A Study on the Approximate Controllability of Hilfer Fractional Evolution Systems

Abstract

In this article, the existence and uniqueness of mild solutions are investigated for Hilfer fractional evolution systems. Particularly, the approximate controllability is also investigated under some essential conditions by applying the sequence method. An example, as an application, is provided to demonstrate the obtained results.

1. Introduction and Main Results

Fractional calculus has gained extensive attention because of its applications in the fields of chemical, biology, mechanics, engineering, neural network model, physics, pure mathematics, etc. Eidelman et al. [1] studied the Cauchy problems of fractional diffusion equations and pointed out that the fractional diffusion equations established a beautiful model for describing the hereditary property of processes.

We first introduce a group of concepts, see [2] for more details.

Definition 1.

Let $\mu \in (0,1)$. The μ-order fractional integral of $\mathcal{F}:[0,+\infty )\to \mathbb{R}$ is defined by

$$I_{0^{+}}^{\mu }\mathcal{F}(\eta )=\frac{1}{\Gamma (\mu )}{\int }_0^{}{(\eta -\theta )}^{\mu -1}\mathcal{F}(\theta )d\theta ,\eta >0,$$

where $\Gamma (\mu )={\int }_0^{\infty }{\eta }^{\mu -1}{e}^{-\eta }d\eta$.

Definition 2.

The μ-order Riemann–Liouville fractional derivative of $\mathcal{F}:[0,+\infty )\to \mathbb{R}$ is written as

$${}^L{D}_{0^{+}}^{\mu }\mathcal{F}(\eta )=\frac{d}{d\eta }{I}_{0^{+}}^{1-\mu }\mathcal{F}(\eta ),\mu \in (0,1),\eta >0.$$

Definition 3.

Let $\mu \in (0,1)$. The μ-order Caputo fractional derivative of $\mathcal{F}\in C^1([0,\infty ),\mathbb{R})$ is given by

$${}^C{D}_{0^{+}}^{\mu }\mathcal{F}(\eta )=I_{0^{+}}^{(1-\mu )}\frac{d}{d\eta }\mathcal{F}(\eta ),\eta >0.$$

Recently, a new concept of fractional derivative was established by Hilfer, named “Hilfer fractional derivative”, see [3,4,5] for further details.

Definition 4

([4,5]). Let $\mu \in (0,1)$ and $\varrho \in [0,1]$. The Hilfer fractional derivative of order μ and type ϱ of $\mathcal{F}:[0,+\infty )\to \mathbb{R}$ is given by

$$D_{0^{+}}^{\varrho ,\mu }\mathcal{F}(\eta )=I_{0^{+}}^{\varrho (1-\mu )}\frac{d}{d\eta }{I}_{0^{+}}^{(1-\varrho )(1-\mu )}\mathcal{F}(\eta ),\eta >0.$$

Remark 1.

(i) If $\mu \in (0,1)$ and $\varrho =0$, we have

$$D_{0^{+}}^{0,\mu }\mathcal{F}(\eta )=\frac{d}{d\eta }{I}_{0^{+}}^{(1-\mu )}\mathcal{F}(\eta ){=}^L{D}_{0^{+}}^{\mu }\mathcal{F}(\eta ),\eta >0.$$

By Definition 2, it is the fractional derivative in the Riemann–Liouville sense.

(ii) If $\mu \in (0,1)$ and $\varrho =1$, we deduce that

$$D_{0^{+}}^{1,\mu }\mathcal{F}(\eta )=I_{0^{+}}^{(1-\mu )}\frac{d}{d\eta }\mathcal{F}(\eta ){=}^C{D}_{0^{+}}^{\mu }\mathcal{F}(\eta ),\eta >0.$$

By Definition 3, it is the Caputo fractional derivative.

Fractional evolution equations are useful mathematical models to describe many physical phenomena. The existence results for fractional evolution equations are obtained widely, see [4,6,7,8,9,10,11,12,13].

Controllability, as an important field of control theory, has been studied by many authors owing to its significant practical value. However, it follows from [14] that the definition of the exact controllability is limited. Approximate controllability is an appropriate extension of the exact controllability. It steers the control system to a small neighborhood of arbitrary final value. For the fractional evolution systems, the approximate controllability has been demonstrated extensively, see [15,16,17,18]. Chang et al. [15], by utilizing the resolvent operator theory, treated the approximate controllability of $\alpha$-order fractional evolution systems of Sobolev-type in the case of $\alpha \in (1,2)$. Ji [16] studied the Caputo fractional evolution nonlocal systems and, by applying the approximate method, obtained the approximate controllability result. Sakthivel et al. [17] built a group of sufficient conditions to guarantee the approximate controllability of Caputo fractional evolution equations. In [18], by utilizing the multivalued analysis, Yang et al. proved the approximate controllability of fractional evolution inclusions involving a Riemann–Liouville derivative. It is worth noting that the compactness of the operator semigroup (or resolvent operator) is assumed in these literature works.

Clearly, the following two questions are raised naturally. (i) May we remove or weaken the compactness conditions of the operator semigroup? (ii) May we remove or weaken the assumption of the approximate controllability of associated linear evolution systems? To answer these questions, many authors have made important contributions. In 1983, Zhou [19] established a new approximate technique and proved the approximate controllability of the problem

$$\left\{\begin{array}{c}{z}^{\prime }(\eta )=Az(\eta )+f(z(\eta ))+Bu(\eta ),\eta \in K^{\prime },\hfill \\ z(0)=z_0,\hfill \end{array}\right.$$

(1)

where $K^{\prime }=(0,b],b>0$, A generates a differentiable semigroup in the Banach space H and f satisfies some assumptions. In 2015, Zhou’s approximate technique was extended by Liu et al. in [20]. They demonstrated the following Riemann–Liouville fractional system

$$\left\{\begin{array}{c}{}^L{D}_{0^{+}}^{\mu }z(\eta )=Az(\eta )+f(\eta ,z(\eta ))+Bu(\eta ),\eta \in K^{\prime },0<\mu \le 1,\hfill \\ I_{0^{+}}^{1-\mu }{z(\eta )|}_{\eta =0}=z_0,\hfill \end{array}\right.$$

(2)

where A generates a differentiable semigroup in H and f is the nonlinear function satisfying some assumptions. Later, Liu et al. [21] applied the technique to the more general case with impulsive effects. Recently, by employing the resolvent operator theory, Yang [2] utilized this technique to treat the Caputo fractional system of Sobolev type

$$\left\{\begin{array}{c}{}^C{D}_{0^{+}}^{\mu }(Ez)(\eta )=Az(\eta )+f(\eta ,z(\eta ))+Bu(\eta ),\eta \in L^{\prime },1<\mu <2,\hfill \\ Ez(0)=z_0-g(z),{(Ez)}^{\prime }(0)=y_0-h(z),\hfill \end{array}\right.$$

(3)

where A and E are linear operators in H and $f,g$ and h are appropriate functions.

