on (see [1,2,3,4,5,6,7,8,9] and the references therein).

Many real-world issues involve processes that can be memorized and inherited. In physics, wave behavior is affected by periodic external influences, which are essential for understanding movement through complex media [10,11]. In economics, call-and-put option valuations are the key to determining insurance premium values, which are crucial in financial mathematics [12,13,14]. In biology, network synchronization stabilizes networks by reducing delay and uncertainty effects using system feedback and impulse sampling [15,16,17]. In chemistry, a significant focus is on understanding and controlling interfaces and grain boundaries to attain desired properties in nanostructured materials [18]. The non-autonomous fractional equation, utilizing fractional derivatives, effectively models memory effects and long-term correlations, making it a powerful mathematical tool. Although the theory of autonomous fractional systems is well-developed, theoretical knowledge of the NFEEs is still incomplete. El-Borai [19] systematically provided the form of the fundamental solution of linear NFEEs with the Caputo derivative by introducing two intermediate operators and studying the existence and uniqueness of the solution. El-Borai, El-Nadi, and El-Akabawy [20] made conditional assumptions to ensure the existence of the resolvent operator of NFEEs. The case of solutions for the NFEEs with delays was studied in [21,22] under different fixed point theories. Compared to [19], the author of [23] improved and deeply studied it, removed the cumbersome intermediate operator, and provided a more concise expression for the classical solution of linear NFEEs.

Controllability is a key issue in engineering and mathematical control. Fractional optimal control is the study of the control scheme that, under certain conditions, causes the realisation criterion to reach the optimal value. It has broad prospects and an increasingly important role (see [24,25,26,27,28,29,30] and the references therein). Complete controllability in Banach spaces has been extensively studied (see [31]). The conditions necessary to achieve complete controllability are overly demanding, leading to an increased interest in approximate controllability as a means of establishing control conditions that are more conducive to the desired outcomes. Zhu Bo and Han Baoyan [32] considered the approximate controllability of mixed-type NFEEs with the Caputo derivative. The authors of [33] considered the approximate controllability for a class of fractional evolution equations with nonlocal conditions in Banach spaces in the Caputo sense.

In light of the aforementioned considerations, we study the existence and controls to semilinear NFEEs in the Banach space X:

$$\left\{\begin{array}{c}{D}_t^{\alpha }x\left(t\right)+A\left(t\right)x\left(t\right)=f(t,x\left(t\right)),t\in (0,a],\hfill \\ J_t^{1-\alpha }x\left(0\right)=x_0,\hfill \end{array}\right.$$

(1)

where $D_t^{\alpha }$ is the Riemann–Liouville fractional derivative of order $0<\alpha <1$, and $J_t^{1-\alpha }$ is the Riemann–Liouville fractional integral of order $1-\alpha$, $A\left(t\right):D\left(A\right(t\left)\right)\subset X\to X$ is a family of closed linear operators, the domain $D\left(A\right(t\left)\right)$ is dense and independent of t, $f:(0,a]\times X\to X$ is a Carathéodory function, and $x_0\in X$ and $a>0$ are constant.

Let us select three key tasks for inclusion in this article. First, the majority of studies have focused on analyzing the Cauchy problem of Caputo derivatives, examining both compact and non-compact semigroup scenarios. In the case where the semigroup is non-compact under the Riemann–Liouville derivative, we discuss some properties of the mild solution. Second, there is little research on the optimal control of NFEEs in the Riemann–Liouville sense. For this reason, we present the result of the optimal control of NFEEs using the Riemann–Liouville derivative. Finally, unlike constructing the control function using Green’s function, we can use the solution operator directly to construct the Gramian operator to obtain the approximate controllability result.

It is worth emphasizing that in the existing body of research concerning NFEEs, assumptions are typically made, and operators adhere to compactness conditions. As a result, in cases where such operators do not satisfy the established compactness conditions, the primary objective of our research shifts to focusing on the examination and comprehension of various techniques for evaluating and addressing the implications and impacts of non-compact operators within the scope of NFEEs.

Section 2 primarily introduces the preliminary concepts and lemmas required for the subsequent argument. The existence and uniqueness of mild solutions to the problem (1) when the semigroup is non-compact are studied in Section 3. In Section 4, we prove the existence of an optimal control for NFEEs by introducing admissible control sets and performance index functions. Section 5 presents an approximate controllability result constructed using a method of building a controllable operator.

2. Preliminaries

In this section, we’ll cover some notations and basic facts needed for this article on fractional calculus, special functions, semigroups, and the measure of non-compactness.

Let X and Y be two separable reflexive Banach spaces with norms $\parallel ·\parallel$ and ${\parallel ·\parallel }_Y$, respectively. Let $a\in {\mathbb{R}}^{+}$. $C\left(\right(0,a],X)$ be the space of all continuous functions normed by ${\parallel x\parallel }_C=sup\left\{\parallel x\left(t\right)\parallel ,t\in (0,a]\right\}$, and let $\mathcal{L}(X,Y)$ be the space of all linear bounded operators from X to Y equipped with norm ${\parallel ·\parallel }_{\mathcal{L}(X,Y)}$ for short ${\parallel ·\parallel }_{\mathcal{L}}$ when linear bounded operators map X onto itself. Let $L^p\left((0,a],X\right)$ be the space of all Bochner integrable functions with norm ${\parallel x\parallel }_p={\left({\int }_0^a{\parallel x\left(t\right)\parallel }^{p}dt\right)}^{{\displaystyle \frac{1}{p}}}$ for $1\le p<\infty$. Let

$$C_{1-\alpha }\left([0,a],X\right)=\left\{x\in C\left((0,a],X\right):\underset{t\to 0+}{lim}{t}^{1-\alpha }x\left(t\right)\mathrm{exists}\mathrm{and}\mathrm{is}\mathrm{finite}\right\},$$

with the norm ${\parallel x\parallel }_{C_{1-\alpha }}={sup}_{t\in [0,a]}{t}^{1-\alpha }\parallel x\left(t\right)\parallel$. Noticeably, $C_{1-\alpha }\left([0,a],X\right)$ is a Banach space.

The Riemann–Liouville fractional integral of order $\alpha \in (0,1)$ is defined by the following:

$$J_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}f\left(s\right)ds,t>0.$$

Furthermore, the Riemann–Liouville fractional derivative of order $\alpha \in (0,1)$ is defined as follows:

$$D_t^{\alpha }f\left(t\right)={\displaystyle \frac{d}{dt}}{J}_t^{1-\alpha }f\left(t\right),t>0.$$

For the Kuratowski measure of non-compactness on a bounded set S, we define it by the following:

$$\rho \left(S\right):=inf\left\{\delta >0:S\subset \bigcup _{k=1}^n{S}_k\mathrm{and}diam\left(S_k\right)\le \delta \mathrm{for}k=1,2,\dots ,n\right\}.$$

As is well known, the Kuratowski measure of non-compactness $\rho (·)$ has some fabulous properties. We refer the reader to [34,35,36,37].

(i) 

$\rho \left(\mathcal{U}\right)\le \rho \left(\mathcal{V}\right)$ if $\mathcal{U}\subset \mathcal{V}$.

(ii) 

$\rho (\mathcal{U}+x)=\rho \left(\mathcal{U}\right)$ for any $x\in X$.

(iii) 

$\rho \left(\iota \mathcal{U}\right)\le \left|\iota \right|\rho \left(\mathcal{U}\right)$, where $\iota \in \mathbb{R}$.

(iv) 

Suppose $\mathfrak{D}$ is a bounded subset of X. Then, there exists a countable set ${\mathfrak{D}}_0\subset \mathfrak{D}$, such that $\rho \left(\mathfrak{D}\right)\le 2\rho \left({\mathfrak{D}}_0\right).$

(v) 

If $\mathfrak{D}={\left\{u_n\right\}}_{n=1}^{\infty }\subset C([0,T],X)$ is a countable set in Banach space X and there exists a function $h\in L^1([0,T],{\mathbb{R}}^{+})$ such that for every $n\in \mathbb{N}$, $\parallel u_n\left(t\right)\parallel \le h\left(t\right),a.e.t\in [0,T].$ Then, $\rho \left(\mathfrak{D}\right(t\left)\right)$ is the Lebesgue integral on $[0,T]$, and

$$\rho \left(\left\{{\int }_0^T{u}_n\left(t\right)dt:n\in \mathbb{N}\right\}\right)\le 2{\int }_0^T\rho \left(\mathfrak{D}\left(t\right)\right)dt.$$

We denote two operators as follows:

$$\begin{array}{cc}\hfill & {\varphi }_t\left(s\right)={\int }_0^{\infty }{\xi }_{\alpha }\left(\theta \right)T_t\left(s^{\alpha }\theta \right)d\theta ,s\ge 0,\hfill \\ \hfill & {\psi }_t\left(s\right)=s^{\alpha -1}{\int }_0^{\infty }\alpha \theta {\xi }_{\alpha }\left(\theta \right)T_t\left(s^{\alpha }\theta \right)d\theta ,s>0,\hfill \end{array}$$

where ${\xi }_{\alpha }(·)$ is the Wright type function

$${\xi }_{\alpha }\left(z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-z)}^n}{n!\Gamma (1-\alpha (n+1\left)\right)}},z\in \mathbb{C},$$

and without losing generality, $-A\left(t\right)$ is the infinitesimal generator of analytic semigroups $\left\{T_t\left(\varsigma \right)\right\},\varsigma \ge 0$ on X.

