On the approximate controllability for fractional evolution inclusions of Sobolev and Clarke subdifferential type

On the approximate controllability for fractional evolution inclusions of Sobolev and Clarke... Abstract This article is concerned with the approximate controllability for some fractional evolution inclusions of Sobolev and Clarke subdifferential type, which can be used to study fractional hemivariational inequalities. Using two characteristic solution operators and their fundamental properties via a fixed point theorem for multi-valued maps, we derive sufficient conditions for approximate controllability of linear and semi-linear controlled systems. Finally, two examples are given to illustrate our theory. 1. Introduction In this article, we study the following fractional evolution inclusion of Sobolev and Clarke subdifferential type: \begin{eqnarray}\label{e1} \left\{ \begin{array}{@{}l} ^CD^{\alpha}_{t}\big(Ex(t)\big) +Ax(t)\in Bu(t)+\partial F(t,x(t)), \quad t\in J=[0,b],\\ x(0)=x_{0}, \end{array} \right. \end{eqnarray} (1.1) where $${}^{C}D^{\alpha}_{t}$$ denotes the Caputo fractional derivative of order $$\alpha$$ and $$\frac{1}{2}<\alpha\leq1$$, $$A:D(A)\subseteq H\rightarrow H$$ on a separable Hilbert space $$H$$, $$E:D(E)\subset H\rightarrow H$$, and $$\partial F$$ denotes the generalized Clarke subdifferential (cf. Clarke (1983)) of a locally Lipschitz function $$F(t,\cdot): H\rightarrow R$$. The control function $$u$$ takes values in $$L^{2}(J,U)$$ and the admissible controls set $$U$$ is a Hilbert space. Finally $$B$$ is a bounded linear operator from $$U$$ into $$H$$. Fractional calculus and fractional dynamic equations (Miller & Ross, 1993; Podlubny, 1999; Kilbas et al., 2006; Lakshmikantham et al., 2009; Diethelm, 2010; Ghomanjani, 2017) arises naturally in phenomena in engineering, physics, science and controllability. For recent work on the existence of mild solutions, controllability and optimal control for some fractional evolution systems (see Hernández et al. (2010), Debbouche et al. (2012), Sakthivel et al. (2016), Wang et al. (2011) and Wang & Zhou (2011)). In Wang et al. (2011), two characteristic solution operators were used to study the existence of mild solutions for some systems governed by fractional equations with initial and non-local conditions. Sobolev-type equation appears in a variety of physical problems, for example the flow of fluid through fissured rocks, thermodynamics, and the propagation of long waves of small amplitude; see Lightbourne & Rankin (1983), Ponce (2014), Kerboua et al. (2014), Kerboua et al. (2013), Li et al. (2012), Benchaabane et al. (2017), Debbouche & Nieto (2014), Debbouche et al. (2015), Fečkan et al. (2013) and Wang et al. (2014). The existence and uniqueness of Hölder continuous solutions of an abstract Sobolev type differential equation was considered in Lightbourne & Rankin (1983) and two new linear operators were given in Ponce (2014). A non-local condition given in the stochastic term fractional derivative was presented in Kerboua et al. (2014), the theory of a propagation family was used to obtain the existence of mild solutions in Li et al. (2012), and existence and uniqueness of mild solutions and optimal multi-controls to Sobolev-type fractional non-local dynamical equations in Banach spaces was presented in Debbouche & Nieto (2014) and Debbouche et al. (2015). Two new characteristic solution operators and their properties was used to prove the controllability of fractional functional evolution equations of Sobolev type in Fečkan et al. (2013) and Wang et al. (2014), and this motivated our research. Hemivariational inequalities were introduced by Panagiotopoulos in 1980s to deal with mechanical problems with non-convex and non-smooth superpotentials; for recent references on existence we refer the reader to Carl et al. (2003), Sakthivel et al. (2013), Liu (2008), Migórski (2001), Migórski & Ochal (2009), Panagiotopoulo (1993) and Panagiotopoulos (1995) and for optimal control problems for hemivariational inequalities (see Liu & Li (2015), Rykaczewski (2012), Migórski & Ochal (2000) and Park et al. (2007)). Specially, in Li et al. (2016), the approximate controllability for fractional evolution hemivariational inequalities was shown. The results in Li et al. (2016) are only valid for fractional evolution inclusions without Sobolev type. It is natural to ask whether we can apply the characteristic solution operators and their properties to deal with approximate controllability of fractional evolution inclusion of Sobolev type in Fečkan et al. (2013) and Wang et al. (2014). To the best of our knowledge, the existence of mild solutions and the approximate controllability for fractional evolution inclusions of Sobolev type have not been investigated and this motivates us to study (1.1). Note $$x\in C(J,H)$$ is a solution of (1.1) if there exists a $$f\in L^2(J,H)$$ such that $$f(t)\in \partial F(t,x(t))$$ and \begin{eqnarray*}\label{e2} \left\{ \begin{array}{@{}l} ^CD^{\alpha}_{t}\big(Ex(t)\big)+Ax(t)= Bu(t)+f(t), a.e. t\in J,\\ x(0)=x_{0}, \end{array} \right. \end{eqnarray*} so we have \begin{eqnarray*}\label{e3} \left\{ \begin{array}{@{}l} \langle-^CD^{\alpha}_{t}\big(Ex(t)\big)-Ax(t)+Bu(t), v\rangle_H+ \langle f(t), v\rangle_H=0,\\ a.e. t\in J, \forall v\in H,\\ x(0)=x_{0}.\\ \end{array} \right. \end{eqnarray*} Since $$f(t)\in \partial F(t,x(t))$$ and $$\langle f(t), v\rangle_H\leq F^0(t,x(t),v)$$, we obtain \begin{eqnarray}\label{e1.2} \left\{ \begin{array}{@{}l} \langle -^CD^{\alpha}_{t}\big(Ex(t)\big)-Ax(t)+Bu(t), v\rangle_H+ F^0(t,x(t),v)\geq 0,\\ a.e. t\in J, \forall v\in H,\\ x(0)=x_{0}. \end{array} \right. \end{eqnarray} (1.2) Therefore to study (1.2), we consider (1.1). The main novelty of this article is that we develop ideas and techniques to establish an effective framework to present approximate controllability results for fractional evolution inclusion of Sobolev and Clarke subdifferential type, which generalizes the known results in Li et al. (2016). We present sufficient conditions to guarantee approximate controllability of linear fractional control system of Sobolev type, which will be used to deal with semi-linear problem via constructing a control function involving the Gramian controllability operator. Concerning semi-linear problem, we transform it to a fixed point problem of a non-linear operator associated with a new constructed control function, which again shows the powerful role of applying a fixed point theorem to deal with non-linear functional analysis problems. In Section 2, we present some preliminaries and in Section 3 sufficient conditions are given for the existence of mild solutions of (1.1). In Section 4, we study the approximate controllability for a linear fractional control system and in Section 5, the approximate controllability of (1.1) is discussed. In Section 6, an example is given to illustrate our theory. 