In this article, we are devoted to study the approximate controllability of the Hilfer fractional system

$$\left\{\begin{array}{c}{D}_{0^{+}}^{\varrho ,\mu }z(\eta )=Az(\eta )+\mathcal{F}(\eta ,z(\eta ))+Bu(\eta ),\eta \in K^{\prime },\hfill \\ I_{0^{+}}^{(1-\varrho )(1-\mu )}{z(\eta )|}_{\eta =0}=z_0,\hfill \end{array}\right.$$

(4)

where $\varrho \in [0,1]$ and $\mu \in (\frac{1}{2},1)$, $D_{0^{+}}^{\varrho ,\mu }$ denotes the Hilfer fractional derivative operator of order $\mu$ and type $\varrho$ with the lower limit zero and $A:D(A)\subset X\to X$ is the infinitesimal generator of a $C_0$-semigroup $T(\eta )(\eta \ge 0)$ of uniformly bounded operator in real Hilbert space X, that is, $\exists M>0,\parallel T(\eta )\parallel \le M$, U is another real Hilbert space, $u(t)\in L^2([0,b],U)$ for $t\in [0,b]$, $B\in \mathcal{L}(U,X)$, where $(\mathcal{L}(U,X),\parallel ·{\parallel }_{\mathcal{L}(U,X)})$ is the Banach space of bounded linear operators, $\mathcal{F}$ is a given function and $x_0\in X$. Denote $M_B:={\parallel B\parallel }_{\mathcal{L}(U,X)}$.

Let $K=[0,b]$. Denote by $C(K,X)$ and $C(K^{\prime },X)$ the sets of continuous functions, respectively. The norm in $C(K,X)$ is given by ${\parallel z\parallel }_C=\underset{\eta \in K}{sup}\parallel z(\eta )\parallel$. Let

$$C_{1-\gamma }(K,X)=\{z\in C(K^{\prime },X):{·}^{1-\gamma }z(·)\in C(K,X)\},$$

where $1-\gamma =(1-\varrho )(1-\mu )$. Then, $(C_{1-\gamma }(K,X),\parallel ·{\parallel }_{C_{1-\gamma }})$ is a Banach space, where

$${\parallel z\parallel }_{C_{1-\gamma }}=\underset{\eta \in K}{sup}\{{\eta }^{1-\gamma }\parallel z(\eta )\parallel \}.$$

Firstly, we introduce the following assumptions:

$(H_A)$ $T(\eta ),\eta >0$, is continuous in the uniform operator topology.

$(H_{f1})$ There is $\varphi \in L^2(K,{\mathbb{R}}^{+})$ and $\kappa >0$ such that, for any $\eta \in K$ and $y\in X$,

$$\parallel \mathcal{F}(\eta ,y)\parallel \le \varphi (\eta )+\kappa {\eta }^{1-\gamma }\parallel y\parallel .$$

$(H_{f2})$ There exists $L_1>0$ such that

$$\parallel \mathcal{F}(\eta ,y_1)-f(\eta ,y_2)\parallel \le L_1\parallel y_1-y_2\parallel$$

for any $\eta \in K$ and $y_1,y_2\in X$,

Theorem 1.

Let $(H_A),(H_{f1})$ and $(H_{f2})$ hold. Then, for every $u\in L^2(J,U)$, the Cauchy problem $(4)$ possesses a unique mild solution in $C_{1-\gamma }(K,X)$.

Let $x^u(\eta ):=x(\eta ;0,z_0,u)$ be the mild solution of (4) from the initial value $z_0\in X$ at time $\eta =0$ associated with $u\in L^2(K,U)$. If the condition $(H_{f2})$ is replaced by $(H_{f2}^{\prime })$ There exists $L_2>0$ such that, for any $\eta \in K$ and $y_1,y_2\in X$,

$$\parallel \mathcal{F}(\eta ,y_1)-\mathcal{F}(\eta ,y_2)\parallel \le L_2{\eta }^{1-\gamma }\parallel y_1-y_2\parallel .$$

The following theorem is obtained by Theorem 1.

Theorem 2.

Let $(H_A),(H_{f1})$ and $(H_{f2}^{\prime })$ be satisfied. Then, for every $u\in L^2(K,U)$, the Cauchy problem $(4)$ possesses a unique mild solution $z^u\in C_{1-\gamma }(K,X)$. Furthermore, $z^u$ satisfies the inequalities

$$\parallel z^u{\parallel }_{C_{1-\gamma }}\le {\mathcal{K}}_1$$

and

$$\parallel z^{v_2}-z^{v_1}{\parallel }_{C_{1-\gamma }}\le {\mathcal{K}}_2{\parallel B{v}_2-B{v}_1\parallel }_{L^2},\forall v_1,v_2\in L^2(K,U),$$

where

$${\mathcal{K}}_1=N_1{E}_{\mu }(M\kappa b^{1-\varrho (1-\mu )}),$$

$${\mathcal{K}}_2=N_2{E}_{\mu }(M{L}_2{b}^{1-\varrho (1-\mu )}),$$

$$N_1=\frac{M}{\Gamma (\gamma )}\parallel z_0\parallel +N_2{(\parallel \varphi \parallel }_{L^2}+{\parallel Bu\parallel }_{L^2}),$$

$$N_2=\frac{M{b}^{\frac{1}{2}-\varrho (1-\mu )}}{\Gamma (\mu )\sqrt{2\mu -1}},$$

where $E_{\mu }(x)={\displaystyle \sum _{n=0}^{\infty }}\frac{x^n}{\Gamma (n\mu +1)}$.

Remark 2.

It follows from pages 48–75 of [22] that if, for $\eta >0$, $T(\eta )$ is differentiable or compact, the condition $(H_A)$ holds. Furthermore, the analytic semigroup is a differentiable semigroup.

Remark 3.

In Theorems 1 and 2, under more general conditions on f, the existence and uniqueness results are obtained without the compactness of $T(\eta )(\eta \ge 0)$. The conclusions extend the related results of [4,13].

Remark 4.

By Remark 1, if $\mu \in (0,1)$ and $\varrho =0$, the problem $(4)$ degenerates to the problem $(1)$. Hence, Theorem 1 contains Theorem 3.2 of [20]. If $\mu \in (0,1)$ and $\varrho =1$, Theorems 1 and 2 are still new.

If $(H_A)$ is replaced by:

$(H_A^{\prime })$ $T(\eta )(\eta \ge 0)$ is a differentiable semigroup.

It follows from Remark 2 and Theorem 2 that the following corollary is easy to verify.

Corollary 1.

Let $(H_A^{\prime }),(H_{f1})$ and $(H_{f2}^{\prime })$ be satisfied. Then, for every $u\in L^2(K,U)$, the Cauchy problem $(4)$ possesses a unique mild solution $z^u\in C_{1-\gamma }(K,X)$.

Denote by ${\mathcal{D}}_b(\mathcal{F}):=\{z^u(b):u\in L^2(K,U)\}$. For the approximate controllability of (4), we understand it in the following sense.

Definition 5.

If $\forall z_0\in X$, $\overline{{\mathcal{D}}_b(\mathcal{F})}=X$, the problem (4) is called approximately controllable on K.