Benefiting from the work in [8], we obtain the definition of the mild solution and several lemmas.

Definition 1. 

Using the mild solution of the nonlocal Cauchy problem (1), the function $x\in C_{1-\alpha }\left([0,a],X\right)$, which satisfies

$$x\left(t\right)={\mathcal{U}}_{\alpha }(t,0)x_0+{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x\left(\tau \right))d\tau ,$$

where

$${\mathcal{U}}_{\alpha }(t,s)={\psi }_s(t-s)+{\int }_s^t{\psi }_{\tau }(t-\tau )R(\tau ,s)d\tau ,$$

and

$$R(t,s)=\sum _{m=1}^{\infty }{R}_m(t,s),$$

inductive to $R_1(t,\tau )=-(A\left(t\right)-A\left(\tau \right)){\psi }_{\tau }(t-\tau )$ and

$$R_{m+1}(t,s)={\int }_s^t{R}_1(t,\tau )R_m(\tau ,s)d\tau ,m\ge 1.$$

Lemma 1. 

If $0\le \varsigma <\sigma \le t\le a,{\mathcal{U}}_{\alpha }(t,\varsigma )$ is bounded, there exists a constant $C>0$ such that

$$\parallel {\mathcal{U}}_{\alpha }{(t,\varsigma )\parallel }_{\mathcal{L}}\le C{(t-\varsigma )}^{\alpha -1}.$$

Going further, ${\mathcal{U}}_{\alpha }$ is continuous in the uniform operator topology on $\mathcal{L}\left(X\right)$, meeting

$$\parallel {\mathcal{U}}_{\alpha }(t,\varsigma )-{\mathcal{U}}_{\alpha }{(\sigma ,\varsigma )\parallel }_{\mathcal{L}}\le C{(t-\sigma )}^{{\alpha }_0}{(\sigma -\varsigma )}^{{\vartheta }_0-1},$$

where ${\alpha }_0=min\{1-\alpha ,\alpha \},{\vartheta }_0=min\{2\alpha -1,\vartheta \}.$

Lemma 2. 

We assume that $A\left(t\right)$ is the infinitesimal generator of a compact analysis semigroup $T_t\left(\varsigma \right)$ for $\varsigma \ge 0$ and every $t\in [0,a]$. Then, operator ${\mathcal{U}}_{\alpha }(t,\varsigma )$ is compact for every $t\in (\varsigma ,a].$

Lemma 3. 

A measurable function $f:[0,a]\to X$ is a Bochner integral if $\parallel f\parallel$ is Lebesgue integrable.

3. Existence and Uniqueness Results

This section is devoted to thinking about the operator $A\left(t\right)$ as a generator of a non-compact semigroup $\left\{T_t\left(\varsigma \right)\right\}(\varsigma \ge 0)$ on X; in other words, for any $t>0$, operator $\left\{T_t\left(\varsigma \right)\right\}$ is non-compact. In the following, we will make the following assumptions based on the data of our problems:

(H1) 

$\left\{T_t\left(\varsigma \right)\right\}(\varsigma \ge 0)$ is equicontinuous;

(H2) 

$f(t,·):X\to X$ is continuous for a.e. $t\in [0,a]$, and $f(·,x):(0,a]\to X$ is measurable for each $x\in X$. There exists a function $\kappa \in L^p\left((0,a],{\mathbb{R}}^{+}\right)({\displaystyle \frac{1}{{\vartheta }_0}}<p<\infty )$, and a constant $0<\varrho <{\displaystyle \frac{\alpha }{Ca}}$, where C is a constant with respect to t, such that

$$\parallel f(t,x)\parallel \le \kappa \left(t\right)+\varrho t^{1-\alpha }\parallel x\parallel \mathrm{for}x\in X\mathrm{and}\mathrm{a}.\mathrm{e}.t\in [0,a];$$

(H3) 

There is a constant $r>0$, which makes

$$r\ge {\displaystyle \frac{\alpha }{\alpha -C\varrho a}}\left[C\parallel x_0\parallel +C{\left({\displaystyle \frac{p-1}{\alpha p-1}}\right)}^{{\displaystyle \frac{p-1}{p}}}{a}^{1-{\displaystyle \frac{1}{p}}}{\parallel \kappa \parallel }_p\right];$$

(H4) 

There exists a constant $M>0$, making any bounded $\mathfrak{D}\subseteq X$

$$\rho \left(f(t,\mathfrak{D})\right)\le M{t}^{1-\alpha }\rho \left(\mathfrak{D}\right),\mathrm{for}\mathrm{a}.\mathrm{e}.t\in [0,a].$$

For any $x\in X$, we define an operator $\Theta$ as shown:

$$(\Theta x)\left(t\right)={\mathcal{U}}_{\alpha }(t,0)x_0+{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x\left(\tau \right))d\tau ,\mathrm{for}t\in (0,a].$$

From the definition of ${\mathcal{U}}_{\alpha }$ and the properties of the ${\xi }_{\alpha }(·)$, we understand that ${lim}_{t\to 0+}{t}^{1-\alpha }(\Theta x)\left(t\right)={\displaystyle \frac{x_0}{\Gamma \left(\alpha \right)}}$.

For $r>0$, we set ${\mathfrak{Q}}_{\alpha }=\left\{x\in C_{1-\alpha }\left([0,a],X\right):{\parallel x\parallel }_{C_{1-\alpha }}\le r\right\}$. Accordingly, ${\mathfrak{Q}}_{\alpha }$ is a bounded closed and convex subset of $C_{1-\alpha }\left([0,a],X\right)$. Let ${\mathfrak{Q}}_r={\{y\in C([0,a],X):\parallel y\parallel }_C\le r\}$ with radius r centered at 0. Thus, ${\mathfrak{Q}}_r$ is the same for $C\left(\right[0,a],X)$.

For any $y\in {\mathfrak{Q}}_r$, set $x\left(t\right)=t^{\alpha -1}y\left(t\right)$ for $t\in (0,a]$. Then, $x\in {\mathfrak{Q}}_{\alpha }$. Define $\Xi$ as follows:

$$\left(\Xi y\right)\left(t\right)=\left\{\begin{array}{cc}{t}^{1-\alpha }\left(\Theta x\right)\left(t\right),\hfill & \hfill \mathrm{for}t\in (0,a],\\ \frac{x_0}{\Gamma \left(\alpha \right)},\hfill & \hfill \mathrm{for}t=0.\end{array}\right.$$

With the preliminary work mentioned above, we will first present the proofs of a number of lemmas.

Lemma 4. 

Let $\alpha >{\displaystyle \frac{1}{2}}$. Assume that (H1)–(H3) are satisfied. Then, $\{\Xi y:y\in {\mathfrak{Q}}_r\}$ is equicontinuous.

Proof. 