2. Preliminaries Let $$I$$ be a Banach space with norm $$\|\cdot\|_I$$. Now $$I^*$$ denotes its dual and $$\langle\cdot,\cdot\rangle_I$$ the duality pairing between $$I^*$$ and $$I$$. As usual $$C(J,I)$$ denotes the Banach space of all continuous functions from $$J=[0,b]$$ into $$I$$ with the norm $$\|x\|_{C(J,I)}=\textrm{sup}_{t\in J}\|x(t)\|_{I}$$. The operators $$A $$ and $$E$$ satisfy: $$(S_1)$$: $$A $$ and $$E$$ are linear operators, and $$A$$ is closed. $$(S_2)$$: $$D(E)\subset D(A)$$ and $$E$$ is bijective. $$(S_3)$$: The linear operator $$E^{-1}$$: $$H\rightarrow D(E)\subset H$$ is compact. From $$(S_3)$$, we know that $$E$$ is closed, because $$E^{-1}$$ is closed and injective, so then the inverse is also closed. From $$(S_1)-(S_3)$$ and the closed graph theorem, we see that the operator $$-AE^{-1}:H\rightarrow H$$ is linear and bounded, so $$-AE^{-1}$$ generates a $$C_0$$ semi-group $$T(t),t\geq 0$$, $$T(t):=e^{-AE^{-1}t}$$, and we suppose $$\textrm{sup}_{t\in[0,\infty)}\|T(t)\|\leq M<+\infty$$. We collect some definitions from fractional calculus; see Kilbas et al. (2006). Definition 2.1 For a given function $$f:[0,+\infty)\rightarrow R$$, the integral $$ I^{\alpha}_{t}f(t)=\frac{1}{{\it{\Gamma}}(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}f(s)ds, \quad t>0, \quad 0<\alpha<1,$$ is called Riemann–Liouville fractional integral of order $$\alpha$$, where $${\it{\Gamma}}$$ is the gamma function. Definition 2.2 For a function $$ f:[0,\infty)\rightarrow R $$, $$[0,\infty)$$, the expression $$^{RL}D^{\alpha}_{t}f(t)=\frac{1}{{\it{\Gamma}}(n-\alpha)}\bigg(\frac{d}{dt}\bigg)^{(n)}\int^{t}_{0}(t-s)^{n-\alpha-1}f(s)dt, $$ where $$n=[\alpha]+1, [\alpha]$$ denotes the integer part of number $$ \alpha$$, is called the Riemann-Liouville fractional derivative of order $$\alpha>0$$. Definition 2.3 Caputo’s derivative for a function $$ f:[0,\infty)\rightarrow R $$ can be written as $$^{C}D^{\alpha}_{t}f(t)= ^{RL}D^{\alpha}_{t}\bigg[f(t)-\sum_{k=0}^{n-1}\frac{t^{k}}{k!}f^{(k)}(0)\bigg], n=[\alpha]+1. $$ Remark 2.4 (i) The Caputo derivative of a constant is equal to zero. (ii) If the function $$f'\in C[0,\infty)$$, then we can get $$D^{\alpha}_{t}f(t)=\frac{1}{{\it{\Gamma}}(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}f'(s)ds=I^{1-\alpha}_{t}f'(t), t>0, 0<\alpha<1.$$ (iii) If $$f$$ is an abstract function with values in $$H$$, then the integrals which appear in Definitions 2.1 and 2.2 are taken in the Bochner’s sense. Next we recall some definitions from multi-valued analysis; see Clarke (1983) and Hu & Papageorgiou (1997): (i) For a given Banach space $$I$$, a multi-valued map $$F:I\rightarrow 2^{I}\setminus\{\emptyset\}:=\mathcal{P}(I)$$ is convex (closed) valued, if $$F(x)$$ is convex (closed) for all $$x \in I.$$ (ii) $$F$$ is called upper semi-continuous (u.s.c.) on $$I$$, if for each $$x\in I$$, the set $$F(x)$$ is a non-empty, closed subset of $$I$$, and if for each open set $$V$$ of $$I$$ containing $$F(x)$$, there exists an open neighbourhood $$N$$ of $$x$$ such that $$F(N)\subseteq V$$. (iii) $$F$$ is said to be completely continuous if $$F(V )$$ is relatively compact, for every bounded subset $$V \subseteq I$$. (iv) Let $$({\it{\Omega}},{\it{\Sigma}})$$ be a measurable space and $$(I,d)$$ a separable metric space. A multi-valued map $$F:J\rightarrow \mathcal{P}(I)$$ is said to be measurable, if for every closed set $$C\subseteq I$$, we have $$F^{-1}(C)=\{t\in J:F(t)\cap C\neq\emptyset\}\in {\it{\Sigma}}$$. We now recall the generalized gradient of Clarke for a locally Lipschitzian functional $$h :I\rightarrow R$$ (cf. Clarke (1983)). We denote by $$h^{0}(y,z)$$ the Clarke generalized directional derivative of $$h$$ at $$y$$ in the direction $$z$$, that is $$h^{0}(y,z):=\limsup_{\lambda\rightarrow0^{+}, \xi\rightarrow y}\frac{h(\xi+\lambda z)-h(\xi)}{\lambda}.$$ Recall also that the generalized Clarke subdifferential of $$h$$ at $$y$$, denoted by $$\partial h(y)$$, is a subset of $$I^{*}$$ given by \[ \partial h(y):=\{y^{*}\in I^{*}:h^{0}(y,z)\geq \langle y^{*},z\rangle, \forall z\in I\}. \] Lemma 2.5 (Proposition 2.1.2 of Clarke (1983)) Let $$h$$ be locally Lipschitz of rank $$K$$ near $$y$$. Then (a) $$\partial h(y)$$ is a non-empty, convex, weak$$^{*}$$-compact subset of $$I^{*}$$ and $$\|y^{*}\|_{I^{*}}\leq K$$ for every $$y^{*}$$ in $$\partial h(y)$$; (b) for every $$z\in I$$, one has $$h^{0}(y,z)=\max\{\langle y^{*},z\rangle: \textrm{for all } y^{*}\in \partial h(y)\}.$$ The following is based on Lemma 3.1 in Fečkan et al. (2013) and Definition 2.5 in Li et al. (2016). Definition 2.6 For each $$u\in L^{2}(J,U)$$, a function $$x\in C(J,H)$$ is a mild solution of (1.1) if $$x(0)=x_{0}$$ and there exists $$f\in L^{2}(J,H)$$ such that $$f(t)\in\partial F(t,x(t))$$ a.e. on $$t\in J$$ and \begin{eqnarray}\label{2.4} x(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f(s)ds\nonumber\\ &&+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds, \quad t\in J. \end{eqnarray} (2.1) where $$\mathscr{T}_E(t):=\int_{0}^{\infty}E^{-1}\xi_{\alpha}(\theta)T(t^{\alpha}\theta)d\theta, \quad \mathscr{L}_E(t):=\alpha\int_{0}^{\infty}E^{-1}\theta\xi_{\alpha}(\theta)T(t^{\alpha}\theta)d\theta,$$ and $$\xi_{\alpha}(\theta)=\frac{1}{\alpha}\theta^{-1-\frac{1}{\alpha}}\varpi_{\alpha}(\theta^{-\frac{1}{\alpha}})\geq0,$$ $\varpi_{\alpha}(\theta)$$=\frac{1}{\pi}\sum_{n=1}^{\infty}(-1)^{n-1}\theta^{-n\alpha-1}\frac{{\it{\Gamma}}(n\alpha+1)}{n!}\sin(n\pi \alpha)$ ,$$\theta\in(0,\infty);$$ here $$\xi_{\alpha}$$ is a probability density function defined on $$(0,\infty),$$ that is $$\xi_{\alpha}(\theta)\geq 0, \theta\in(0,\infty)$$, and $$\int_{0}^{\infty}\xi_{\alpha}(\theta)d\theta=1.