Let us define $\mathcal{R}$ in $C_{1-\gamma }(K,X)$ as

$$\mathcal{R}(z)(·)=\mathcal{F}(·,z(·)).$$

From $(H_{f1})$, $\mathcal{R}$ maps $C_{1-\gamma }(K,X)$ to $L^2(K,X)$. Define an operator G by

$$Gh={\int }_0^b{(b-\tau )}^{\mu -1}{P}_{\mu }(b-\tau )h(\tau )d\tau ,\forall h\in L^2(K,X),$$

(5)

where $P_{\mu }(·)$ is defined as (7) in the following. Then, $G:L^2(K,X)\to X$ is bounded. To prove the approximate controllability of (4), by Definition 5, we find $u_{ϵ}\in L^2(K,U)$ such that, for every $\xi \in X$ and any $ϵ>0$,

$$\parallel \xi -S_{\varrho ,\mu }(b)z_0-G\mathcal{R}(z^{u_{ϵ}})-GB{u}_{ϵ}\parallel <ϵ,$$

where $S_{\varrho ,\mu }(·)$ is defined as (6) in the following.

Now, let us introduce the result of the approximate controllability of (4).

Theorem 3.

Let $(H_A^{\prime }),(H_{f1}),(H_{f2}^{\prime })$ and $(H_G)$ be fulfilled, where $(H_G)$ is expressed by

$(H_G)$ $\forall ϵ>0,h\in L^2(K,X)$, there is a $u\in L^2(K,U)$ satisfying

$$\parallel GBu-Gh\parallel <ϵ$$

and

$${\parallel Bu\parallel }_{L^2}<\Lambda {\parallel h\parallel }_{L^2},$$

where $\Lambda >0$ is a constant and

$$\frac{M\Lambda L_2{E}_{\mu }(M{L}_2{b}^{1-\varrho (1-\mu )})b^{1-\varrho (1-\mu )}}{\Gamma (\mu )\sqrt{2\mu -1}}<1.$$

Then, the control system $(4)$ is approximately controllable on K.

Remark 5.

By Theorem 3.3 of [19], if the range of B is dense in $L^2(K,X)$, $(H_G)$ is fulfilled.

Remark 6.

In view of Remark 1, Theorem 3 is a natural improvement of [19,20,21].

Remark 7.

In Theorem 3, the approximate controllability of $(4)$ is proved without the compactness of $T(\eta )(\eta \ge 0)$. Hence, Theorem 3 improves some existing research works.

2. Preliminaries

Let $M_{\mu }(\theta )$ be Mainardi’s Wright type function given by

$$M_{\mu }(\theta )=\sum _{n=1}^{\infty }\frac{{(-\theta )}^{n-1}}{(n-1)!\Gamma (1-n\mu )},\mu \in (0,1).$$

By direct calculations, we have

$$M_{\mu }(\theta )\ge 0,\forall \theta >0,$$

$${\int }_0^{\infty }{\theta }^{\alpha }{M}_{\mu }(\theta )d\theta =\frac{\Gamma (1+\alpha )}{\Gamma (1+\mu \alpha )},\forall -1<\alpha <\infty .$$

Let ${\left\{S_{\varrho ,\mu }(\eta )\right\}}_{\eta \ge 0}$ and ${\left\{P_{\mu }(\eta )\right\}}_{\eta \ge 0}$ be two operator families given by

$$S_{\varrho ,\mu }(\eta )=I_{0^{+}}^{\varrho (1-\mu )}({\eta }^{\mu -1}{P}_{\mu }(\eta )),$$

(6)

$$P_{\mu }(\eta )={\int }_0^{\infty }\mu \theta M_{\mu }(\theta )T({\eta }^{\mu }\theta )d\theta .$$

(7)

Lemma 1

([4]). For $\eta \ge 0$, $S_{\varrho ,\mu }(\eta )$, $P_{\mu }(\eta )$ are linear operators and

$$\parallel S_{\varrho ,\mu }(\eta )y\parallel \le \frac{M{\eta }^{\gamma -1}}{\Gamma (\gamma )}\parallel y\parallel ,\eta \in K^{\prime },y\in X,$$

$$\parallel P_{\mu }(\eta )y\parallel \le \frac{M}{\Gamma (\mu )}\parallel y\parallel ,\eta \in K,y\in X.$$

Lemma 2.

For any $\eta \ge 0$, $P_{\mu }(\eta )$ is strongly continuous, that is, for any $0\le h_1<h_2\le b$,

$$\parallel P_{\mu }(h_2)y-P_{\mu }(h_1)y\parallel \to 0(h_2-h_1\to 0),\forall y\in X.$$

Proof 

For any $y\in X$ and $0\le h_1<h_2\le b$, we have

$$\parallel P_{\mu }(h_2)y-P_{\mu }(h_1)y\parallel \le {\int }_0^{\infty }\mu \theta M_{\mu }(\theta )\parallel T(h_2^{\mu }\theta )y-T(h_1^{\mu }\theta )y\parallel d\theta .$$

Since

$${\int }_0^{\infty }\mu \theta M_{\mu }(\theta )\parallel T(h_2^{\mu }\theta )y-T(h_1^{\mu }\theta )y\parallel d\theta \le \frac{2M}{\Gamma (\mu )}\parallel y\parallel ,$$

and $T(\eta )(\eta \ge 0)$ is strongly continuous, it follows that

$$\parallel P_{\mu }(h_2)y-P_{\mu }(h_1)y\parallel \to 0(h_2-h_1\to 0),$$

which is the desired conclusion. □

Lemma 3

([4]). Let $(H_A)$ hold. Then, for $\eta >0$, $P_{\mu }(\eta )$ is continuous in the uniform operator topology.

Lemma 4.

Let $(H_A^{\prime })$ hold. Then, we deduce that, for any $y\in X$,

$$P_{\mu }(\eta )y\in D(A),\forall \eta >0,$$

$$P_{\mu }(\eta )P_{\mu }(s)y=P_{\mu }(s)P_{\mu }(\eta )y,\forall \eta ,s\ge 0,$$

$$\frac{d{P}_{\mu }^2(\eta )}{d\eta }y=2P_{\mu }(\eta )\frac{d{P}_{\mu }(\eta )x}{d\eta }y,\forall \eta >0.$$

Proof 

According to $(H_A^{\prime })$ and the definition of $P_{\mu }(\eta )$ for $\eta \ge 0$, the conclusions are easily obtained by a direct calculation. □

For the mild solution of (4), according to Lemma 2.11 and Lemma 2.12 of [4], we adopt the following concept.

Definition 6

([4]). For any $u\in L^2(K,U)$, if $z\in C_{1-\gamma }(K,X)$ satisfies the integral equation

$$\left\{\begin{array}{c}z(\eta )=S_{\varrho ,\mu }(\eta )z_0+{\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}{P}_{\mu }(\eta -\tau )[\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau ,\eta \in K^{\prime },\hfill \\ I_{0^{+}}^{(1-\gamma )}{z(\eta )|}_{\eta =0}=z_0,\hfill \end{array}\right.$$

We call it the mild solution of $(4)$.