For $t_1=0,0<t_2\le a$, given the form of ${\mathcal{U}}_{\alpha }$ and ${\xi }_{\alpha }(·)$, we have

$$\begin{array}{cc}\hfill ∥\left(\Xi y\right)\left(t_2\right)-\left(\Xi y\right)\left(0\right)∥=& ∥t_2^{1-\alpha }(\Theta x)\left(t_2\right)-{\displaystyle \frac{x_0}{\Gamma \left(\alpha \right)}}∥\hfill \\ \hfill \le & ∥t_2^{1-\alpha }{\mathcal{U}}_{\alpha }\left(t_2,0\right)x_0-{\displaystyle \frac{x_0}{\Gamma \left(\alpha \right)}}∥+t_2^{1-\alpha }{\int }_0^{t_2}∥{\mathcal{U}}_{\alpha }\left(t_2,\tau \right)f(\tau ,x\left(\tau \right))∥d\tau \hfill \\ \hfill \le & ∥t_2^{1-\alpha }\left({\psi }_0\left(t_2\right)+{\int }_0^{t_2}{\psi }_{\tau }(t_2-\tau )R(\tau ,0)d\tau \right)x_0-{\displaystyle \frac{x_0}{\Gamma \left(\alpha \right)}}∥\hfill \\ \hfill & +t_2^{1-\alpha }{\int }_0^{t_2}∥{\mathcal{U}}_{\alpha }\left(t_2,\tau \right)f(\tau ,x\left(\tau \right))∥d\tau \hfill \\ \hfill \le & ∥{\int }_0^{\infty }\alpha \theta {\xi }_{\alpha }\left(\theta \right)T_0\left(t_2^{\alpha }\theta \right)d\theta ·x_0-{\displaystyle \frac{x_0}{\Gamma \left(\alpha \right)}}∥\hfill \\ \hfill & +t_2^{1-\alpha }{x}_0{\int }_0^{t_2}\parallel {\psi }_{\tau }(t-\tau )R(\tau ,0)\parallel d\tau \hfill \\ \hfill & +t_2^{1-\alpha }{\int }_0^{t_2}∥{\mathcal{U}}_{\alpha }\left(t_2,\tau \right)f(\tau ,x\left(\tau \right))∥d\tau \hfill \\ \hfill \to & 0,\mathrm{as}{t}_2\to 0.\hfill \end{array}$$

For $0<t_1<t_2\le a$, we obtain the following:

$$\begin{array}{cc}\hfill ∥(\Xi y)\left(t_2\right)-(\Xi y)\left(t_1\right)∥\le & \left|t_2^{1-\alpha }-t_1^{1-\alpha }\right|∥(\Theta x)\left(t_2\right)∥\hfill \\ \hfill & +t_1^{1-\alpha }∥{\mathcal{U}}_{\alpha }\left(t_2,0\right)x_0-{\mathcal{U}}_{\alpha }\left(t_1,0\right)x_0∥\hfill \\ \hfill & +t_1^{1-\alpha }{\int }_{t_1}^{t_2}∥{\mathcal{U}}_{\alpha }\left(t_2,\tau \right)f(\tau ,x\left(\tau \right))∥d\tau \hfill \\ \hfill & +t_1^{1-\alpha }{\int }_0^{t_1}∥\left({\mathcal{U}}_{\alpha }\left(t_2,\tau \right)-{\mathcal{U}}_{\alpha }\left(t_1,\tau \right)\right)f(\tau ,x\left(\tau \right))∥d\tau \hfill \\ \hfill =:& J_1+J_2+J_3+J_4,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & J_1=\left|t_2^{1-\alpha }-t_1^{1-\alpha }\right|∥(\Theta x)\left(t_2\right)∥;\hfill \\ \hfill & J_2=t_1^{1-\alpha }∥{\mathcal{U}}_{\alpha }\left(t_2,0\right)x_0-{\mathcal{U}}_{\alpha }\left(t_1,0\right)x_0∥;\hfill \\ \hfill & J_3=t_1^{1-\alpha }{\int }_{t_1}^{t_2}∥{\mathcal{U}}_{\alpha }\left(t_2,\tau \right)f(\tau ,x\left(\tau \right))∥d\tau ;\hfill \\ \hfill & J_4=t_1^{1-\alpha }{\int }_0^{t_1}∥\left({\mathcal{U}}_{\alpha }\left(t_2,\tau \right)-{\mathcal{U}}_{\alpha }\left(t_1,\tau \right)\right)f(\tau ,x\left(\tau \right))∥d\tau .\hfill \end{array}$$

By the continuity of $t^{1-\alpha }$ and $\Theta x\in {\mathfrak{Q}}_{\alpha }$, $J_1\to 0$, as $t_2\to t_1$, is easily obtained. Using Lemma 1, we can obtain that $J_2\to 0$ when $t_2\to t_1$. As for $J_3$, by the property of ${\mathcal{U}}_{\alpha }$ and assumptions about f, we know the following:

$$\begin{array}{cc}\hfill J_3\le & C{t}_1^{1-\alpha }{\int }_{t_1}^{t_2}{\left(t_2-\tau \right)}^{\alpha -1}\left(\kappa \left(\tau \right)+\varrho {\tau }^{\alpha -1}\parallel x\left(\tau \right)\parallel \right)d\tau \hfill \\ \hfill \le & C{t}_1^{1-\alpha }{\left({\displaystyle \frac{p-1}{\alpha p-1}}\right)}^{{\displaystyle \frac{p-1}{p}}}{\left(t_2-t_1\right)}^{\alpha -{\displaystyle \frac{1}{p}}}{∥\kappa ∥}_p\hfill \\ \hfill & +C{t}_1^{1-\alpha }{\int }_{t_1}^{t_2}{\left(t_2-\tau \right)}^{\alpha -1}\varrho {\parallel x\parallel }_{C_{1-\alpha }}d\tau \hfill \\ \hfill \le & C{\left({\displaystyle \frac{p-1}{\alpha p-1}}\right)}^{{\displaystyle \frac{p-1}{p}}}{\left(t_2-t_1\right)}^{\alpha -{\displaystyle \frac{1}{p}}}{∥\kappa ∥}_p+{\displaystyle \frac{Cr\varrho {\left(t_2-t_1\right)}^{\alpha }}{\alpha }}\hfill \\ \hfill \to & 0,\mathrm{as}{t}_2\to t_1.\hfill \end{array}$$

Let us use the Hölder inequality and Lemma 1 to show that

$$\begin{array}{cc}\hfill J_4& \le C{t}_1^{1-\alpha }{\left(t_2-t_1\right)}^{{\alpha }_0}{\int }_0^{t_1}{\left(t_1-\tau \right)}^{{\vartheta }_0-1}\left(\kappa \left(\tau \right)+\varrho {\tau }^{\alpha -1}\parallel x\left(\tau \right)\parallel \right)d\tau \hfill \\ \hfill & \le C{\left({\displaystyle \frac{p-1}{{\vartheta }_{0}p-1}}\right)}^{{\displaystyle \frac{p-1}{p}}}{\left(t_2-t_1\right)}^{{\alpha }_0}{t}_1^{{\vartheta }_0-{\displaystyle \frac{1}{p}}}{∥\kappa ∥}_p+{\left(t_2-t_1\right)}^{{\alpha }_0}{\displaystyle \frac{C\varrho r{t}_1^{{\vartheta }_0}}{{\vartheta }_0}}\hfill \\ \hfill & \to 0,\mathrm{as}{t}_2\to t_1.\hfill \end{array}$$

In summary, $∥\left(\Xi y\right)\left(t_2\right)-\left(\Xi y\right)\left(t_1\right)∥$ tends toward zero as $t_2\to t_1$. This suggests that $\left\{\Xi y:y\in \right.{\mathfrak{Q}}_r\}$ is equicontinuous. □

Lemma 5. 

Let $\alpha >{\displaystyle \frac{1}{2}}$. We assume that (H1)–(H3) are satisfied. Then, Ξ maps ${\mathfrak{Q}}_r$ onto ${\mathfrak{Q}}_r$, and Ξ is continuous in ${\mathfrak{Q}}_r$.

Proof. 

Claim I. $\Xi$ maps ${\mathfrak{Q}}_r$ onto ${\mathfrak{Q}}_r$.

For $t\in [0,a]$, in consideration of (H1)–(H3), we have the following:

$$\begin{array}{cc}\hfill \parallel (\Xi y)\left(t\right)\parallel & \le \parallel t^{1-\alpha }{\mathcal{U}}_{\alpha }(t,0)x_0\parallel +t^{1-\alpha }{\int }_0^t\parallel {\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x\left(\tau \right))\parallel d\tau \hfill \\ \hfill & \le C{t}^{1-\alpha }{t}^{\alpha -1}\parallel x_0\parallel +C{t}^{1-\alpha }{\int }_0^t{(t-\tau )}^{\alpha -1}\left(\kappa \left(\tau \right)+\varrho {\tau }^{\alpha -1}\parallel x\left(\tau \right)\parallel \right)d\tau \hfill \\ \hfill & \le C\parallel x_0\parallel +C{\left({\displaystyle \frac{p-1}{\alpha p-1}}\right)}^{{\displaystyle \frac{p-1}{p}}}{t}^{1-{\displaystyle \frac{1}{p}}}{\parallel \kappa \parallel }_p+{\displaystyle \frac{C\varrho ar}{\alpha }}\le r.\hfill \end{array}$$

(2)

Hence, ${\parallel \Xi y\parallel }_C\le r$ for any $y\in {\mathfrak{Q}}_r$.

Claim II. $\Xi$ is continuous in ${\mathfrak{Q}}_r$.