$$ Let $$K_{b}(F)=\{x(b)\in H: x(\cdot)$$ is a mild solution of system (1.1) corresponding to a control $$u\in L^{2}(J,U)$$ with initial value $$x_{0}\in H\}$$, which is called the reachable set of (1.1). If $$F\equiv0$$, then this system is called the corresponding linear system of (1.1). In this case, $$K_{b}(0)$$ denotes the reachable set of the linear system. Next recall (see Definitions 2.4 and 2.5 in Kumar & Sukavanam (2012)): Definition 2.7 The system (1.1) is said to be approximately controllable on $$J=[0,b]$$, if $$\overline{K_{b}(F)}=H$$, where $$\overline{K_{b}(F)}$$ denotes the closure of $$K_{b}(F)$$. Clearly, the corresponding linear system is approximately controllable on $$J$$, if $$\overline{K_{b}(0)}=H$$. Lemma 2.8 (Lemma 3.2 of Wang et al. (2014)) Under $$(S_1)$$-$$(S_3)$$, the operators $$\mathscr{T}_E(t)$$ and $$\mathscr{L}_E(t)$$ have the following properties: $${\rm (i)}$$ For any fixed $$t\geq 0, \mathscr{T}_E(t)$$ and $$\mathscr{L}_E(t)$$ are linear operators and bounded operators, i.e. for any $$x \in X$$, $$\big\|\mathscr{T}_E(t)x\big\|\leq M\big\|E^{-1}\big\|\|x\| \textrm{and} \|\mathscr{L}_E(t)x\|\leq \frac{M\|E^{-1}\big\|}{{\it{\Gamma}}(\alpha)}\|x\|.$$ $${\rm (ii)}$$$$\{\mathscr{T}_E(t),t\geq 0\}$$ and $$\{\mathscr{L}_E(t),t\geq 0\}$$ are compact. Finally we recall the following well known fixed point theorem (see Ma (1972)). Theorem 2.9 Let $$I$$ be a Banach space and $$\digamma:I\rightarrow 2^{I} $$ be a compact convex valued, u.s.c. multi-valued map such that there exists a closed neighbourhood $$V$$ of $$0$$ for which $$\digamma(V)$$ is a relatively compact set. If the set $${\it{\Omega}}=\{x\in I:\lambda x\in \digamma(x)\,\,\textrm{for some}\,\,\lambda>1\}$$ is bounded, then $$\digamma$$ has a fixed point. 3. Existence of mild solutions In this section, we study the existence of mild solutions for (1.1). We assume the following: $$H(F):$$$$F:J\times H\rightarrow R$$ is a function such that: (i) the function $$t\mapsto F(t,x)$$ is measurable for all $$x\in H$$; (ii) the function $$x\mapsto F(t,x)$$ is locally Lipschitz for a.e. $$t\in J$$; (iii) there exist a function $$a\in L^{2}(J,R^{+})$$ and a constant $$c>0$$ such that $$\|\partial F(t,x)\|_H =\textrm{sup}\{\|f\|_H :f\in\partial F(t,x)\}\leq a(t)+c\|x\|_{H},$$ for a.e. $$t\in J\,\,\textrm{and all}\,\,x\in H.$$ Now, we define an operator $$\mathcal{N}: L^{2}(J,H)\rightarrow 2^{L^{2}(J,H)}$$ as follows $$\mathcal{N}(x)=\{w\in L^{2}(J,H): w(t)\in \partial F(t;x(t)) \textrm{a.e.} t \in J\}, \textrm{ for } x\in L^{2}(J,H).$$ Lemma 3.1 (Lemma 5.3 of Migórski et al. (2013)) If $$H(F)$$ holds, then for $$x\in L^{2}(J,H),$$ the set $$\mathcal{N}(x)$$ has non-empty, convex and weakly compact values. Lemma 3.2 (Lemma 11 of Migórski & Ochal (2009)) If $$H(F)$$ holds, the operator $$\mathcal{N}$$ satisfies: if $$x_{n}\rightarrow x$$ in $$L^{2}(J,H)$$, $$w_{n}\rightarrow w $$ weakly in $$L^{2}(J,H)$$ and $$ w_{n}\in \mathcal{N}(x_{n})$$, then we have $$w\in \mathcal{N}(x)$$. Theorem 3.3 Assume $$(S_1)$$-$$(S_3)$$ are satisfied. For each $$u\in L^2(J,U)$$, if $$H(F)$$ holds, then (1.1) has a mild solution on $$J$$. Proof. For any $$x\in C(J,H)\subset L^2(J,H)$$, from Lemma 3.1 we can consider the map $$\digamma:C(J,H)\rightarrow 2^{C(J,H)} $$ as follows \begin{eqnarray} \digamma(x)&=&\bigg{\{} h\in C(J,H): h(t)= \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f(s)ds\nonumber\\& & +\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds, f\in \mathcal{N}(x)\bigg{\}}, \textrm{ for } x\in C(J,H). \nonumber\end{eqnarray} We will show $$\digamma$$ has a fixed point using Theorem 2.9. Note $$\digamma(x)$$ is convex from the convexity of $$\mathcal{N}(x)$$. We divide the proof into five steps. Step 1: $$\digamma$$ maps bounded sets into bounded sets in $$C(J,H)$$. For any $$ x\in B_{r}=\{x\in C(J,H):\|x\|_C\leq r\}, r>0$$ and $$\varphi\in\digamma(x)$$, we choose a $$f\in \mathcal{N}(x)$$ with \begin{eqnarray*} \varphi(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f(s)ds\\&& +\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds, \quad t\in J. \nonumber \nonumber\end{eqnarray*} From $$H(F)(iii)$$, Lemma 2.8(i) and the Hölder inequality, we have \begin{eqnarray*} \|\varphi(t)\|_H&\leq& M\|E^{-1}\|\|x_{0}\|_H+\frac{M\|E^{-1}\|cb^{\alpha}r}{{\it{\Gamma}}(1+\alpha)}\\&& +\frac{M\|E^{-1}\|b^{\alpha-\frac{1}{2}}}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)}(\|a\|_{L^{2}(J,R^+)} +\|B\|\|u\|_{L^{2}(J,U)}). \nonumber\end{eqnarray*} Thus $$\digamma(B_{r})$$ is bounded in $$C(J,H)$$. Step 2. $$\{\digamma(x):x\in B_{r}\}$$ is equicontinuous (for all $$r>0$$). For any $$x\in B_{r}, \varphi\in\digamma(x)$$, there exists $$f\in \mathcal{N}(x)$$ such that \begin{eqnarray} \varphi(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)[f(s)+Bu(s)]ds, \quad t\in J. \nonumber \nonumber\end{eqnarray} From $$H(F)(iii)$$ and Lemma 2.8(i), for $$\forall t\in J$$, we have \begin{eqnarray*} &&\|\varphi(t)-\varphi(0)\|_H \\ &\leq &\|\mathscr{T}_E(t)Ex_{0}-x_{0}\|_H+\frac{M\|E^{-1}\|t^{\alpha-\frac{1}{2}}}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)} (\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^2(J,U)})\nonumber \\&&+\frac{M\|E^{-1}\|crt^\alpha}{{\it{\Gamma}}(1+\alpha)}. \nonumber\end{eqnarray*} Thus, for $$\forall\varepsilon>0$$ and for sufficiently small $$\delta_1>0$$, with $$0<t\leq\delta_1$$, we have $$\|\varphi(t)-\varphi(0)\|_H<\frac{\varepsilon}{2}$$. Hence, for $$\forall\varepsilon>0, \forall \tau_1, \tau_2\in [0, \delta_1]$$ and $$\forall\varphi\in\digamma(B_r)$$, we have $$\|\varphi(\tau_2)-\varphi(\tau_1)\|_H<\varepsilon$$. For any $$x\in B_{r}$$ and $$\frac{\delta_1}{2}\leq\tau_{1}<\tau_{2}\leq b$$, we obtain \begin{eqnarray*} & &\|\varphi(\tau_{2})-\varphi(\tau_{1})\|_H\nonumber\\&\leq& \|\mathscr{T}_E(\tau_{2})Ex_{0}-\mathscr{T}_E(\tau_{1})Ex_{0}\|_H\nonumber \\& &+\|\int_{0}^{\tau_{1}}[(\tau_{2}-s)^{\alpha-1}-(\tau_{1}-s)^{\alpha-1}] \mathscr{L}_E(\tau_{2}-s)[f(s)+Bu(s)]ds\|_H\nonumber \\& & +\|\int_{0}^{\tau_{1}}(\tau_{1}-s)^{\alpha-1}[\mathscr{L}_E(\tau_{2}-s)-\mathscr{L}_E(\tau_{1}-s)][f(s)+Bu(s)]ds\|_H\nonumber \\& & +\|\int_{\tau_{1}}^{\tau_{2}}(\tau_{2}-s)^{\alpha-1}\mathscr{L}_E(\tau_{2}-s)[f(s)+Bu(s)]ds\|_H\nonumber \\& & := Q_{1}+Q_{2}+Q_{3}+Q_{4}. \nonumber\end{eqnarray*} Clearly, \begin{eqnarray*} Q_1\leq\|Ex_{0}\|_H\|\mathscr{T}_E(\tau_{2})-\mathscr{T}_E(\tau_{1})\|_H. \end{eqnarray*} From $$H(F)(iii)$$, Lemma 2.8(i) and the Hölder’s inequality, we obtain \begin{eqnarray*} Q_{2}&\leq &\frac{M\|E^{-1}\|(\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^{2}(J,U)})}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)} [\tau_{1}^{2\alpha-1}-\tau_{2}^{2\alpha-1}-(\tau_{2}-\tau_{1})^{2\alpha-1}]^{\frac{1}{2}}\nonumber \\& &+ \frac{M\|E^{-1}\|cr}{{\it{\Gamma}}(1+\alpha)}[\tau_{2}^{\alpha}-\tau_{1}^{\alpha}+(\tau_{2}-\tau_{1})^{\alpha}]. \nonumber\end{eqnarray*} Taking $$\delta_2>0$$ small enough, we have \begin{eqnarray*} &&Q_{3}\\&\leq&\textrm{sup}_{s\in[0,\tau_{1}-\delta_2]}\|\mathscr{L}_E(\tau_{2}-s)-\mathscr{L}_E(\tau_{1}-s)\|\\&&\times \bigg{[}\frac{\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^{2}(J,U)}}{\sqrt{2\alpha-1}} (\tau_{1}^{\alpha-\frac{1}{2}}-\delta_2^{\alpha-\frac{1}{2}})\nonumber +\frac{cr}{\alpha}(\tau_{1}^{\alpha}-\delta_2^{\alpha})\bigg{]} \\ &&+\frac{2M\|E^{-1}\|(\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^{2}(J,U)})}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)} \delta_2^{\alpha-\frac{1}{2}}+\frac{2M\|E^{-1}\|cr}{{\it{\Gamma}}(1+\alpha)}\delta_2^{\alpha}. \nonumber\end{eqnarray*} Also, we have \begin{eqnarray*} Q_{4}&\leq&\frac{M\|E^{-1}\|(\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^{2}(J,U)})}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)} (\tau_{2}-\tau_{1})^{\alpha-\frac{1}{2}}\\&&+\frac{M\|E^{-1}\|cr}{{\it{\Gamma}}(1+\alpha)}(\tau_{2}-\tau_{1})^{\alpha}. \nonumber\end{eqnarray*} From the compactness of $$T(t)(t>0)$$ and Definition 2.6, we see that the operator $$\mathscr{L}_E(t)(t>0)$$ is continuous in the uniform operator topology, and thus, for any $$x\in B_{r}$$, $$Q_{3}$$ tends to zero as $$\tau_{2}\rightarrow \tau_{1}, \delta_2\rightarrow0$$. Also note $$Q_{i}(i=1,2,4)$$ tends to zero as $$\tau_{2}\rightarrow\tau_{1}$$. Thus we get that $$\|\varphi(\tau_{2})-\varphi(\tau_{1})\|_H$$ tends to zero as $$\tau_{2}\rightarrow \tau_{1}$$ and $$\delta_2\rightarrow0$$. Let $$\delta=\min\{\delta_1, \delta_2\}$$. For $$\forall \varepsilon>0, \forall \tau_1, \tau_2\in [0, b], |\tau_1-\tau_2|<\delta,\forall \varphi\in\digamma(B_r)$$, one sees that $$\|\varphi(\tau_{2})-\varphi(\tau_{1})\|_H<\varepsilon$$ independently of $$x\in B_r$$. Therefore, we deduce that $$\{\digamma(x):x\in B_{r}\}$$ is an equicontinuous family of functions in $$C(J,H)$$. Step 3: $$\digamma$$ is completely continuous. We prove that for $$\forall t\in J$$, $$r>0$$, the set $${\it{\Pi}}(t)=\{\varphi(t):\varphi\in \digamma(B_{r})\}$$ is relatively compact in $$H$$. Obviously, $${\it{\Pi}}(0)=\{x_{0}\}$$ is compact, so we only need to consider $$t>0$$. Let $$0<t\leq b$$ be fixed. For any $$x\in B_{r}, \varphi\in\digamma(x)$$, we choose $$f\in \mathcal{N}(x)$$ with \begin{eqnarray} \varphi(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)[f(s)+Bu(s)]ds, \quad t\in J. \nonumber \nonumber\end{eqnarray} For each $$\epsilon\in(0,t), t\in(0,b], x\in B_{r}$$ and any $$\delta>0$$, we define \begin{eqnarray*} \varphi^{\epsilon,\delta}(t)&=&\mathscr{T}_E(t)Ex_{0}+\alpha \int_{0}^{t-\epsilon}\int_{\delta}^{\infty} E^{-1}\theta(t-s)^{\alpha-1}\xi_{\alpha}(\theta)\\&&\times T((t-s)^{\alpha}\theta)[f(s)+Bu(s)]d\theta ds. \nonumber\\ &=&\mathscr{T}_E(t)Ex_{0}+\alpha T(\epsilon^{\alpha}\delta)\int_{0}^{t-\epsilon}\int_{\delta}^{\infty} E^{-1}\theta(t-s)^{\alpha-1}\xi_{\alpha}(\theta)\\&&\times T((t-s)^{\alpha}\theta-\epsilon^{\alpha}\delta)[f(s)+Bu(s)]d\theta ds. \nonumber \nonumber\end{eqnarray*} From the compactness of $$\mathscr{T}_E(t)(t>0)$$ (see Lemma 2.8(ii)) we see that the set $${\it{\Pi}}_{\epsilon,\delta}(t)=\{\varphi^{\epsilon,\delta}(t):\varphi\in \digamma(B_{r})\},$$ is relatively compact in $$H$$ for each $$\epsilon\in(0,t)$$ and $$\delta>0$$. Moreover, we have \begin{eqnarray} & &\|\varphi(t)-\varphi^{\epsilon,\delta}(t)\|_H\nonumber\\ &\leq&\frac{\alpha M\|E^{-1}\|}{\sqrt{2\alpha-1}}\bigg{(}\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^{2}(J,U)}\bigg{)} \bigg{[}b^{\alpha-\frac{1}{2}}\int_{0}^{\delta}\theta\xi_{\alpha}(\theta)d\theta\nonumber\\& & +\frac{1}{{\it{\Gamma}}(1+\alpha)}\epsilon^{\alpha-\frac{1}{2}}\bigg{]} +M\|E^{-1}\|cr\bigg{[}\frac{1}{{\it{\Gamma}}(1+\alpha)}\epsilon^{\alpha} +b^{\alpha}\int_{0}^{\delta}\theta\xi_{\alpha}(\theta)d\theta\bigg{]}. \nonumber\end{eqnarray} Now since $$0\leq\int_{0}^{\delta}\theta\xi_{\alpha}(\theta)d\theta\leq \int_{0}^{\infty}\theta\xi_{\alpha}(\theta)d\theta=\frac{1}{{\it{\Gamma}}(1+\alpha)}$$, the inequality above tends to zero when $$\epsilon\rightarrow 0$$ and $$\delta\rightarrow 0.$$ Therefore, the set $${\it{\Pi}}(t) (t>0)$$ is totally bounded, i.e., relatively compact in $$H$$. From above (and Step 2) and the Ascoli–Arzela Theorem, we see that $$\digamma$$ is completely continuous. Step 4: $$\digamma$$ has a closed graph. Let $$x_{n}\rightarrow x_{*}$$ as $$n\rightarrow \infty $$ in $$C(J,H)$$, $$\varphi_{n}\in \digamma(x_{n})$$ and $$\varphi_{n}\rightarrow \varphi_{*}$$ as $$n\rightarrow \infty $$ in $$C(J,H)$$. We prove that $$\varphi_{*}\in \digamma(x_{*}).$$ Now, $$\varphi_{n}\in \digamma(x_{n})$$ so there exists $$f_{n}\in \mathcal{N}(x_{n})$$ with \begin{eqnarray} \label{3.4} \varphi_{n}(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f_{n}(s)ds\nonumber \\&&+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds. \end{eqnarray} (3.1) From $$H(F)(iii)$$, $$\{f_{n}\}_{n\geq1}\subseteq L^{2}(J,H)$$ is bounded. Hence we assume that \begin{eqnarray}\label{3.5} f_{n}\rightarrow f_{*}, \textrm{ weakly in } L^{2}(J,H), \end{eqnarray} (3.2) From (3.1), (3.2) and the compactness of the operator $$\mathscr{L}_E(t)$$, we have that \begin{eqnarray}\label{3.6} \varphi_{n}(t)&\rightarrow & \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f_{*}(s)ds\nonumber\\&& +\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds. \end{eqnarray} (3.3) Note that $$\varphi_{n}\rightarrow \varphi_{*}$$ in $$C(J,H)$$ and $$f_{n}\in \mathcal{N}(x_{n})$$. From Lemma 3.2, we obtain $$f_{*}\in \mathcal{N}(x_{*})$$. Hence, $$\varphi_{*}\in \digamma(x_{*}),$$ which implies $$\digamma$$ has closed graph. From Proposition 3.3.12(2) of Migórski et al. (2013), $$\digamma$$ is u.s.c. Step 5: Apriori estimate. From Steps 1–4, we have that $$\digamma$$ is u.s.c. and is compact convex valued and $$\digamma(B_r)$$ is a relatively compact set (here $$r>0$$). We now prove that the set $${\it{\Omega}}=\{x\in C(J,H): \lambda x\in \digamma (x), \lambda>1\}$$ is bounded. For $$\forall x\in{\it{\Omega}}$$, there exists $$f\in \mathcal{N}(x)$$ with \begin{eqnarray*} x(t)&=& \lambda^{-1}\mathscr{T}_E(t)Ex_{0}+\lambda^{-1}\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f(s)ds\\&& +\lambda^{-1}\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds. \nonumber\end{eqnarray*} Then from assumption $$H(F)(iii)$$, we derive \begin{eqnarray}\label{3.7} \|x(t)\|_H\leq \rho+\frac{M\|E^{-1}\|c}{{\it{\Gamma}}(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\|x(s)\|_Hds, \nonumber\end{eqnarray} where \begin{eqnarray} \rho=M\|E^{-1}\|\|x_0\|_H +\frac{M\|E^{-1}\|b^{\alpha-\frac{1}{2}}}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)}\bigg{(}\|a\|_{L^2(J,R^+)}+\|B\|\|u\|_{L^2(J,U)}\bigg{)}. \nonumber\end{eqnarray} It follows from (3.4) and Corollary 2 in Ye et al. (2007) that $$\|x(t)\|_H\leq \rho E_\alpha(M\|E^{-1}\|ct^\alpha).