The following Gronwall inequality (see [23]) is needed in the following.

Lemma 5

([23]). Let $\beta >0$, $a(\theta )$ a nondecreasing and nonnegative function which is locally integrable on $[0,b)$ for some $b\le +\infty$, and $g(\theta )$, defined on $[0,b)$, a nondecreasing and nonnegative continuous function satisfying $g(\theta )\le N^{*}$(constant). Assume that $u(\theta )$ is locally integrable and nonnegative on $[0,b)$ satisfying

$$u(\eta )\le a(\eta )+g(\eta ){\int }_0^{\eta }{(\eta -\tau )}^{\beta -1}u(\tau )ds\tau ,\forall \eta \in [0,b).$$

Then,

$$u(\eta )\le a(\eta )E_{\beta }(g(\eta )\Gamma (\beta ){\eta }^{\beta }).$$

Lemma 6.

Let Φ be a mapping from a Banach space X to itself. If $\exists n\in \mathbb{Z}$ such that ${\Phi }^n$ is a contraction mapping, Φ possesses a unique fixed point in X.

3. Proof of the Main Results

Proof of Theorem 1.

We introduce $\Phi$ in $C_{1-\gamma }(K,X)$ by

$$(\Phi z)(\eta )=S_{\varrho ,\mu }(\eta )z_0+{\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}{P}_{\mu }(\eta -\tau )[\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau ,\eta \in K^{\prime }.$$

(8)

We first show that $\Phi$ maps $C_{1-\gamma }(K,X)$ into itself. For each $z\in C_{1-\gamma }(K,X)$, denote $y(·):={·}^{1-\gamma }z(·)\in C(K,X)$. Define $\Psi$ in $C(K,X)$ as

$$(\Psi y)(\eta )={\eta }^{1-\gamma }(\Phi z)(\eta ),\eta \in K^{\prime }.$$

(9)

Then, by (8) and (9), we have

$$\underset{\eta \to 0^{+}}{lim}(\Psi y)(\eta )=\underset{\eta \to 0^{+}}{lim}{\eta }^{1-\gamma }{S}_{\varrho ,\mu }(\eta )z_0=\frac{z_0}{\Gamma (\gamma )}.$$

Hence, we can define $(\Psi y)(0)=\frac{z_0}{\Gamma (\gamma )}$. To prove $\Phi$ maps $C_{1-\gamma }(K,X)$ into itself, we just prove that $\Psi$ maps $C(K,X)$ into itself.

Step I. $\underset{\eta \in K}{sup}\parallel (\Psi y)(\eta )\parallel <+\infty$.

For $z\in C_{1-\gamma }(K,X)$, by Hölder’s inequality and $(H_{f1})$, we get

$$\begin{array}{ccc}\hfill \parallel (\Psi y)(\eta )\parallel & \le & \parallel {\eta }^{1-\gamma }{S}_{\varrho ,\mu }(\eta )z_0\parallel \hfill \\ & & +{\eta }^{1-\gamma }\parallel {\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}{P}_{\mu }(\eta -\tau )[\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau \parallel \hfill \\ & \le & \frac{M}{\Gamma (\gamma )}\parallel z_0\parallel +\frac{M{\eta }^{1-\gamma }}{\Gamma (\mu )}{\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}[\varphi (\eta )+\kappa {\tau }^{1-\gamma }\parallel z(\tau )\parallel ]d\tau \hfill \\ & & +\frac{M{\eta }^{1-\gamma }}{\Gamma (\mu )}{\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}\parallel Bu(\tau )\parallel d\tau \hfill \\ & \le & \frac{M}{\Gamma (\gamma )}\parallel z_0\parallel +\frac{M{b}^{\frac{1}{2}-\varrho (1-\mu )}}{\Gamma (\mu )\sqrt{2\mu -1}}{(\parallel \varphi \parallel }_{L^2}+{\parallel Bu\parallel }_{L^2})\hfill \\ & & +\frac{M\kappa b^{1-\varrho (1-\mu )}}{\Gamma (\mu +1)}{\parallel z\parallel }_{C_{1-\gamma }}\hfill \\ & <& +\infty ,\hfill \end{array}$$

which implies $\underset{\eta \in K}{sup}\parallel (\Psi y)(\eta )\parallel <+\infty$.

Step II. $\eta \mapsto (\Psi y)(\eta )$ is continuous on K.

If ${\eta }_1\equiv 0$ and $0<{\eta }_2\le b$, we have

$$\begin{array}{ccc}& & \parallel (\Psi y)({\eta }_2)-(\Psi y)(0)\parallel \hfill \\ & \le & \parallel {\eta }_2^{1-\gamma }{S}_{\varrho ,\mu }({\eta }_2)z_0-\frac{z_0}{\Gamma (\gamma )}\parallel \hfill \\ & & +\parallel {\eta }_2^{1-\gamma }{\int }_0^{{\eta }_2}{({\eta }_2-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )[\mathcal{F}(\tau ,z(\tau )+Bu(\tau ))]d\tau \parallel \hfill \\ & \le & \parallel {\eta }_2^{1-\gamma }{S}_{\varrho ,\mu }({\eta }_2)z_0-\frac{z_0}{\Gamma (\gamma )}\parallel \hfill \\ & & +\frac{M{\eta }_2^{\frac{1}{2}-\varrho (1-\mu )}}{\Gamma (\mu )\sqrt{2\mu -1}}{(\parallel \varphi \parallel }_{L^2}+{\parallel Bu\parallel }_{L^2})+\frac{M\kappa {\eta }_2^{1-\varrho (1-\mu )}}{\Gamma (\mu +1)}{\parallel x\parallel }_{C_{1-\gamma }}\hfill \\ & \to & 0({\eta }_2\to 0).\hfill \end{array}$$

If $0<{\eta }_1<{\eta }_2\le b$, we have

$$\begin{array}{ccc}& & \parallel (\Psi y)({\eta }_2)-(\Psi y)({\eta }_1)\parallel \hfill \\ & \le & \parallel ({\eta }_2^{1-\gamma }-{\eta }_1^{1-\gamma })S_{\varrho ,\mu }({\eta }_2)z_0\parallel \hfill \\ & & +\parallel {\eta }_1^{1-\gamma }[S_{\varrho ,\mu }({\eta }_2)-S_{\varrho ,\mu }({\eta }_1)]z_0\parallel \hfill \\ & & +\parallel ({\eta }_2^{1-\gamma }-{\eta }_1^{1-\gamma }){\int }_0^{{\eta }_2}{({\eta }_2-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )[\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau \parallel \hfill \\ & & +\parallel {\eta }_1^{1-\gamma }{\int }_{{\eta }_1}^{{\eta }_2}{({\eta }_2-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )[\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau \parallel \hfill \\ & & +\parallel {\eta }_1^{1-\gamma }{\int }_0^{{\eta }_1}[{({\eta }_2-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )-{({\eta }_1-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )][\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau \parallel \hfill \\ & & +\parallel {\eta }_1^{1-\gamma }{\int }_0^{{\eta }_1}[{({\eta }_1-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )-{({\eta }_1-\tau )}^{\mu -1}{P}_{\mu }({\eta }_1-\tau )][f(\tau ,x(\tau ))+Bu(\tau )]d\tau \parallel \hfill \\ & :=& \sum _{i=1}^6{I}_i.\hfill \end{array}$$