${\left\{y_n\right\}}_{n=1}^{\infty }\subset {\mathfrak{Q}}_r,y\in {\mathfrak{Q}}_r$ with $y_n\to y$ as $n\to \infty$. Given the continuity of f, we have

$$\underset{n\to \infty }{lim}f\left(t,x_n\left(t\right)\right)=\underset{n\to \infty }{lim}f\left(t,t^{\alpha -1}{y}_n\left(t\right)\right)=f\left(t,t^{\alpha -1}y\left(t\right)\right)=f(t,x\left(t\right)).$$

Simultaneously, for each $t\in (0,a]$, using (H2), we have

$${(t-s)}^{\alpha -1}∥f\left(s,x_n\left(s\right)\right)-f(s,x\left(s\right))∥\le 2{(t-s)}^{\alpha -1}\left(\kappa \left(s\right)+\varrho s^{1-\alpha }\parallel x\left(s\right)\parallel \right),\mathrm{a}.\mathrm{e}.\mathrm{in}[0,t].$$

At the same time, the function $2{(t-s)}^{\alpha -1}\left(\kappa \left(s\right)+\varrho s^{1-\alpha }\parallel x\left(s\right)\parallel \right)$ is integrable for $s\in [0,t]$ and $t\in (0,a]$. Using the Lebesgue dominated convergence theorem, we obtain the following:

$${\int }_0^t{(t-s)}^{\alpha -1}\left|f\left(s,x_n\left(s\right)\right)-f(s,x\left(s\right))\right|ds\to 0,\mathrm{as}n\to 0.$$

To sum up, for $t\in [0,a]$,

$$\begin{array}{cc}\hfill ∥\left(\Xi y_n\right)\left(t\right)-(\Xi y)\left(t\right)∥& =∥t^{1-\alpha }\left(\Theta x_n\left(t\right)-\Theta x\left(t\right)\right)∥\hfill \\ \hfill & \le t^{1-\alpha }{\int }_0^t∥{\mathcal{U}}_{\alpha }(t,\tau )\left(f\left(\tau ,x_n\left(\tau \right)\right)-f(\tau ,x\left(\tau \right))\right)∥d\tau \hfill \\ \hfill & \le C{\int }_0^t{(t-\tau )}^{\alpha -1}∥f\left(\tau ,x_n\left(\tau \right)\right)-f(\tau ,x\left(\tau \right))∥d\tau \hfill \\ \hfill & \to 0,\mathrm{as}n\to \infty ,\hfill \end{array}$$

which infer that

$${∥\left(\Xi y_n\right)-(\Xi y)∥}_C\to 0,\mathrm{as}n\to \infty .$$

On this account, operator $\Xi$ is continuous. □

Theorem 1. 

Let $\alpha >{\displaystyle \frac{1}{2}}$. Assume that (H1)–(H4) are satisfied. Then, Problem (1) has at least one mild solution in ${\mathfrak{Q}}_{\alpha }$.

Proof. 

For any $y\in {\mathfrak{Q}}_r$, $x\left(t\right)=t^{\alpha -1}y\left(t\right),t\in (0,a]$. Then, $x\in {\mathfrak{Q}}_{\alpha }$. Problem (1) in ${\mathfrak{Q}}_{\alpha }$ has a solution if and only if the operator equation $\Xi y=y$ has a fixed point in ${\mathfrak{Q}}_r$. Using Lemmas 4 and 5, we know that $\Xi :{\mathfrak{Q}}_r\to {\mathfrak{Q}}_r$ is bounded and continuous and $\left\{\Xi y:y\in {\mathfrak{Q}}_r\right\}$ is equicontinuous. Next, we show that $\Xi$ is relatively compact in ${\mathfrak{Q}}_r$.

Let $y_0=t^{1-\alpha }{\mathcal{U}}_{\alpha }(t,0)x_0$ and $y_{n+1}=\Xi y_n,n=0,1,2,\cdots .$ Using Lemma 5, $\Xi y_n\in {\mathfrak{Q}}_r$ for $y_n\in {\mathfrak{Q}}_r$ and is bounded. Consider the set $H={\left\{\Xi y_n:y_n\in {\mathfrak{Q}}_r\right\}}_{n=0}^{\infty }$. Next, we will show that H is a relatively compact set. Given Lemma 4, the set H is equicontinuous. We just need to verify that $H\left(t\right)={\left\{\left(\Xi y_n\right)\left(t\right),y_n\in {\mathfrak{Q}}_r\right\}}_{n=0}^{\infty }$ is relatively compact in X for $t\in [0,a]$. Due to the nature of the measure of non-compactness, for any $t\in [0,a]$, we have

$$\rho \left({\left\{y_n\left(t\right)\right\}}_{n=0}^{\infty }\right)=\rho \left(\left\{y_0\left(t\right)\right\}\cup {\left\{y_n\left(t\right)\right\}}_{n=1}^{\infty }\right)=\rho \left({\left\{y_n\left(t\right)\right\}}_{n=1}^{\infty }\right)=\rho \left(H\left(t\right)\right).$$

(3)

Using the condition (H4) and properties (i)–(v), we have the following:

$$\begin{array}{cc}\hfill \rho \left(H\left(t\right)\right)=& \rho \left({\left\{\left(\Xi y_n\right)\left(t\right)\right\}}_{n=0}^{\infty }\right)\hfill \\ \hfill =& \rho \left({\left\{t^{1-\alpha }{\mathcal{U}}_{\alpha }(t,s)x_0+t^{1-\alpha }{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x_n\left(\tau \right))d\tau \right\}}_{n=0}^{\infty }\right)\hfill \\ \hfill =& \rho \left({\left\{t^{1-\alpha }{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x_n\left(\tau \right))d\tau \right\}}_{n=0}^{\infty }\right)\hfill \\ \hfill \le & 2C{t}^{1-\alpha }{\int }_0^t{(t-\tau )}^{\alpha -1}\rho \left(f\left(\tau ,{\left\{t^{\alpha -1}{y}_n\left(\tau \right)\right\}}_{n=0}^{\infty }\right)\right)d\tau \hfill \\ \hfill \le & 2CM{\int }_0^t{(t-\tau )}^{\alpha -1}{t}^{1-\alpha }\rho \left({\left\{t^{\alpha -1}{y}_n\left(\tau \right)\right\}}_{n=0}^{\infty }\right)d\tau \hfill \\ \hfill \le & 2CM{\int }_0^t{(t-\tau )}^{\alpha -1}\rho \left({\left\{y_n\left(\tau \right)\right\}}_{n=0}^{\infty }\right)d\tau .\hfill \end{array}$$

Judging by (3), we obtain

$$\rho \left(H\left(t\right)\right)\le 2CM{\int }_0^t{(t-\tau )}^{\alpha -1}\rho \left(H\left(\tau \right)\right)d\tau .$$

Therefore, using the generalized Gronwall inequality, $\rho \left(H\right(t\left)\right)=0$ can be obtained. We can further deduce that $H\left(t\right)$ is relatively compact. Thus, using the Arzela–Ascoli theorem, the set H is relatively compact. There is no doubt that ${lim}_{n\to \infty }{y}_n=y^{*}\in {\mathfrak{Q}}_r$. So, using the operator $\Xi$, we obtain the following:

$$y^{*}=\underset{n\to \infty }{lim}{y}_n=\underset{n\to \infty }{lim}\Xi y_{n-1}=\Xi \left(\underset{n\to \infty }{lim}{y}_{n-1}\right)=\Xi y^{*}.$$

Let $x^{*}\left(t\right)=t^{\alpha -1}{y}^{*}\left(t\right)$. In summary, $x^{*}$ is a mild solution to Problem (1) in ${\mathfrak{Q}}_{\alpha }$. □

The following is the proof of the uniqueness of a mild solution to Problem (1).

(H5) 

There exists a function $L\in C\left([0,a],{\mathbb{R}}^{+}\right)$, such that $J_{0+}^{\alpha }L\in C\left([0,a],{\mathbb{R}}^{+}\right)$,

$$∥f\left(t,x_1\left(t\right)\right)-f\left(t,x_2\left(t\right)\right)∥\le L\left(t\right){∥x_1-x_2∥}_{C_{1-\alpha }},\mathrm{for}\mathrm{any}{x}_1,x_2\in X,$$

and

$$\underset{t\in [0,a]}{sup}C{a}^{\alpha -1}{\int }_0^t{(t-s)}^{\alpha -1}L\left(s\right)ds\le M<1,$$

where C is defined in Lemma 1.

Theorem 2. 

Let $\alpha >{\displaystyle \frac{1}{2}}$. Suppose that conditions (H1)–(H4) and (H5) are satisfied. Then, Problem (1) has a unique mild solution in ${\mathfrak{Q}}_{\alpha }$.

Proof. 

We mainly want to show that $\Xi$ refers to contraction mapping.