$$ Hence, $$\|x\|_C=\sup_{t\in J}\|x(t)\|_H\leq \rho E_\alpha(M\|E^{-1}\|cb^\alpha)$$, which implies the set $${\it{\Omega}}$$ is bounded. From Theorem 2.9, $$\digamma$$ has a fixed point, i.e., (1.1) has a mild solution. The proof is complete. □ 4. Approximate controllability of a linear system In this section, we consider the approximate controllability of the following linear fractional control system of Sobolev type: \begin{eqnarray}\label{4.1} \left\{\begin{array}{@{}llll} ^{C}D^{\alpha}_{t}(Ex(t))+ Ax(t)=Bu(t), t\in J=[0,b], \frac{1}{2}<\alpha<1,\\ x(0)=x_{0}. \end{array}\right. \end{eqnarray} (4.1) Define a bounded linear operator $$\mathcal {K}:L^{2}(J,H)\rightarrow H$$ as follows: $$\mathcal {K}h=\int_{0}^{b}(b-s)^{\alpha-1}\mathscr{L}_E(b-s)h(s)ds, \quad h(\cdot)\in L^{2}(J,H).$$ We assume the following: $$H(B):$$ For each $$h(\cdot)\in L^{2}(J,H)$$, there exists a function $$g(\cdot)\in \overline{R(B)}$$, such that $$\mathcal{K}h=\mathcal{K}g,$$ where $$R(B)$$ denotes the range of operator $$B$$ and $$\overline{R(B)}$$ is the closure of $$R(B)$$. Theorem 4.1 Suppose $$H(B)$$ is satisfied. Then (4.1) is approximately controllable on $$J$$ provided that $$T(t)$$ is a differentiable semi-group. Proof. The idea comes from Kumar & Sukavanam (2012). Since the domain $$D(A)$$ of the operator $$A$$ is dense in $$H$$, it is sufficient to show that $$D(A)\subset \overline{K_{b}(0)},$$ i.e. for any $$\epsilon>0$$ and $$\eta\in D(A)$$, there exists a control function $$u\in L^{2}(J,U)$$, such that \begin{eqnarray}\label{4.2} \|\eta-\mathscr{T}_E(b)Ex_{0}-\mathcal{K}Bu\|_{H}<\epsilon. \end{eqnarray} (4.2) For any $$x_{0}\in H$$, $$T(t)$$ is a differentiable semi-group, so $$T(t)Ex_{0}\in D(A)$$, such that $$\mathscr{T}_E(b)Ex_{0}\in D(A)$$. For a given $$ \eta\in D(A)$$, we see that there exists a function $$h(\cdot)\in L^{2}(J,H)$$, such that $$\mathcal{K}h=\eta-\mathscr{T}_E(b)Ex_{0}$$. for example, one can choose $$h(t)=\frac{E^2[{\it{\Gamma}}(\alpha)]^{2}(b-t)^{1-\alpha}}{b}\bigg{[}\mathscr{L}_E(b-t)+2t \frac{d \mathscr{L}_E(b-t)}{dt}\bigg{]}[\eta-\mathscr{T}_E(b)Ex_{0}];$$ note that, $$\mathcal{K}h=\frac{E^2[{\it{\Gamma}}(\alpha)]^{2}}{b}\big[\eta-\mathscr{T}_E(b)Ex_{0}\big](b\mathscr{L}_E^{2}(0)).\nonumber$$ From Definition 2.6, $$\mathscr{L}_E(0)=\frac{E^{-1}}{{\it{\Gamma}}(\alpha)}$$. Further, $$\mathcal{K}h=\eta-\mathscr{T}_E(b)Ex_{0}.$$ Next, we prove there is a control function $$u_\varepsilon\in L^{2}(J,U)$$ such that (4.2) holds. From assumption $$H(B)$$, for $$h(\cdot)\in L^{2}(J,H)$$, there exists a function $$g\in \overline{R(B)}$$ with $$\mathcal{K}h=\int_{0}^{b}(b-s)^{\alpha-1}\mathscr{L}_E(b-s)h(s)ds=\int_{0}^{b}(b-s)^{\alpha-1}\mathscr{L}_E(b-s)g(s)ds.$$ Since $$g\in \overline{R(B)}$$, for a given $$\varepsilon>0$$, there exists a control function $$u_\varepsilon\in L^{2}(J,U)$$ such that $$\|Bu_{\varepsilon}-g\|_{L^{2}(J,U)}<\frac{{\it{\Gamma}}(\alpha)}{M\|E^{-1}\|}\sqrt{2\alpha-1} b^{\frac{1}{2}-\alpha}\varepsilon.$$ Then for $$\varepsilon>0, u_\varepsilon\in L^{2}(J,U)$$, from the above arguments, we have \begin{eqnarray*} \|\eta-\mathscr{T}_E(b)Ex_{0}-\mathcal{K}Bu_\varepsilon\|_{H} &=&\|\mathcal{K}h-\mathcal{K}Bu_\varepsilon\|_{H} \nonumber\\ &\leq& \frac{M\|E^{-1}\| b^{\alpha-\frac{1}{2}}}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)}\|Bu_{\varepsilon}-g\|_{L^2(J,U)}<\varepsilon. \nonumber\end{eqnarray*} Since $$\varepsilon$$ is arbitrary, we deduce that $$D(A)\subset \overline{K_{b}(0)}.$$ The denseness of the domain $$D(A)$$ in $$H$$ implies the approximate controllability of (4.1) on $$J$$. This completes the proof. □ For Section 5, we consider two relevant operators associated with (4.1): $${\it{\Gamma}}_{0}^{b}=\int_{0}^{b}(b-s)^{\alpha-1}\mathscr{L}_E(b-s)BB^{*}\mathscr{L}_E^{*}(b-s)ds$$ and $$R(\varepsilon,{\it{\Gamma}}_{0}^{b})=(\varepsilon {\mathbb I}+{\it{\Gamma}}_{0}^{b})^{-1}, \varepsilon>0,$$ respectively, where $$B^{*}$$ denotes the adjoint of $$B$$ and $$\mathscr{L}_E^{*}(t)$$ is the adjoint of $$\mathscr{L}_E(t)$$. Lemma 4.2 (Lemma 2.10 of Liu & Li (2015)). The linear fractional control system (4.1) is approximately controllable on $$J$$ if and only if $$\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})\rightarrow 0$$ as $$\varepsilon\rightarrow 0^{+}$$ in the strong operator topology. 5. Approximate controllability for the semi-linear case In this section, we consider the approximate controllability of (1.1). For any $$x\in C(J,H)\subset L^2(J,H)$$, from Lemma 3.1, we know that $$\mathcal{N}(x)\not=\emptyset$$. Now, for any $$\varepsilon>0$$ and $$x_{1}\in H$$ is fixed, we consider the map $$\digamma_{\varepsilon}:C(J,H)\rightarrow 2^{C(J,H)} $$ given by: \begin{eqnarray} \digamma_{\varepsilon}(x)&=&\bigg{\{} h\in C(J,H): h(t)= \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f(s)ds\nonumber\\& & +\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu_{\varepsilon}(s)ds, f\in \mathcal{N}(x)\bigg{\}}, \nonumber\end{eqnarray} where $$ u_{\varepsilon}(t)=B^{*}\mathscr{L}_E^{*}(b-t) R(\varepsilon,{\it{\Gamma}}_{0}^{b})\bigg{(}x_{1}-\mathscr{T}_E(b)Ex_{0} -\int_{0}^{b}(b-\tau)^{\alpha-1}\mathscr{L}_E(b-\tau)f(\tau)d\tau\bigg{)}. $$ Theorem 5.1 Suppose $$H(F)$$ (i), (ii) are satisfied and there exists a function $$\psi\in L^{2}(J,R^{+})$$ such that $$\|\partial F(t,x)\|_{H}\leq \psi(t),\,\,\textrm{for a.e.}\,\,t\in J,\,\,\textrm{all}\,\,x\in H.