By $(H_{f1})$ and Lemma 1, we easily get

$$\begin{array}{ccc}\hfill I_1& =& \parallel ({\eta }_2^{1-\gamma }-{\eta }_1^{1-\gamma })S_{\varrho ,\mu }({\eta }_2)z_0\parallel \hfill \\ & \le & \frac{M{\eta }_2^{\gamma -1}}{\Gamma (\gamma )}\parallel ({\eta }_2^{1-\gamma }-{\eta }_1^{1-\gamma })z_0\parallel \hfill \\ & \to & 0({\eta }_2-{\eta }_1\to 0),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_3& =& \parallel ({\eta }_2^{1-\gamma }-{\eta }_1^{1-\gamma }){\int }_0^{{\eta }_2}{({\eta }_2-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )[\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau \parallel \hfill \\ & \le & |{\eta }_2^{1-\gamma }-{\eta }_1^{1-\gamma }|\left[\frac{M{\eta }_2^{\mu -\frac{1}{2}}}{\Gamma (\mu )\sqrt{2\mu -1}}{(\parallel \varphi \parallel }_{L^2}+{\parallel Bu\parallel }_{L^2})+\frac{M\kappa {\eta }_2^{\mu }}{\Gamma (\mu +1)}{\parallel z\parallel }_{C_{1-\gamma }}\right]\hfill \\ & \to & 0({\eta }_2-{\eta }_1\to 0),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_4& =& \parallel {\eta }_1^{1-\gamma }{\int }_{{\eta }_1}^{{\eta }_2}{({\eta }_2-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )[\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau \parallel \hfill \\ & \le & \frac{M{\eta }_1^{1-\gamma }}{\Gamma (\mu )}\left[\frac{{({\eta }_2-{\eta }_1)}^{\mu -\frac{1}{2}}}{\sqrt{2\mu -1}}{(\parallel \varphi \parallel }_{L^2}+{\parallel Bu\parallel }_{L^2})+\frac{\kappa {({\eta }_2-{\eta }_1)}^{\mu }}{\mu }{\parallel z\parallel }_{C_{1-\gamma }}\right]\hfill \\ & \to & 0({\eta }_2-{\eta }_1\to 0),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_5& =& \parallel {\eta }_1^{1-\gamma }{\int }_0^{{\eta }_1}[{({\eta }_2-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )-{({\eta }_1-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )][\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau \parallel \hfill \\ & \le & \frac{M{\eta }_1^{1-\gamma }}{\Gamma (\mu )}{\int }_0^{{\eta }_1}|{({\eta }_2-\tau )}^{\mu -1}-{({\eta }_1-\tau )}^{\mu -1}|(\parallel \mathcal{F}(\tau ,z(\tau ))\parallel +\parallel Bu(\tau )\parallel )d\tau \hfill \\ & \le & \frac{M{\eta }_1^{1-\gamma }}{\Gamma (\mu )}{\int }_0^{{\eta }_1}|{({\eta }_2-\tau )}^{\mu -1}-{({\eta }_1-\tau )}^{\mu -1}|(\varphi (\tau )+\kappa {\tau }^{1-\gamma }\parallel z(\tau )\parallel +\parallel Bu(\tau )\parallel )d\tau \hfill \\ & \to & 0({\eta }_2-{\eta }_1\to 0).\hfill \end{array}$$

By Lemma 3, we have

$$I_2=\parallel {\eta }_1^{1-\gamma }[S_{\varrho ,\mu }({\eta }_2)-S_{\varrho ,\mu }({\eta }_1)]z_0\parallel \to 0({\eta }_2-{\eta }_1\to 0).$$

For any $ϵ\in (0,t_1)$, by Lemma 3, we have

$$\begin{array}{ccc}\hfill I_6& =& \parallel {\eta }_1^{1-\gamma }{\int }_0^{{\eta }_1}[{({\eta }_1-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )-{({\eta }_1-\tau )}^{\mu -1}{P}_{\mu }({\eta }_1-\tau )][\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau \parallel \hfill \\ & \le & \parallel {\eta }_1^{1-\gamma }{\int }_0^{{\eta }_1-ϵ}[{({\eta }_1-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )-{({\eta }_1-\tau )}^{\mu -1}{P}_{\mu }({\eta }_1-\tau )][\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau \parallel \hfill \\ & & +\parallel {\eta }_1^{1-\gamma }{\int }_{{\eta }_1-ϵ}^{{\eta }_1}[{({\eta }_1-\tau )}^{\mu -1}{P}_{\mu }({\eta }_2-\tau )-{({\eta }_1-\tau )}^{\mu -1}{P}_{\mu }({\eta }_1-\tau )][\mathcal{F}(\tau ,z(\tau ))+Bu(\tau )]d\tau \parallel \hfill \\ & \le & {\eta }_1^{1-\gamma }\underset{\tau \in [0,{\eta }_1-ϵ]}{sup}\parallel P_{\mu }({\eta }_2-\tau )-P_{\mu }({\eta }_1-\tau )\parallel {\int }_0^{{\eta }_1-ϵ}{({\eta }_1-\tau )}^{\mu -1}\parallel \mathcal{F}(\tau ,z(\tau ))+Bu(\tau )\parallel d\tau \hfill \\ & & +\frac{2M{\eta }_1^{1-\gamma }}{\Gamma (\mu )}{\int }_{{\eta }_1-ϵ}^{{\eta }_1}{({\eta }_1-\tau )}^{\mu -1}\parallel \mathcal{F}(\tau ,z(\tau ))+Bu(\tau )\parallel d\tau \hfill \\ & \le & M^{*}\underset{\tau \in [0,{\eta }_1-ϵ]}{sup}\parallel P_{\mu }({\eta }_2-\tau )-P_{\mu }({\eta }_1-\tau )\parallel \hfill \\ & & +\frac{2M{\eta }_1^{1-\gamma }}{\Gamma (\mu )}\left[\frac{{ϵ}^{\mu -\frac{1}{2}}}{\sqrt{2\mu -1}}{(\parallel \varphi \parallel }_{L^2}+{\parallel Bu\parallel }_{L^2})+\frac{\kappa {ϵ}^{\mu }}{\mu }{\parallel z\parallel }_{C_{1-\gamma }}\right]\hfill \\ & \to & 0\hfill \end{array}$$

as ${\eta }_2-{\eta }_1\to 0$ and $ϵ\to 0$, where $M^{*}=\left[\frac{{\eta }_1^{1-\gamma }{({\eta }_1^{2\mu -1}-{ϵ}^{2\mu -1})}^{\frac{1}{2}}}{\sqrt{2\mu -1}}{(\parallel \varphi \parallel }_{L^2}+{\parallel Bu\parallel }_{L^2})+\frac{\kappa {\eta }_1^{1-\gamma }({\eta }_1^{\mu }-{ϵ}^{\mu })}{\mu }{\parallel z\parallel }_{C_{1-\gamma }}\right]$.