From Lemma 5, we know that $\Xi {\mathfrak{Q}}_r\subset {\mathfrak{Q}}_r$. For any $y_1,y_2\in {\mathfrak{Q}}_r,t\in [0,a]$, we have

$$\begin{array}{cc}\hfill & ∥\left(\Xi y_1\right)\left(t\right)-\left(\Xi y_2\right)\left(t\right)∥\hfill \\ \hfill \le & t^{1-\alpha }{\int }_0^t∥{\mathcal{U}}_{\alpha }(t,\tau )\left(f\left(\tau ,x_1\left(\tau \right)\right)-f\left(\tau ,x_2\left(\tau \right)\right)\right)∥d\tau \hfill \\ \hfill \le & C{a}^{1-\alpha }{\int }_0^t{(t-\tau )}^{\alpha -1}∥f\left(\tau ,x_1\left(\tau \right)\right)-f\left(\tau ,x_2\left(\tau \right)\right)∥d\tau \hfill \\ \hfill \le & C{a}^{1-\alpha }{\int }_0^t{(t-\tau )}^{\alpha -1}L\left(\tau \right){∥x_1-x_2∥}_{C_{1-\alpha }}d\tau \hfill \\ \hfill \le & M∥y_1\left(t\right)-y_2\left(t\right)∥.\hfill \end{array}$$

Thus,

$${∥\left(\Xi y_1\right)-\left(\Xi y_2\right)∥}_C\le M{∥y_1-y_2∥}_C.$$

It follows from the proof that $\Xi$ refers to contraction mapping. Using the contraction mapping principle, it can be obtained that $\Xi$ has a unique fixed point $y^{*}\in {\mathfrak{Q}}_r$. Therefore, $x^{*}\left(t\right)=t^{\alpha -1}{y}^{*}\left(t\right)$. Thus, $x^{*}$ is a unique mild solution to Problem (1) in ${\mathfrak{Q}}_{\alpha }$. □

4. Optimal Control

We denote by ${\mathcal{P}}_f\left(Y\right)$ the class of non-empty closed convex subsets of Y. ${\mathfrak{U}}_{ad}=\left\{u\in L^p\left([0,a],Z\right)\mid u\left(t\right)\in \mathfrak{U},a.e.t\in [0,a]\right\}$ is an admissible set, where $1<p<\infty$, $\mathfrak{U}:[0,a]⟶{\mathcal{P}}_f\left(Y\right)$ is a measurable multifunction and $Z\subset Y$ is a bounded set. Using Lemma 3.2 of [38], we can understand that ${\mathfrak{U}}_{ad}\ne \varnothing$.

For the fractional non-autonomous evolution equation with control, we prove optimal control in this section:

$$\left\{\begin{array}{c}{D}^{\alpha }x\left(t\right)+A\left(t\right)x\left(t\right)=f(t,x\left(t\right))+B\left(t\right)u\left(t\right),t\in (0,a],u\in {\mathfrak{U}}_{ad},\hfill \\ J_t^{1-\alpha }x\left(0\right)=x_0.\hfill \end{array}\right.$$

(4)

For Problem (4) to obtain the result of optimal control, the cost function

$$\mathcal{F}(x,u):={\int }_0^{a}P(t,x\left(t\right),u\left(t\right))dt$$

can be minimized as long as a state-control pair $\left(x^0,u^0\right)\in C_{1-\alpha }\left([0,a],X\right)\times {\mathfrak{U}}_{\mathrm{ad}}$ can be found.

To prove that the needs of the below, we introduce the following commonly used assumptions:

(H6) 

The function $P:[0,a]\times X\times Y\to \mathbb{R}\cup \{\infty \}$ satisfies the following:

(i) The function $P:[0,a]\times X\times Y\to \mathbb{R}\cup \{\infty \}$ is Borel measurable;

(ii) $P(t,·,·)$ is sequentially lower semicontinuous on $X\times Y$ for almost all $t\in [0,a]$;

(iii) $P(t,x,·)$ is convex on Y for each $x\in X$ and almost all $t\in [0,a]$;

(iv) Constants $c\ge 0,d>0,\phi$ are non-negative, and $\phi \in L^1\left([0,a],\mathbb{R}\right)$ such that

$$P(t,x,u)\ge \phi \left(t\right)+{c\parallel x\parallel }_X+d{\parallel u\parallel }_Y^p.$$

(H7) 

$B\in L^{\infty }\left([0,a],\mathcal{L}(Y,X)\right)$, where $L^{\infty }\left([0,a],\mathcal{L}(Y,X)\right)$ is a Banach space with norm ${\parallel ·\parallel }_{\infty }$.

Theorem 3. 

Let $\alpha >{\displaystyle \frac{1}{2}}$. Under the assumptions of Theorem 1, assumption (H7) also holds. For every $u\in {\mathfrak{U}}_{ad}$, Problem (4) has a mild solution corresponding to u provided by the following:

$$x\left(t\right)={\mathcal{U}}_{\alpha }(t,0)x_0+{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x\left(\tau \right))d\tau +{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )B\left(\tau \right)u\left(\tau \right)d\tau .$$

(5)

Proof. 

In contrast to Theorem 1, the key consideration is the term that contains the control function. Let

$$\Psi \left(t\right)={\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )B\left(\tau \right)u\left(\tau \right)d\tau .$$

Then, using the Hölder inequality, the definition of ${\mathfrak{U}}_{ad}$, and condition (H7), we have

$$\begin{array}{cc}\hfill \parallel \Psi \left(t\right)\parallel & \le {\int }_0^t∥{\mathcal{U}}_{\alpha }(t,\tau )B\left(\tau \right)u\left(\tau \right)∥d\tau \hfill \\ \hfill & \le C{\int }_0^t{(t-\tau )}^{\alpha -1}\parallel B\left(\tau \right)u\left(\tau \right)\parallel d\tau \hfill \\ \hfill & \le {C\parallel B\parallel }_{\infty }{\int }_0^t{(t-\tau )}^{\alpha -1}{\parallel u\left(\tau \right)\parallel }_{Y}d\tau \hfill \\ \hfill & \le {C\parallel B\parallel }_{\infty }{\left({\int }_0^t{(t-\tau )}^{{\displaystyle \frac{p}{p-1}}(\alpha -1)}d\tau \right)}^{{\displaystyle \frac{p-1}{p}}}{\left({\int }_0^t{\parallel u\left(\tau \right)\parallel }_Y^{p}d\tau \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {C\parallel B\parallel }_{\infty }{\left({\displaystyle \frac{p-1}{p\alpha -1}}\right)}^{{\displaystyle \frac{p-1}{p}}}{a}^{{\displaystyle \frac{p\alpha -1}{p}}}{\parallel u\parallel }_{L^p([0,a],Y)}.\hfill \end{array}$$

Thus, $t\in [0,a]$, $∥{\mathcal{U}}_{\alpha }(t,\tau )B\left(\tau \right)u\left(\tau \right)∥$ is Lebesgue integrable with respect to $\tau \in [0,t]$. Using Lemma 3, for all $t\in [0,a]$, it follows that ${\mathcal{U}}_{\alpha }(t,\tau )B\left(\tau \right)u\left(\tau \right)$ is the Bochner integral with respect to $\tau \in [0,t]$. Hence, $\Psi \left(t\right)\in C_{1-\alpha }$. The procedure of the proof is analogous to Theorem 1; therefore, we will omit it. □

Lemma 6. 

Let $\alpha >{\displaystyle \frac{1}{2}}$. Suppose that conditions (H1)–(H4) and (H7) are satisfied. Then, for fixed $u\in {\mathfrak{U}}_{ad}$, there exists a number $ϵ>0$ such that ${\parallel x\parallel }_{C_{1-\alpha }}\le ϵ$.

Proof. 