$$ Then $$\digamma_\varepsilon$$ has a fixed point on $$J$$. The proof is similar to that in Theorem 3.3 so we omit it. Theorem 5.2 Assume that the assumptions of Theorem 5.1 hold, and in addition, suppose (4.1) is approximately controllable. Then the system (1.1) is approximately controllable on $$J$$. Proof. Let $$x_{\varepsilon}$$ be a fixed point of $$\digamma_\varepsilon(x)$$ in $$C(J,H)$$. Then, there exists $$f_{\varepsilon}\in \mathcal{N}(x_{\varepsilon})$$ such that for each $$t\in J,$$ \begin{eqnarray*} x_{\varepsilon}(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f_{\varepsilon}(s)ds \\&&+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)BB^{*}\mathscr{L}_E^{*}(b-s)\nonumber\\& &\times R(\varepsilon,{\it{\Gamma}}_{0}^{b})\bigg{(}x_{1}-\mathscr{T}_E(b)Ex_{0} -\int_{0}^{b}(b-\tau)^{\alpha-1}\mathscr{L}_E(b-\tau)f_{\varepsilon}(\tau)d\tau\bigg{)}ds. \nonumber \nonumber\end{eqnarray*} Since $${\mathbb I}-{\it{\Gamma}}_{0}^{b}R(\varepsilon,{\it{\Gamma}}_{0}^{b})=\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b}),$$ we get $$x_\varepsilon(b)=x_{1}-\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})G(f_{\varepsilon}),$$ where $$G(f_{\varepsilon})=x_{1}-\mathscr{T}_E(b)Ex_{0} -\int_{0}^{b}(b-\tau)^{\alpha-1}\mathscr{L}_E(b-\tau)f_{\varepsilon}(\tau)d\tau.$$ Since $$\|\partial F(t,x)\|_H\leq\psi(t)$$, we have $$\int_{0}^{b}\|f_{\varepsilon}(s)\|ds\leq \|\psi\|_{L^{2}(J,R^+)}\sqrt{b}.$$ Thus, the sequence $$\{f_{\varepsilon}\}$$ is bounded in $$L^{2}(J,H)$$, so there is a subsequence, still denoted by $$\{f_{\varepsilon}\}$$, which converges weakly to $$f$$, that is $$f_{\varepsilon}\xrightarrow{w} f$$. Let $$h=x_{1}-\mathscr{T}_E(b)Ex_{0}-\int_{0}^{b}(b-\tau)^{\alpha-1}\mathscr{L}_E(b-\tau)f(\tau)d\tau.$$ The linear system (4.1) is approximately controllable, so from Lemma 4.2, as $$\varepsilon \to 0^{+}$$, we have $$\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})\to 0.$$ It follows that \begin{eqnarray} \|G(f_{\varepsilon})-h\| & \leq&\sup_{0\leq t\leq b}\|\int_{0}^{t}(t-\tau)^{\alpha-1}\mathscr{L}_E(t-\tau)\big[f_{\varepsilon}(\tau)-f(\tau)\big]d\tau\|. \nonumber\end{eqnarray} The compactness of the operator $$\mathscr{L}_E(t),t>0$$ guarantees that $$\|G(f_{\varepsilon})-h\|\to 0$$ as $$\varepsilon\rightarrow 0^{+}$$. From the above arguments, we also get \begin{eqnarray} \|x_{\varepsilon}(b)-x_{1}\| &\leq&\|\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})(h)\|+\|\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})[G(f_{\varepsilon})-h]\|\nonumber\\ &\leq&\|\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})(h)\|+\|G(f_{\varepsilon})-h\|\rightarrow0,\,\,\textrm{as}\,\,\varepsilon\rightarrow 0^{+}, \nonumber\end{eqnarray} and this proves the approximately controllability of (1.1) on $$J$$. □ Remark 5.3 From the above conclusion, we find that it is generalizes the results in Li et al. (2016). In fact, if $$E={\mathbb I}$$, the results are similar to Li et al. (2016). 6. Examples Example 6.1 Motivated by an example in Fečkan et al. (2013), we consider the following fractional heat equation: \begin{eqnarray}\label{6.1} \left\{\begin{array}{@{}llll} ^CD^{3/4}_{t}(x(t,y)-x_{yy}(t,y))=x_{yy}(t,y)+Bu(t,y)+\phi(t,y),\\ 0<t<1, 0<y<\pi, x(t,0)=x(t,\pi)=0, 0<t<1,\\ x(0,y)=x_0(y), 0<y<\pi, \end{array}\right. \end{eqnarray} (6.1) where $$x(t,y)$$ represents the temperature at the point $$y\in (0,\pi)$$ and time $$t\in (0,1)$$. Now, set $$H=L^2[0,\pi]$$ and $$e_n(y)=\sqrt{2/\pi}\sin(ny), n=1,2,\cdots.$$ Then $$\{e_n(y)\}$$ is an orthonormal base for $$H$$. Define $$A:D(A)\subset X\rightarrow H$$ by $$Ax=-x_{yy}$$ and $$E:D(E)\subset X\rightarrow H$$ by $$Ex:=x-x_{yy}$$ with domain \begin{eqnarray}\label{6.1-dom} \{x\in H: x, x'\textrm{ are absolutely continuous}, x''\in H, x(0)=x(\pi)=0\}. \end{eqnarray} (6.2) Then $$Ax=\sum_{n=1}^\infty n^2\langle x,e_n\rangle e_n, x\in D(A)$$ and $$Ex=\sum_{n=1}^\infty(1+n^2)\langle x,e_n\rangle e_n, x\in D(A).$$ Furthermore, for $$x\in X$$, we have $$E^{-1}x=\sum_{n=1}^\infty\frac{1}{1+n^2}\langle x,e_n\rangle e_n, \quad -AE^{-1}x=\sum_{n=1}^\infty\frac{-n^2}{1+n^2}\langle x,e_n\rangle e_n x\in D(A).$$ It is well known that $$-AE^{-1}$$ generates a compact semi-group $$T(t)(t>0)$$ on $$H$$ and \begin{eqnarray}\label{6.1-T} T(t)x=\sum_{n=1}^\infty e^{\frac{-n^2}{1+n^2}t}\langle x,e_n\rangle e_n, x\in H. \end{eqnarray} (6.3) It is easy to see that $$T(t)$$ is compact with $$\|T(t)\|\leq e^{-t}\leq 1$$, $$E^{-1}$$ is also compact and bounded with $$\|E^{-1}\|\leq1$$. Moreover, the two operators $$\mathscr{T}_E(t)$$ and $$\mathscr{L}_E(t)$$ can be defined by $$\mathscr{T}_E(t)=\int_{0}^{\infty}E^{-1}\xi_{\frac{3}{4}}(\theta)T(t^{\frac{3}{4}}\theta)d\theta, \quad \mathscr{L}_E(t)=\frac{3}{4}\int_{0}^{\infty}E^{-1}\theta\xi_{\frac{3}{4}}(\theta)T(t^{\frac{3}{4}}\theta)d\theta.$$ We easily get, $$\|\mathscr{T}_E(t)\|\leq1, \quad \|\mathscr{L}_E(t)\|\leq\frac{1}{{\it{\Gamma}}(\frac{3}{4})}.$$ Let the infinite dimensional Hilbert space $$U$$ be defined by $$U:=\{u:u=\sum_{n=2}^\infty u_ne_n, \sum_{n=2}^\infty u_{n}^2<\infty\}.$$ The norm in $$U$$ is defined by $$\|u\|_U=(\sum_{n=2}^\infty u_n^2)^{1/2}$$. Define a mapping $$B\in \mathcal{L}(U,H)$$ as follows: $$ Bu=2u_2e_1+\sum_{n=2}^\infty u_ne_n \quad \textrm{ for } u=\sum_{n=2}^\infty u_ne_n\in U, $$ and for $$v=\sum_{n=1}^\infty v_ne_n\in H$$, inner product $$\langle Bu,v\rangle=\langle u,B^*v\rangle$$, thus $$ B^*v=(2v_1+v_2)e_2+\sum_{n=3}^\infty v_ne_n$$ and $$ B^*T^*(t)x=(2x_1e^{-t}+x_2e^{-4t})e_2+\sum_{n=3}^\infty e^{\frac{-n^2}{1+n^2}}x_ne_n. $$ It follows that $$\|B^*T^*(t)x\|=0$$ for some $$t\in J$$, which implies that $$x=0$$, and thus the linear part of (6.1) is approximate controllable on $$J$$ (see Theorem 4.1.7 of Curtain & Zwart (1995)). Then all hypotheses in Theorem 5.2 are satisfied, so (6.1) is approximate controllable. Example 6.2 Consider the partial differential system of the form \begin{eqnarray}\label{6.2} \left\{\begin{array}{@{}llll} ^CD^{4/5}_{t}(x(t,\theta)-x_{\theta\theta}(t,\theta))=x_{\theta\theta}(t,y)+\bar{b}(\theta)u(t)+\phi(t,\theta),\\ 0<t<1, 0<\theta<\pi, x(t,0)=x(t,\pi)=0, 0<t<1,\\ x(0,\theta)=x_0(\theta), \quad 0<\theta<\pi, \end{array}\right. \end{eqnarray} (6.4) where $$u\in H=L^2[0,\pi]$$. Let $$e_n(\theta)=\sqrt{2/\pi}\sin(n\theta), n=1,2,\cdots.$$ Then $$\{e_n(\theta)\}$$ is an orthonormal base for $$H$$. Let $$B$$ can be define as $$(Bu)(\theta)=\bar{b}(\theta)u,$$$$(B^*v)(\theta)=\sum_{n=0}^{\infty}\langle \bar{b},e_n\rangle \langle v,e_n\rangle.