Therefore, $\eta \mapsto (\Psi y)(\eta )$ is continuous on K. Steps I–II yield $\Psi :C(K,X)\to C(K,X)$ and then $\Phi$ maps $C_{1-\gamma }(K,X)$ into itself.

Secondly, we claim that $\exists n\in \mathbb{Z}$ such that ${\Phi }^n$ is a contraction mapping in $C_{1-\gamma }(K,X)$. In fact, for $z_1,z_2\in C_{1-\gamma }(K,X)$, we have

$$\begin{array}{ccc}& & {\eta }^{1-\gamma }\parallel (\Phi z_1)(\eta )-(\Phi z_2)(\eta )\parallel \hfill \\ & =& {\eta }^{1-\gamma }\parallel {\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}{P}_{\mu }(\eta -\tau )[\mathcal{F}(\tau ,z_1(\tau ))-\mathcal{F}(\tau ,z_2(\tau ))]d\tau \parallel \hfill \\ & \le & \frac{M{\eta }^{1-\gamma }}{\Gamma (\mu )}{\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}\parallel \mathcal{F}(\tau ,z_1(\tau ))-\mathcal{F}(\tau ,z_2(\tau ))\parallel d\tau \hfill \\ & \le & \frac{M{L}_1{\eta }^{1-\gamma }}{\Gamma (\mu )}{\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}\parallel z_1(\tau )-z_2(\tau )\parallel d\tau \hfill \\ & \le & \frac{M{L}_1{\eta }^{1-\gamma }}{\Gamma (\mu )}{\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}{\tau }^{\gamma -1}{\tau }^{1-\gamma }\parallel z_1(\tau )-z_2(\tau )\parallel d\tau \hfill \\ & =& \frac{\Gamma (\gamma )M{L}_1{\eta }^{\mu }}{\Gamma (\mu +\gamma )}{\parallel z_1-z_2\parallel }_{C_{1-\gamma }}.\hfill \end{array}$$

By inductions, we obtain that

$$\parallel {\Phi }^n{z}_1-{\Phi }^n{z}_2{\parallel }_{C_{1-\gamma }}\le \frac{\Gamma (\gamma ){(M{L}_1{b}^{\mu })}^n}{\Gamma (n\mu +\gamma )}{\parallel z_1-z_2\parallel }_{C_{1-\gamma }}.$$

Since the Mittag-Leffer series $E_{\mu ,\gamma }(M{L}_1{b}^{\mu })={\displaystyle \sum _{n=0}^{\infty }}\frac{{(M{L}_1{b}^{\mu })}^n}{\Gamma (n\mu +\gamma )}$ has uniform convergence, it follows that

$$\frac{\Gamma (\gamma ){(M{L}_1{b}^{\mu })}^n}{\Gamma (n\mu +\gamma )}<1.$$

Therefore, all conditions of Lemma 6 hold. By Lemma 6, the Cauchy problem (4) possesses a unique mild solution in $C_{1-\gamma }(K,X)$. □

Proof of Theorem 2.

Since $(H_{f2}^{\prime })\Rightarrow (H_{f2})$, by Theorem 1, for every $u\in L^2(K,U)$, the Cauchy problem (4) possesses a mild solution $z^u\in C_{1-\gamma }(K,X)$ associated with the control function u given by

$$z^u(\eta )=S_{\varrho ,\mu }(\eta )z_0+{\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}{P}_{\mu }(\eta -\tau )[\mathcal{F}(\tau ,z^u(\tau ))+Bu(\tau )]d\tau ,\eta \in K^{\prime }.$$

By $(H_{f1})$ and Lemma 1, we get

$$\begin{array}{ccc}\hfill {\eta }^{1-\gamma }\parallel z^u(\eta )\parallel & \le & \frac{M}{\Gamma (\gamma )}\parallel z_0\parallel +\frac{M{\eta }^{1-\gamma }}{\Gamma (\mu )}{\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}\parallel \mathcal{F}(\tau ,z^u(\tau ))+Bu(\tau )\parallel d\tau \hfill \\ & \le & \frac{M}{\Gamma (\gamma )}\parallel z_0\parallel +\frac{M{\eta }^{1-\gamma }}{\Gamma (\mu )}{\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}[\varphi (\tau )+\kappa {\tau }^{1-\gamma }\parallel z^u(\tau )\parallel +\parallel Bu(\tau )\parallel ]d\tau \hfill \\ & \le & \frac{M}{\Gamma (\gamma )}\parallel z_0\parallel +\frac{M{b}^{\frac{1}{2}-\varrho (1-\mu )}}{\Gamma (\mu )\sqrt{2\mu -1}}{(\parallel \varphi \parallel }_{L^2}+{\parallel Bu\parallel }_{L^2})\hfill \\ & & +\frac{M\kappa {\eta }^{1-\gamma }}{\Gamma (\mu )}{\int }_0^{\eta }{(\eta -\tau )}^{\mu -1}{\tau }^{1-\gamma }\parallel z^u(\tau )\parallel d\tau .\hfill \end{array}$$

Then, Lemma 5 yields

$${\eta }^{1-\gamma }\parallel z^u(\eta )\parallel \le N_1{E}_{\mu }(M\kappa b^{1-\varrho (1-\mu )}).$$

This fact implies

$$\parallel z^u{\parallel }_{C_{1-\gamma }}\le {\mathcal{K}}_1.$$

Similarly, for any $v_1,v_2\in L^2(K,U)$, by applying Lemma 5 again and using $(H_{f2}^{\prime })$, we can get

$$\parallel z^{v_2}-z^{v_1}{\parallel }_{C_{1-\gamma }}\le {\mathcal{K}}_2{\parallel B{v}_2-B{v}_1\parallel }_{L^2}.$$

Then, the proof is finished. □

Proof of Theorem 3.