On the basis of Theorem 3, we have

$$x\left(t\right)={\mathcal{U}}_{\alpha }(t,0)x_0+{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )(f(\tau ,x\left(\tau \right))+B\left(\tau \right)u\left(\tau \right))\mathrm{d}\tau ,t\in (0,a].$$

With Lemma 1 and conditions (H2) and (H7), we obtain the following:

$$\begin{array}{cc}\hfill t^{1-\alpha }\parallel x\left(t\right)\parallel \le & ∥t^{1-\alpha }{\mathcal{U}}_{\alpha }(t,0)x_0∥+t^{1-\alpha }∥{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x\left(\tau \right))\mathrm{d}\tau ∥\hfill \\ \hfill & +t^{1-\alpha }∥{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )B\left(\tau \right)u\left(\tau \right))\mathrm{d}\tau ∥\hfill \\ \hfill \le & C∥x_0∥+C{a}^{1-\alpha }{\int }_0^t{(t-\tau )}^{\alpha -1}\left(\parallel \kappa \left(\tau \right)\parallel +\varrho {\tau }^{\alpha -1}\parallel x\left(\tau \right)\parallel +\parallel B\left(\tau \right)u\left(\tau \right)\parallel \right)\mathrm{d}\tau \hfill \\ \hfill \le & C∥x_0∥+C{\displaystyle \frac{\varrho ra}{\alpha }}+C{a}^{1-{\displaystyle \frac{1}{p}}}{\left({\displaystyle \frac{p-1}{\alpha p-1}}\right)}^{1-{\displaystyle \frac{1}{p}}}\left({\parallel \kappa \parallel }_p+{\parallel B\parallel }_{\infty }{\parallel u\parallel }_{L^p([0,a],Y)}\right)=:ϵ.\hfill \end{array}$$

This indicates that ${\parallel x\parallel }_{C_{1-\alpha }}\le ϵ$. □

With respect to the existence of an optimal control for Problem (4), we have the following result.

Theorem 4. 

Let $\alpha >{\displaystyle \frac{1}{2}}$. Assuming that both Theorem 3 and assumption (H6) hold, the functional admits at least one optimal pair.

Proof. 

Claim I. Obviously, $inf\left\{\mathcal{F}(x,u):(x,u)\in C_{1-\alpha }([0,a],X)\times {\mathfrak{U}}_{\mathrm{ad}}\right\}=+\infty$, so the conclusion is valid.

Claim II. Not breaking general, we conjecture that

$$inf\{\mathcal{F}(x,u):\left.(x,u)\in C_{1-\alpha }([0,a],X)\times {\mathfrak{U}}_{\mathrm{ad}}\right\}=\varpi <+\infty .$$

$\varpi >-\infty$ can be determined by H(6)(iv). By the definition of infimum, there exists a minimizing sequence pair $\left\{\left(x^m,u^m\right)\right\}$, which makes $\mathcal{F}\left(x^m,u^m\right)\to \varpi$$m\to +\infty$. Because $\left\{u^m\right\}\subseteq {\mathfrak{U}}_{\mathrm{ad}}(m=1,2,\dots )$, $\left\{u^m\right\}$ is bound to $L^p\left([0,a],Y\right)$. So, a subsequence, relabeled $\left\{u^m\right\}$, can be found from $\left\{u^m\right\}$, such that

$$u^m\stackrel{w}{⟶}{u}^0\mathrm{in}{L}^p\left([0,a],Y\right),$$

where $u^0\in L^p\left([0,a],Y\right)$. Combining the convexity and closedness of ${\mathfrak{U}}_{\mathrm{ad}}$ with Mazur’s lemma, it follows that $u^0\in {\mathfrak{U}}_{\mathrm{ad}}$.

Next, in terms of $\left\{u^m\right\}$, we can denote the corresponding sequence of solutions $\left\{x^m\right\}$ of Problem (4):

$$\begin{array}{cc}\hfill x^m\left(t\right)=& {\mathcal{U}}_{\alpha }(t,0)x_0+{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )B\left(\tau \right)u^m\left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f\left(\tau ,x^m\left(\tau \right)\right)d\tau .\hfill \end{array}$$

(6)

We must demonstrate that $\left\{x^m\right\}$ is relatively compact on $C_{1-\alpha }\left([0,a],X\right)$.

In the first step, the uniform boundedness of ${∥x^m∥}_{C_{1-\alpha }}$ can be deduced from the boundedness of $\left\{u^m\right\}$ and Lemma 6.

Additionally, let $y^m\left(t\right)=t^{1-\alpha }{x}^m\left(t\right)$. As proof of Theorem 1, $\left\{y^m:y^m=t^{1-\alpha }{x}^m\left(t\right)\right\}$ is equicontinuous and a relatively compact subset of $C\left(\right[0,a],X)$. Correspondingly, $\left\{x^m\right\}$ is relatively compact on $C_{1-\alpha }\left([0,a],X\right)$. So, function $x^0\in C_{1-\alpha }\left([0,a],X\right)$ ensures

$$x^m⟶x^0\mathrm{in}{C}_{1-\alpha }\left([0,a],X\right)$$

(7)

is established.

In addition, from (H5), there is

$$∥f\left(t,x^m\left(t\right)\right)-f\left(t,x^0\left(t\right)\right)∥\le L\left(t\right){∥x^m-x^0∥}_{C_{1-\alpha }},$$

followed by (7). We have

$$f\left(t,x^m\left(t\right)\right)⟶f\left(t,x^0\left(t\right)\right),\mathrm{a}.\mathrm{e}.t\in [0,a],$$

and

$$∥f\left(t,x^m\left(t\right)\right)∥\le \kappa \left(t\right)+\varrho t^{1-\alpha }\parallel x^m\left(t\right)\parallel ,$$

can be obtained from (H2). The flexible application of the dominated convergence theorem can be derived as follows:

$$\begin{array}{cc}\hfill & {\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f\left(\tau ,x^m\left(\tau \right)\right)d\tau \hfill \\ \hfill \to & {\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f\left(\tau ,x^0\left(\tau \right)\right)d\tau ,\mathrm{a}.\mathrm{e}.t\in [0,a].\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}\hfill & {\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )B\left(\tau \right)u^m\left(\tau \right)d\tau \hfill \\ \hfill \to & {\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )B\left(\tau \right)u^0\left(\tau \right)d\tau ,\mathrm{a}.\mathrm{e}.t\in [0,a].\hfill \end{array}$$

Hence, it follows from (6) that

$$\begin{array}{cc}\hfill x^0\left(t\right)=& {\mathcal{U}}_{\alpha }(t,0)x_0+{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )B\left(\tau \right)u^0\left(\tau \right)d\tau \hfill \\ \hfill & +{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f\left(\tau ,x^0\left(\tau \right)\right)d\tau .\hfill \end{array}$$

Notice that all of Balder’s assumptions (see Theorem 2.1 in [39]) are valid with (H6). Naturally, with Balder’s theorem and (H6)(iv), $\mathcal{F}>-\infty ,\mathcal{F}$ reaches its infimum at $\left(x^0,u^0\right)\in C_{1-\alpha }\left([0,a],X\right)\times {\mathfrak{U}}_{\mathrm{ad}}$, i.e.,

$$\begin{array}{cc}\hfill \varpi & =\underset{m\to \infty }{lim}{\int }_0^{a}P\left(t,x^m\left(t\right),u^m\left(t\right)\right)dt\hfill \\ \hfill & \ge {\int }_0^{a}P\left(t,x^0\left(t\right),u^0\left(t\right)\right)dt=\mathcal{F}\left(x^0,u^0\right)\ge \varpi .\hfill \end{array}$$

The proof is completed. □

5. Approximate Controllability

In this section, we prove the approximate controllability of NFEEs control systems:

$$\left\{\begin{array}{c}{D}^{\alpha }x\left(t\right)+A\left(t\right)x\left(t\right)=f(t,x\left(t\right))+Bu\left(t\right),t\in (0,a],u\in Y,\hfill \\ J_t^{1-\alpha }x\left(0\right)=x_0,\hfill \end{array}\right.$$

(8)

where B is a linear bounded operator from Y to X. For convenience, we introduce $N_B=\parallel B\parallel$.

Let $x(a,u)$ be the state of Problem (8) at $t=a$ with respect to admissible control u. Then, we introduce the set $\Pi (a,\zeta )=\left\{x(a,u):u(·)\in L^p\left([0,a],Y\right),x(0,u)=\zeta \right\}$, which is called the reachable set of system problem (8), and its closure in X is designated by $\overline{\Pi }$.

Definition 2. 

Problem (8) is said to be approximately controllable on $[0,a]$ if for all $x(0,u)=\zeta \in X$ the closure of the reachable set $\overline{\Pi }(a,\zeta )=X$, i.e., for any given $\epsilon >0$, the problem can be steered from initial state ζ to a ε neighborhood of any of state in X at time a.

To prove what we want to say, let us introduce two relevant operators:

$${\Gamma }_0^a={\int }_0^a{\mathcal{U}}_{\alpha }(a,\tau )B{B}^{*}{\mathcal{U}}_{\alpha }^{*}(a,\tau )d\tau ,$$

and

$$\mathfrak{R}\left(\gamma ,{\Gamma }_0^a\right)={\left(\gamma I+{\Gamma }_0^a\right)}^{-1},\gamma >0,$$

where $B^{*}$ denotes the adjoint of B, and ${\mathcal{U}}_{\alpha }^{*}$ is the adjoint of ${\mathcal{U}}_{\alpha }$. It is straightforward that the operator ${\Gamma }_0^a$ is a linear bounded operator.