$$ Define $$A:D(A)\subset X\rightarrow H$$ by $$Ax=-x_{\theta\theta}$$ and $$E:D(E)\subset X\rightarrow H$$ by $$Ex:=x-x_{\theta\theta}$$ with the domain (6.2). Like Example 6.1, $$-AE^{-1}$$ generates a compact semi-group $$T(t)(t>0)$$ on $$H$$ given in (6.3) and $$\mathscr{L}_E(t)$$ can be defined by $$\mathscr{L}_E(t)x=\frac{4}{5}\sum_{n=1}^\infty \int_{0}^{\infty}E^{-1}\theta\xi_{\frac{4}{5}}(\theta)e^{\frac{-n^2}{1+n^2}t^{\frac{4}{5}}\theta}d\theta\langle x,e_n\rangle e_n.$$ To prove the system (6.4) is approximately controllable, we will show that $$(b-s)^{\alpha-1}B^*\mathscr{L}_E(b-s)x=0\Rightarrow x=0$$. In fact, \begin{eqnarray} &&(b-s)^{\alpha-1}B^*\mathscr{L}_E(b-s)x\nonumber\\&=&(b-s)^{\alpha-1}\frac{4}{5}\sum_{n=1}^\infty \int_{0}^{\infty}E^{-1}\theta\xi_{\frac{4}{5}}(\theta)e^{\frac{-n^2}{1+n^2}(b-s)^{\frac{4}{5}}\theta}d\theta\langle \bar{b},e_n\rangle \langle x,e_n\rangle=0. \nonumber\end{eqnarray} Thus, provided that $$\langle \bar{b},e_n\rangle= \int_0^{\pi}\bar{b}e_n(\theta)d\theta \neq 0$$, so $$\langle x,e_n\rangle=0\Rightarrow x=0$$. Therefore, the system (6.4) is approximately controllable (Curtain & Zwart, 1995) provided that $$\langle \bar{b},e_n\rangle \neq 0$$ for $$n=1,2,3\cdots$$. 7. Conclusion Approximate controllability for fractional evolution inclusions of Sobolev and Clarke subdifferential type has been investigated. Utilizing two characteristic solution operators and their fundamental properties via a fixed point theorem for multi-valued maps, sufficient conditions for the mild solution of (6.1) are proved, and using two related operators, we studied the approximate controllability. In the future, we will study the controllability of relative problems by using the techniques of the measure of non-compactness. For example, one can consider approximation controllability of impulsive fractional problems (Wang et al., 2016), fractional damped equations (Wang et al., 2017), fractional delay problems (Li & Wang, 2017b) and fractional Navier-Stokes equations (Zhou & Peng, 2017a,b). 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On the approximate controllability for fractional evolution inclusions of Sobolev and Clarke subdifferential type

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Abstract

Abstract This article is concerned with the approximate controllability for some fractional evolution inclusions of Sobolev and Clarke subdifferential type, which can be used to study fractional hemivariational inequalities. Using two characteristic solution operators and their fundamental properties via a fixed point theorem for multi-valued maps, we derive sufficient conditions for approximate controllability of linear and semi-linear controlled systems. Finally, two examples are given to illustrate our theory. 1. Introduction In this article, we study the following fractional evolution inclusion of Sobolev and Clarke subdifferential type: \begin{eqnarray}\label{e1} \left\{ \begin{array}{@{}l} ^CD^{\alpha}_{t}\big(Ex(t)\big) +Ax(t)\in Bu(t)+\partial F(t,x(t)), \quad t\in J=[0,b],\\ x(0)=x_{0}, \end{array} \right. \end{eqnarray} (1.1) where $${}^{C}D^{\alpha}_{t}$$ denotes the Caputo fractional derivative of order $$\alpha$$ and $$\frac{1}{2}<\alpha\leq1$$, $$A:D(A)\subseteq H\rightarrow H$$ on a separable Hilbert space $$H$$, $$E:D(E)\subset H\rightarrow H$$, and $$\partial F$$ denotes the generalized Clarke subdifferential (cf. Clarke (1983)) of a locally Lipschitz function $$F(t,\cdot): H\rightarrow R$$. The control function $$u$$ takes values in $$L^{2}(J,U)$$ and the admissible controls set $$U$$ is a Hilbert space. Finally $$B$$ is a bounded linear operator from $$U$$ into $$H$$. Fractional calculus and fractional dynamic equations (Miller & Ross, 1993; Podlubny, 1999; Kilbas et al., 2006; Lakshmikantham et al., 2009; Diethelm, 2010; Ghomanjani, 2017) arises naturally in phenomena in engineering, physics, science and controllability. For recent work on the existence of mild solutions, controllability and optimal control for some fractional evolution systems (see Hernández et al. (2010), Debbouche et al. (2012), Sakthivel et al. (2016), Wang et al. (2011) and Wang & Zhou (2011)). In Wang et al. (2011), two characteristic solution operators were used to study the existence of mild solutions for some systems governed by fractional equations with initial and non-local conditions. Sobolev-type equation appears in a variety of physical problems, for example the flow of fluid through fissured rocks, thermodynamics, and the propagation of long waves of small amplitude; see Lightbourne & Rankin (1983), Ponce (2014), Kerboua et al. (2014), Kerboua et al. (2013), Li et al. (2012), Benchaabane et al. (2017), Debbouche & Nieto (2014), Debbouche et al. (2015), Fečkan et al. (2013) and Wang et al. (2014). The existence and uniqueness of Hölder continuous solutions of an abstract Sobolev type differential equation was considered in Lightbourne & Rankin (1983) and two new linear operators were given in Ponce (2014). A non-local condition given in the stochastic term fractional derivative was presented in Kerboua et al. (2014), the theory of a propagation family was used to obtain the existence of mild solutions in Li et al. (2012), and existence and uniqueness of mild solutions and optimal multi-controls to Sobolev-type fractional non-local dynamical equations in Banach spaces was presented in Debbouche & Nieto (2014) and Debbouche et al. (2015). Two new characteristic solution operators and their properties was used to prove the controllability of fractional functional evolution equations of Sobolev type in Fečkan et al. (2013) and Wang et al. (2014), and this motivated our research. Hemivariational inequalities were introduced by Panagiotopoulos in 1980s to deal with mechanical problems with non-convex and non-smooth superpotentials; for recent references on existence we refer the reader to Carl et al. (2003), Sakthivel et al. (2013), Liu (2008), Migórski (2001), Migórski & Ochal (2009), Panagiotopoulo (1993) and Panagiotopoulos (1995) and for optimal control problems for hemivariational inequalities (see Liu & Li (2015), Rykaczewski (2012), Migórski & Ochal (2000) and Park et al. (2007)). Specially, in Li et al. (2016), the approximate controllability for fractional evolution hemivariational inequalities was shown. The results in Li et al. (2016) are only valid for