Since $\overline{D(A)}=X$, in order to prove $\overline{{\mathcal{D}}_b(\mathcal{F})}=X$, we show that $D(A)\subset {\mathcal{D}}_b(\mathcal{F})$, that is, for $\forall ϵ>0$ and $\xi \in D(A)$, there is $u_{ϵ}\in L^2(K,U)$ with

$$\parallel \xi -S_{\varrho ,\mu }(b)z_0-G\mathcal{R}(z^{u_{ϵ}})-GB{u}_{ϵ}\parallel <ϵ.$$

(10)

For any $z_0\in X$, by Lemma 4, $S_{\varrho ,\mu }(b)z_0\in D(A)$ and $P_{\mu }(b)z_0\in D(A)$. Then, for $\forall \xi \in D(A)$, we can find $\psi \in L^2(K,X)$ such that

$$G\psi =\xi -S_{\varrho ,\mu }(b)z_0.$$

For instance, we choose

$$\psi (\eta )=\frac{{\Gamma }^2(\mu ){(b-\eta )}^{1-\mu }}{b}[P_{\mu }(b-\eta )-2\eta \frac{d{P}_{\mu }(b-\eta )}{d\eta }](\xi -S_{\varrho ,\mu }(b)z_0).$$

In fact, by (5) and Lemma 4, we have

$$\begin{array}{ccc}\hfill G\psi & =& \frac{{\Gamma }^2(\mu )}{b}{\int }_0^b[P_{\mu }^2(b-\tau )-2\tau P_{\mu }(b-\tau )\frac{d{P}_{\mu }(b-\tau )}{d\tau }]d\tau (\xi -S_{\varrho ,\mu }(b)z_0)\hfill \\ & =& \frac{{\Gamma }^2(\mu )}{b}[\tau P_{\mu }^2(b-\tau )]{|}_0^b(\xi -S_{\varrho ,\mu }(b)z_0)\hfill \\ & =& \xi -S_{\nu ,\mu }(b)z_0.\hfill \end{array}$$

Hence, for any $ϵ>0$ and given $u_1\in L^2(K,U)$, by the definition of $\mathcal{R}$ and the assumption $(H_G)$, there is a $u_2\in L^2(K,U)$ satisfying

$$\parallel \xi -S_{\varrho ,\mu }(b)z_0-G\mathcal{R}(z^{u_1})-GB{u}_2\parallel <\frac{ϵ}{2^2}.$$

By employing $(H_G)$ again, there is a $w_2\in L^2(K,U)$ satisfying

$$\parallel G(\mathcal{R}(z^{u_2})-\mathcal{R}(z^{u_1}))-GB{w}_2\parallel <\frac{ϵ}{2^3},$$

and

$$\begin{array}{ccc}\hfill \parallel B{w}_2{\parallel }_{L^2}& <& \Lambda \parallel \mathcal{R}(z^{u_2})(·)-\mathcal{R}(z^{u_1})(·){\parallel }_{L^2}\hfill \\ & =& \Lambda ({\int }_0^b\parallel \mathcal{F}(\tau ,z^{u_2}(\tau ))-\mathcal{F}(\tau ,z^{u_1}(\tau )){{\parallel }^{2}d\tau )}^{\frac{1}{2}}\hfill \\ & \le & \Lambda L_2({\int }_0^b({\tau }^{1-\gamma }\parallel z^{u_2}(\tau )-z^{u_1}{(\tau )\parallel )}^2{d\tau )}^{\frac{1}{2}}\hfill \\ & \le & \Lambda L_2{b}^{\frac{1}{2}}{\parallel z^{u_2}-z^{u_1}\parallel }_{C_{1-\gamma }}.\hfill \end{array}$$

This fact together with Theorem 2 yields

$$\parallel B{w}_2{\parallel }_{L^2}<\Lambda L_2{b}^{\frac{1}{2}}{\mathcal{K}}_2{\parallel B{u}_2-B{u}_1\parallel }_{L^2}.$$

Now, we choose

$$u_3(\eta )=u_2(\eta )-w_2(\eta ),\eta \in K.$$

Then, $u_3\in L^2(K,U)$ and

$$\begin{array}{ccc}& & \parallel \xi -S_{\varrho ,\mu }(b)z_0-G\mathcal{R}(z^{u_2})-GB{u}_3\parallel \hfill \\ & \le & \parallel \xi -S_{\varrho ,\mu }(b)z_0-G\mathcal{R}(z^{u_1})-GB{u}_2\parallel \hfill \\ & & +\parallel GB{w}_2-G(\mathcal{R}(z^{u_2})-\mathcal{R}(z^{u_1}))\parallel \hfill \\ & <& (\frac{1}{2^2}+\frac{1}{2^3})ϵ,\hfill \end{array}$$

and

$$\parallel B{u}_3-B{u}_2{\parallel }_{L^2}<\Lambda L_2{b}^{\frac{1}{2}}{\mathcal{K}}_2{\parallel B{u}_2-B{u}_1\parallel }_{L^2}.$$

By inductions, we can get a sequence $\left\{u_n\right\}\subset L^2(K,U)$ satisfying

$$\parallel \xi -S_{\varrho ,\mu }(b)z_0-G\mathcal{R}(z^{u_n})-GB{u}_{n+1}\parallel <(\frac{1}{2^2}+\cdots +\frac{1}{2^{n+1}})ϵ$$

and

$$\parallel B{u}_{n+1}-B{u}_n{\parallel }_{L^2}<\Lambda L_2{b}^{\frac{1}{2}}{\mathcal{K}}_2{\parallel B{u}_n-B{u}_{n-1}\parallel }_{L^2}.$$

Thus, $\left\{B{u}_n\right\}$ is a Cauchy sequence in $L^2(K,X)$. For $\forall ϵ>0$, there is a $N>0$ such that

$$\parallel GB{u}_{N+1}-GB{u}_N\parallel <\frac{ϵ}{2}.$$

Therefore,

$$\begin{array}{ccc}& & \parallel \xi -S_{\varrho ,\mu }(b)z_0-G\mathcal{R}(z^{u_N})-GB{u}_N\parallel \hfill \\ & \le & \parallel \xi -S_{\varrho ,\mu }(b)z_0-G\mathcal{R}(z^{u_N})-GB{u}_{N+1}\parallel +\parallel GB{u}_{N+1}-GB{u}_N\parallel \hfill \\ & <& (\frac{1}{2^2}+\cdots +\frac{1}{2^{N+1}})ϵ+\frac{ϵ}{2}\hfill \\ & <& ϵ.\hfill \end{array}$$

This fact implies (10) and hence the approximate controllability of (4) on K. □

4. An Example

Consider the Hilfer fractional partial differential equation

$$\left\{\begin{array}{c}{D}_{0^{+}}^{\frac{2}{3},\frac{2}{3}}z(\eta ,\omega )=\frac{{\partial }^2}{\partial {\omega }^2}z(\eta ,\omega )+\mathcal{H}(\eta ,z(\eta ,\omega ))+Bu(\eta ,\omega ),\forall \eta \in (0,1],\omega \in [0,\pi ],\hfill \\ z(\eta ,0)=z(\eta ,\pi )=0,\eta \in [0,1],\hfill \\ I_{0^{+}}^{\frac{1}{9}}{z(\eta ,\omega )|}_{\eta =0}=z_0(\omega ),\omega \in [0,\pi ],\hfill \end{array}\right.$$

(11)

where $u(·,·)\in L^2([0,1]\times [0,\pi ],\mathbb{R})$ is the control.