Before stating and proving the main results, we impose the following hypotheses on data of the problem:

(H8) 

$\gamma \mathfrak{R}\left(\gamma ,{\Gamma }_0^a\right)\to 0$, as $\gamma \to 0^{+}$ in the strong operator topology.

Theorem 5. 

Let $\alpha >{\displaystyle \frac{1}{2}}$. Suppose that (H1)–(H3) are satisfied. For every $u\in Y$, Problem (8) has a mild solution corresponding to u, provided by

$$x\left(t\right)={\mathcal{U}}_{\alpha }(t,0)x_0+{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x\left(\tau \right))d\tau +{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )Bu\left(\tau \right)d\tau .$$

(9)

Proof. 

According to the above, Problem (1) has at least one mild solution. Then, the key consideration is the term containing the nonlinear function. For all $x\in X$ and almost all $t\in [0,a]$, we have

$$\begin{array}{cc}\hfill \parallel f(t,x)+Bu\left(t\right)\parallel & \le \kappa \left(t\right)+\varrho t^{1-\alpha }\parallel x\parallel +\parallel Bu\left(t\right)\parallel \hfill \\ \hfill & \le \kappa \left(t\right)+\varrho t^{1-\alpha }\parallel x\parallel +N_B{\parallel u\left(t\right)\parallel }_Y\hfill \\ \hfill & =\left(\kappa \left(t\right)+N_B{\parallel u\left(t\right)\parallel }_Y\right)+\varrho t^{1-\alpha }\parallel x\parallel .\hfill \end{array}$$

With the help of assumption (H2), it can be easily seen that $\kappa \left(t\right)+N_B{\parallel u\left(t\right)\parallel }_Y\in L^p\left((0,a],{\mathbb{R}}^{+}\right)$$({\displaystyle \frac{1}{{\vartheta }_0}}<p<\infty )$.

Similarly to Theorem 3, we have

$$\begin{array}{cc}\hfill ∥{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )Bu\left(\tau \right)d\tau ∥& \le {\int }_0^t∥{\mathcal{U}}_{\alpha }(t,\tau )Bu\left(\tau \right)∥d\tau \hfill \\ \hfill & \le C{\int }_0^t{(t-\tau )}^{\alpha -1}\parallel Bu\left(\tau \right)\parallel d\tau \hfill \\ \hfill & \le C{N}_B{\int }_0^t{(t-\tau )}^{\alpha -1}{\parallel u\left(\tau \right)\parallel }_{Y}d\tau \hfill \\ \hfill & \le C{N}_B{\left({\int }_0^t{(t-\tau )}^{{\displaystyle \frac{p}{p-1}}(\alpha -1)}d\tau \right)}^{{\displaystyle \frac{p-1}{p}}}{\left({\int }_0^t{\parallel u\left(\tau \right)\parallel }_Y^{p}d\tau \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le C{N}_B{\left({\displaystyle \frac{p-1}{p\alpha -1}}\right)}^{{\displaystyle \frac{p-1}{p}}}{a}^{{\displaystyle \frac{p\alpha -1}{p}}}{\parallel u\parallel }_{L^p([0,a],Y)}.\hfill \end{array}$$

(10)

Using Lemma 3, (9) is well defined. The rest of the process is similar to Theorem 3 and so we will omit it. □

Below we present our conclusion regarding approximate controllability.

For every $\gamma >0$ and final state $x_a\in X$, there exists a continuous function $u\in Y$ such that

$$\begin{array}{c}\hfill x\left(t\right)={\mathcal{U}}_{\alpha }(t,0)x_0+{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x\left(\tau \right))d\tau +{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )Bu\left(\tau \right)d\tau ,\end{array}$$

(11)

where u is the control function defined by

$$u\left(t\right)=B^{*}{\mathcal{U}}_{\alpha }^{*}(a,t)\mathfrak{R}\left(\gamma ,{\Gamma }_0^a\right)\eta \left(x(·)\right),$$

(12)

with

$$\eta \left(x(·)\right)=x_a-{\mathcal{U}}_{\alpha }(·,0)x_0-{\int }_0^t{\mathcal{U}}_{\alpha }(·,\tau )f(\tau ,x\left(\tau \right))d\tau .$$

(13)

Theorem 6. 

Let $\alpha >{\displaystyle \frac{1}{2}}$ and assume that (H1)–(H3) and (H8) are satisfied. Then, Problem (8) is approximately controllable on $(0,a]$.

Proof. 

Under the above assumptions and by Theorem 5, we know that Problem (8) has at least one mild solution $x_{\gamma }\in {\mathfrak{Q}}_{\alpha }$, which means that

$$\begin{array}{c}\hfill x_{\gamma }\left(t\right)={\mathcal{U}}_{\alpha }(t,0)x_0+{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x_{\gamma }\left(\tau \right))d\tau +{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )B{u}_{\gamma }\left(\tau \right)d\tau ,\end{array}$$

(14)

with

$$u_{\gamma }\left(t\right)=B^{*}{\mathcal{U}}_{\alpha }^{*}(a,t)\mathfrak{R}\left(\gamma ,{\Gamma }_0^a\right)\eta \left(x_{\gamma }(·)\right),$$

(15)

and

$$\eta \left(x_{\gamma }\left(t\right)\right)=x_a-{\mathcal{U}}_{\alpha }(t,0)x_0-{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x_{\gamma }\left(\tau \right))d\tau .$$

(16)

Combining (14)–(16), we have

$$\begin{array}{cc}\hfill x_{\gamma }\left(a\right)=& {\mathcal{U}}_{\alpha }(a,0)x_0+{\int }_0^a{\mathcal{U}}_{\alpha }(a,\tau )f(\tau ,x_{\gamma }\left(\tau \right))d\tau +{\int }_0^t{\mathcal{U}}_{\alpha }(a,\tau )B{u}_{\gamma }\left(\tau \right)d\tau \hfill \\ \hfill =& {\mathcal{U}}_{\alpha }(a,0)x_0+{\int }_0^a{\mathcal{U}}_{\alpha }(a,\tau )f(\tau ,x_{\gamma }\left(\tau \right))d\tau \hfill \\ \hfill & +{\int }_0^a{\mathcal{U}}_{\alpha }(a,\tau )B\left(\tau \right)B^{*}{\mathcal{U}}_{\alpha }^{*}(a,\tau )\mathfrak{R}\left(\gamma ,{\Gamma }_0^a\right)\eta \left(x_{\gamma }\left(a\right)\right)d\tau \hfill \\ \hfill =& {\mathcal{U}}_{\alpha }(a,0)x_0+{\int }_0^a{\mathcal{U}}_{\alpha }(a,\tau )f(\tau ,x_{\gamma }\left(\tau \right))d\tau +{\Gamma }_0^a\mathfrak{R}\left(\gamma ,{\Gamma }_0^a\right)\eta \left(x_{\gamma }\left(a\right)\right)\hfill \\ \hfill =& {\mathcal{U}}_{\alpha }(a,0)x_0+{\int }_0^a{\mathcal{U}}_{\alpha }(a,\tau )f(\tau ,x_{\gamma }\left(\tau \right))d\tau +\eta \left(x_{\gamma }\left(a\right)\right)-\gamma \mathfrak{R}\left(\gamma ,{\Gamma }_0^a\right)\eta \left(x_{\gamma }\left(a\right)\right)\hfill \\ \hfill =& x_a-\gamma \mathfrak{R}\left(\gamma ,{\Gamma }_0^a\right)\eta \left(x_{\gamma }\left(a\right)\right).\hfill \end{array}$$

(17)

Assumption (H2), Lemma 6, and the Minkowski inequality imply the following:

$$\begin{array}{cc}\hfill {\left({\int }_0^a{∥f(\tau ,x_{\gamma }\left(\tau \right))∥}^{p}d\tau \right)}^{{\displaystyle \frac{1}{p}}}& \le {\left({\int }_0^a{\left(\kappa \left(\tau \right)+\varrho {\tau }^{1-\alpha }\parallel x_{\gamma }\parallel \right)}^{p}d\tau \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {\left({\int }_0^a{\left(\kappa \left(\tau \right)\right)}^{p}d\tau \right)}^{{\displaystyle \frac{1}{p}}}+{\left({\int }_0^a{\left(\varrho {\tau }^{1-\alpha }\parallel x_{\gamma }\parallel \right)}^{p}d\tau \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {\parallel \kappa \parallel }_p+\varrho \parallel x_{\gamma }\parallel {\left({\displaystyle \frac{a^{(1-\alpha )p+1}}{(1-\alpha )p+1}}\right)}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

Using formula (2) and (10), the solution $x_{\gamma }$ corresponding to the control $u_{\gamma }$ is bounded, which implies that the sequence $\left\{f(·,x_{\gamma }(·))\right\}$ is bounded in $L^p\left((0,a],X\right)$. Therefore, there exists a subsequence of $\left\{f(·,x_{\gamma }(·))\right\}$, still denoted by $\left\{f(·,x_{\gamma }(·))\right\}$, which converges weakly to some point $\mathfrak{F}\in L^p\left((0,a],X\right)$.