Let $X=L^2([0,\pi ],\mathbb{R})=U$. Take $A=\frac{{\partial }^2}{\partial z^2}$ and

$$D(A)=\{v\in X:v(0)=v(\pi )=0,v^{\prime },v^{″}\in X\}.$$

Denote $e_i(\omega )=\sqrt{\frac{2}{\pi }}sini\omega ,i=1,2,\dots$, which is an orthonormal basis of X. Thus, A can be written as

$$Av=-\sum _{i=1}^{\infty }{i}^2\langle v,e_i\rangle e_i,v\in D(A).$$

Then, A generates a compact analytic semigroup $T(\eta )(\eta \ge 0)$ in X given by

$$T(\eta )v=\sum _{i=1}^{\infty }{e}^{-i^2\eta }\langle v,e_i\rangle e_i,v\in X.$$

(12)

By Remark 2, $(H_A)$ and $(H_A^{\prime })$ are satisfied. Obviously, formula (12) implies

$$\parallel T(\eta )\parallel \le 1,\forall \eta \ge 0.$$

For every $u\in L^2(K,U)$ given by $u(\eta )={\displaystyle \sum _{i=1}^{\infty }}{u}_i(\eta )e_i$, where $u_i(\eta )=\langle u(\eta ),e_i\rangle$, we define the operator B by

$$Bu(\eta )=\sum _{i=1}^{\infty }{\tilde{u}}_i(\eta )e_i,$$

where ${\tilde{u}}_i(\eta )(i=1,2,\dots )$ are given as

$${\tilde{u}}_i(\eta )=\left\{\begin{array}{c}0,0\le \eta <1-\frac{1}{i^2},\hfill \\ u_i(\eta ),1-\frac{1}{i^2}\le \eta \le 1.\hfill \end{array}\right.$$

Then, $\parallel Bu(·)\parallel \le \parallel u(·)\parallel$ and $M_B=1$.

Theorem 4.

Let $\mathcal{H}:[0,1]\times L^2([0,\pi ],\mathbb{R})\to L^2([0,\pi ],\mathbb{R})$ satisfy the following conditions:

$(P_1)$ $\exists a\ge 0$, $b>0$ satisfying

$$\parallel \mathcal{H}(\eta ,\wp )\parallel \le a+b{\eta }^{\frac{1}{9}}\parallel \wp \parallel ,\forall \eta \in [0,1],\wp \in L^2([0,\pi ],\mathbb{R}).$$

$(P_2)$ $\exists c>0$ satisfying

$$\parallel \mathcal{H}(\eta ,{\wp }_1)-\mathcal{H}(\eta ,{\wp }_2)\parallel \le c{\eta }^{\frac{1}{9}}\parallel {\wp }_1-{\wp }_2\parallel ,\forall \eta \in [0,1],{\wp }_1,{\wp }_2\in L^2([0,\pi ],\mathbb{R}).$$

Then, the problem $(11)$ possesses a unique mild solution on $[0,1]$.

Proof. 

By $(P_1)$ and $(P_2)$, we know that $(H_{f1})$ and $(H_{f2}^{\prime })$ hold with $\varphi (t)\equiv a,\kappa =b$ and $L_2=c$. Hence, the conclusion of Theorem 4 is proved by Theorem 2. □

Next, we check that the assumption $(H_G)$ is satisfied.

By the above definition of B, the corresponding linear problem of (11) can be expressed by

$$\left\{\begin{array}{c}{D}_{0^{+}}^{\frac{2}{3},\frac{2}{3}}{y}_i(\eta )+i^2{y}_i(\eta )={\tilde{u}}_i(\eta ),1-\frac{1}{i^2}<\eta <1,\hfill \\ I_{0^{+}}^{\frac{1}{9}}{y}_i{(\eta )|}_{\eta =1-\frac{1}{i^2}}=0.\hfill \end{array}\right.$$

For every $h\in L^2([0,1],X)$, let

$$g={\int }_0^1{(1-\vartheta )}^{-\frac{1}{3}}{P}_{\frac{2}{3}}(1-\vartheta )h(\vartheta )d\vartheta =\sum _{i=1}^{\infty }{g}_i{e}_i,$$

where $h={\displaystyle \sum _{i=1}^{\infty }}{h}_i{e}_i$ and

$$h_i=\langle h,e_i\rangle ,g_i=\langle g,e_i\rangle .$$

Then, we can choose $\widehat{u}(t)$ as

$$\widehat{u}(\eta )=\frac{2i^2}{1-e^{-2}}{h}_i{e}^{-2^2(1-\eta )},1-\frac{1}{i^2}\le \eta \le 1,$$

where

$$h_i={\int }_{1-\frac{1}{i^2}}^1{(1-\vartheta )}^{-\frac{1}{3}}{P}_{\frac{2}{3}}(1-\vartheta ){\widehat{u}}_i(\vartheta )d\vartheta .$$

Define

$$u(\eta )=\sum _{i=1}^{\infty }{u}_i(\eta )e_i,$$

where $u_i(\eta )(i=1,2,\dots )$ is given as

$$u_i(\eta )=\left\{\begin{array}{c}0,0\le \eta <1-\frac{1}{i^2},\hfill \\ {\widehat{u}}_i(\eta ),1-\frac{1}{i^2}\le \eta \le 1.\hfill \end{array}\right.$$

Thus, $u\in L^2([0,1],U)$ and

$${\int }_0^1{(1-\vartheta )}^{-\frac{1}{3}}{P}_{\frac{2}{3}}(1-\vartheta )Bu(\vartheta )d\vartheta ={\int }_0^1{(1-\vartheta )}^{-\frac{1}{3}}{P}_{\frac{2}{3}}(1-\vartheta )h(\vartheta )d\vartheta .$$

Furthermore, we have

$$\begin{array}{ccc}\hfill {\parallel Bu\parallel }_{L^2}^2& =& \sum _{i=1}^{\infty }{\int }_{1-\frac{1}{i^2}}^1{|{\widehat{u}}_i(\vartheta )|}^{2}d\vartheta \hfill \\ & =& \frac{1}{1-e^{-2}}\sum _{i=1}^{\infty }2i^2{h}_i^2\hfill \\ & =& \frac{1}{1-e^{-2}}\sum _{i=1}^{\infty }(1-e^{-2i^2}){\int }_0^1{|g_i(\vartheta )|}^{2}d\vartheta \hfill \\ & \le & \frac{1}{1-e^{-2}}{\parallel g\parallel }_{L^2}^2.\hfill \end{array}$$

Therefore, the assumption $(H_G)$ is fulfilled with $\Lambda =\frac{1}{1-e^{-2}}$ satisfying

$$\frac{3c{E}_{\frac{2}{3}}(c)}{\sqrt{3}\Gamma (\frac{2}{3})\frac{1}{1-e^{-2}}}<1.$$

Therefore, by Theorem 3, the following theorem is obvious.

Theorem 5.

Suppose that $(P_1)$ and $(P_2)$ are fulfilled. Then, problem $(11)$ is approximately controllable on $[0,1]$.

5. Conclusions

In the present work, the existence and uniqueness results of Hilfer fractional evolution equations were obtained by utilizing a generalized type of Banach’s fixed-point theorem (see Lemma 6)). The approximate controllability was also investigated by applying the sequence approach. The conclusions were proved without the compactness conditions on $T(\eta )(\eta \ge 0)$. The condition $(H_G)$ was presented to reduce the related conditions. Therefore, the obtained results greatly generalized previous research works.