We write

$$v:=x_a-{\int }_0^a{\mathcal{U}}_{\alpha }(a,\tau )\mathfrak{F}\left(\tau \right)d\tau .$$

(18)

Hence, from (16) and (18), one obtains that

$$\begin{array}{cc}\hfill ∥\eta \left(x_{\gamma }\right)-v∥& =∥x_a-{\mathcal{U}}_{\alpha }(t,0)x_0-{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x_{\gamma }\left(\tau \right))d\tau -x_a+{\int }_0^a{\mathcal{U}}_{\alpha }(a,\tau )\mathfrak{F}\left(\tau \right)d\tau ∥\hfill \\ \hfill & \le ∥{\int }_0^a{\mathcal{U}}_{\alpha }(a,\tau )\left[f(\tau ,x_{\gamma }\left(\tau \right))-\mathfrak{F}\left(\tau \right)\right]d\tau ∥.\hfill \end{array}$$

(19)

From Lemma 2, ${\mathcal{U}}_{\alpha }$ is compact. Then, the mapping is as follows:

$$f\left(t\right)\to {\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f\left(\tau \right)d\tau ,$$

and is it compact for $t\in (0,a]$, which implies that

$${\int }_0^a{\mathcal{U}}_{\alpha }(a,\tau )\left[f(\tau ,x_{\gamma }\left(\tau \right))-\mathfrak{F}\left(\tau \right)\right]d\tau \to 0\mathrm{as}\gamma \to 0^{+}.$$

(20)

Therefore, from (19) and (20), we know that

$$∥\eta \left(x_{\gamma }\right)-v∥\to 0\mathrm{as}\gamma \to 0^{+}.$$

(21)

Hence, (17), (21), and assumption (H8) imply the following:

$$\begin{array}{cc}\hfill ∥x_{\gamma }\left(a\right)-x_a∥& \le ∥\gamma \mathfrak{R}\left(\gamma ,{\Gamma }_0^a\right)\eta \left(x_{\gamma }\left(a\right)\right)∥\hfill \\ \hfill & \le ∥\gamma \mathfrak{R}\left(\gamma ,{\Gamma }_0^a\right)v∥+∥\gamma \mathfrak{R}\left(\gamma ,{\Gamma }_0^a\right)∥·∥\eta \left(x_{\gamma }\right)-v∥\hfill \\ \hfill & \to 0\mathrm{as}\gamma \to 0^{+}.\hfill \end{array}$$

Therefore, Problem (8) is approximately controllable. □

The following is a conclusion about the continuous dependence of the solution on the control.

Theorem 7. 

Let $\alpha >{\displaystyle \frac{1}{2}}$. Suppose that conditions (H1)–(H3) and (H5) are satisfied. If $x_1$ and $x_2$ are two mild solutions of Problem (8) corresponding to the control functions $u_1,u_2\in Y$, respectively, then for any $\omega >0$, if ${∥u_1\left(t\right)-u_2\left(t\right)∥}_Y\le \delta$, we have

$$\parallel x_1\left(t\right)-x_2\left(t\right)\parallel \le \omega .$$

Proof. 

Let $x_1$ and $x_2$ be two mild solutions of Problem (8). Then, using condition (H5), we have

$$\begin{array}{cc}\hfill ∥x_1\left(t\right)-x_2\left(t\right)∥=& ∥{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x_1\left(\tau \right))d\tau -{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )f(\tau ,x_2\left(\tau \right))d\tau \hfill \\ \hfill & +{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )B{u}_1\left(\tau \right)d\tau -{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )B{u}_2\left(\tau \right)d\tau ∥\hfill \\ \hfill \le & {\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )∥f(\tau ,x_1\left(\tau \right))-f(\tau ,x_2\left(\tau \right))∥d\tau \hfill \\ \hfill & +{\int }_0^t{\mathcal{U}}_{\alpha }(t,\tau )B∥u_1\left(\tau \right)-u_2\left(\tau \right)∥d\tau \hfill \\ \hfill \le & C{t}^{1-\alpha }{\int }_0^t{(t-\tau )}^{\alpha -1}L\left(\tau \right)∥x_1\left(\tau \right)-x_2\left(\tau \right)∥d\tau \hfill \\ \hfill & +C{N}_B{\int }_0^t{(t-\tau )}^{\alpha -1}∥u_1\left(\tau \right)-u_2\left(\tau \right)∥d\tau \hfill \\ \hfill \le & C{a}^{1-\alpha }{\int }_0^t{(t-\tau )}^{\alpha -1}∥x_1\left(\tau \right)-x_2\left(\tau \right)∥d\tau \hfill \\ \hfill & +C{N}_B{\int }_0^t{(t-\tau )}^{\alpha -1}∥u_1\left(\tau \right)-u_2\left(\tau \right)∥d\tau \hfill \\ \hfill \le & C{\int }_0^t{(t-\tau )}^{\alpha -1}\left[∥x_1\left(\tau \right)-x_2\left(\tau \right)∥-∥u_1\left(\tau \right)-u_2\left(\tau \right)∥\right]d\tau .\hfill \end{array}$$

On account of $\alpha >{\displaystyle \frac{1}{2}}$, by Schwartz’s inequality, we obtain the following:

$$\begin{array}{cc}\hfill & {∥x_1\left(t\right)-x_2\left(t\right)∥}^2\hfill \\ \hfill \le C& \left({\int }_0^t{(t-\tau )}^{2(\alpha -1)}d\tau \right)\left({\int }_0^t{\left[∥x_1\left(\tau \right)-x_2\left(\tau \right)∥+∥u_1\left(\tau \right)-u_2\left(\tau \right)∥\right]}^{2}d\tau \right)\hfill \\ \hfill \le C& \left({\int }_0^t{(t-\tau )}^{2(\alpha -1)}d\tau \right)\left({\int }_0^t{∥x_1\left(\tau \right)-x_2\left(\tau \right)∥}^{2}d\tau +{\int }_0^t{∥u_1\left(\tau \right)-u_2\left(\tau \right)∥}^{2}d\tau \right)\hfill \\ \hfill \le C& \left({\int }_0^t{(t-\tau )}^{2(\alpha -1)}d\tau \right){\int }_0^t{∥x_1\left(\tau \right)-x_2\left(\tau \right)∥}^{2}d\tau \hfill \\ \hfill +& C\left({\int }_0^t{(t-\tau )}^{2(\alpha -1)}d\tau \right){\int }_0^t{∥u_1\left(\tau \right)-u_2\left(\tau \right)∥}^{2}d\tau .\hfill \end{array}$$

We know that the integral ${\int }_0^t{(t-\tau )}^{2(\alpha -1)}d\tau$ can be integrated and has bounds. Therefore, if ${∥u_1\left(t\right)-u_2\left(t\right)∥}_Y\le \delta$, we have

$${∥x_1\left(t\right)-x_2\left(t\right)∥}^2\le C{\int }_0^t{∥x_1\left(\tau \right)-x_2\left(\tau \right)∥}^{2}d\tau +Ca{\delta }^2.$$

According to the Gronwall inequality, we can see that

$${∥x_1\left(t\right)-x_2\left(t\right)∥}^2\le C{\delta }^{2}a{e}^{Ca},\mathrm{where}\delta ={\displaystyle \frac{\omega }{Ca{e}^{Ca}+1}}.$$

Therefore, $∥x_1\left(t\right)-x_2\left(t\right)∥<\omega$ holds. □

6. Conclusions

In this study, our primary focus was examining the existence and controllability of mild solutions to NFEEs. We employed Kuratowski’s non-compactness measure along with the fixed point theorem as fundamental tools in our analysis. Through this approach, we established the existence of a mild solution to the equation, notably without relying on the compactness of the nonlinear term, the Lipschitz continuity assumption, or the compactness assumption of the operator. The findings presented herein serve to extend certain previously established research results in this domain.

The complexity of NFEEs increases the difficulty of theoretical analysis. Additionally, research on these equations is fragmented, lacking a unified theoretical framework across different fields and equation types and hindering systematic development.

In parallel with the theory of fractional autonomous evolution equations, we defined solutions for NFEEs to further address the well-posedness of more general non-autonomous problems. Building upon the foundational theory of NFEE systems, we aimed to deepen research into inverse problems, weak solutions, regularity, and similar topics.