fractional evolution inclusions without Sobolev type. It is natural to ask whether we can apply the characteristic solution operators and their properties to deal with approximate controllability of fractional evolution inclusion of Sobolev type in Fečkan et al. (2013) and Wang et al. (2014). To the best of our knowledge, the existence of mild solutions and the approximate controllability for fractional evolution inclusions of Sobolev type have not been investigated and this motivates us to study (1.1). Note $$x\in C(J,H)$$ is a solution of (1.1) if there exists a $$f\in L^2(J,H)$$ such that $$f(t)\in \partial F(t,x(t))$$ and \begin{eqnarray*}\label{e2} \left\{ \begin{array}{@{}l} ^CD^{\alpha}_{t}\big(Ex(t)\big)+Ax(t)= Bu(t)+f(t), a.e. t\in J,\\ x(0)=x_{0}, \end{array} \right. \end{eqnarray*} so we have \begin{eqnarray*}\label{e3} \left\{ \begin{array}{@{}l} \langle-^CD^{\alpha}_{t}\big(Ex(t)\big)-Ax(t)+Bu(t), v\rangle_H+ \langle f(t), v\rangle_H=0,\\ a.e. t\in J, \forall v\in H,\\ x(0)=x_{0}.\\ \end{array} \right. \end{eqnarray*} Since $$f(t)\in \partial F(t,x(t))$$ and $$\langle f(t), v\rangle_H\leq F^0(t,x(t),v)$$, we obtain \begin{eqnarray}\label{e1.2} \left\{ \begin{array}{@{}l} \langle -^CD^{\alpha}_{t}\big(Ex(t)\big)-Ax(t)+Bu(t), v\rangle_H+ F^0(t,x(t),v)\geq 0,\\ a.e. t\in J, \forall v\in H,\\ x(0)=x_{0}. \end{array} \right. \end{eqnarray} (1.2) Therefore to study (1.2), we consider (1.1). The main novelty of this article is that we develop ideas and techniques to establish an effective framework to present approximate controllability results for fractional evolution inclusion of Sobolev and Clarke subdifferential type, which generalizes the known results in Li et al. (2016). We present sufficient conditions to guarantee approximate controllability of linear fractional control system of Sobolev type, which will be used to deal with semi-linear problem via constructing a control function involving the Gramian controllability operator. Concerning semi-linear problem, we transform it to a fixed point problem of a non-linear operator associated with a new constructed control function, which again shows the powerful role of applying a fixed point theorem to deal with non-linear functional analysis problems. In Section 2, we present some preliminaries and in Section 3 sufficient conditions are given for the existence of mild solutions of (1.1). In Section 4, we study the approximate controllability for a linear fractional control system and in Section 5, the approximate controllability of (1.1) is discussed. In Section 6, an example is given to illustrate our theory. 2. Preliminaries Let $$I$$ be a Banach space with norm $$\|\cdot\|_I$$. Now $$I^*$$ denotes its dual and $$\langle\cdot,\cdot\rangle_I$$ the duality pairing between $$I^*$$ and $$I$$. As usual $$C(J,I)$$ denotes the Banach space of all continuous functions from $$J=[0,b]$$ into $$I$$ with the norm $$\|x\|_{C(J,I)}=\textrm{sup}_{t\in J}\|x(t)\|_{I}$$. The operators $$A $$ and $$E$$ satisfy: $$(S_1)$$: $$A $$ and $$E$$ are linear operators, and $$A$$ is closed. $$(S_2)$$: $$D(E)\subset D(A)$$ and $$E$$ is bijective. $$(S_3)$$: The linear operator $$E^{-1}$$: $$H\rightarrow D(E)\subset H$$ is compact. From $$(S_3)$$, we know that $$E$$ is closed, because $$E^{-1}$$ is closed and injective, so then the inverse is also closed. From $$(S_1)-(S_3)$$ and the closed graph theorem, we see that the operator $$-AE^{-1}:H\rightarrow H$$ is linear and bounded, so $$-AE^{-1}$$ generates a $$C_0$$ semi-group $$T(t),t\geq 0$$, $$T(t):=e^{-AE^{-1}t}$$, and we suppose $$\textrm{sup}_{t\in[0,\infty)}\|T(t)\|\leq M<+\infty$$. We collect some definitions from fractional calculus; see Kilbas et al. (2006). Definition 2.1 For a given function $$f:[0,+\infty)\rightarrow R$$, the integral $$ I^{\alpha}_{t}f(t)=\frac{1}{{\it{\Gamma}}(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}f(s)ds, \quad t>0, \quad 0<\alpha<1,$$ is called Riemann–Liouville fractional integral of order $$\alpha$$, where $${\it{\Gamma}}$$ is the gamma function. Definition 2.2 For a function $$ f:[0,\infty)\rightarrow R $$, $$[0,\infty)$$, the expression $$^{RL}D^{\alpha}_{t}f(t)=\frac{1}{{\it{\Gamma}}(n-\alpha)}\bigg(\frac{d}{dt}\bigg)^{(n)}\int^{t}_{0}(t-s)^{n-\alpha-1}f(s)dt, $$ where $$n=[\alpha]+1, [\alpha]$$ denotes the integer part of number $$ \alpha$$, is called the Riemann-Liouville fractional derivative of order $$\alpha>0$$. Definition 2.3 Caputo’s derivative for a function $$ f:[0,\infty)\rightarrow R $$ can be written as $$^{C}D^{\alpha}_{t}f(t)= ^{RL}D^{\alpha}_{t}\bigg[f(t)-\sum_{k=0}^{n-1}\frac{t^{k}}{k!}f^{(k)}(0)\bigg], n=[\alpha]+1. $$ Remark 2.4 (i) The Caputo derivative of a constant is equal to zero. (ii) If the function $$f'\in C[0,\infty)$$, then we can get $$D^{\alpha}_{t}f(t)=\frac{1}{{\it{\Gamma}}(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}f'(s)ds=I^{1-\alpha}_{t}f'(t), t>0, 0<\alpha<1.$$ (iii) If $$f$$ is an abstract function with values in $$H$$, then the integrals which appear in Definitions 2.1 and 2.2 are taken in the Bochner’s sense. Next we recall some definitions from multi-valued analysis; see Clarke (1983) and Hu & Papageorgiou (1997): (i) For a given Banach space $$I$$, a multi-valued map $$F:I\rightarrow 2^{I}\setminus\{\emptyset\}:=\mathcal{P}(I)$$ is convex (closed) valued, if $$F(x)$$ is convex (closed) for all $$x \in I.$$ (ii) $$F$$ is called upper semi-continuous (u.s.c.) on $$I$$, if for each $$x\in I$$, the set $$F(x)$$ is a non-empty, closed subset of $$I$$, and if for each open set $$V$$ of $$I$$ containing $$F(x)$$, there exists an open neighbourhood $$N$$ of $$x$$ such that $$F(N)\subseteq V$$. (iii) $$F$$ is said to be completely continuous if $$F(V )$$ is relatively compact, for every bounded subset $$V \subseteq I$$. (iv) Let $$({\it{\Omega}},{\it{\Sigma}})$$ be a measurable space and $$(I,d)$$ a separable metric space. A multi-valued map $$F:J\rightarrow \mathcal{P}(I)$$ is said to be measurable, if for every closed set $$C\subseteq I$$, we have $$F^{-1}(C)=\{t\in J:F(t)\cap C\neq\emptyset\}\in {\it{\Sigma}}$$. We now recall the generalized gradient of Clarke for a locally Lipschitzian functional $$h :I\rightarrow R$$ (cf. Clarke (1983)). We denote by $$h^{0}(y,z)$$ the Clarke generalized directional derivative of $$h$$ at $$y$$ in the direction $$z$$, that is $$h^{0}(y,z):=\limsup_{\lambda\rightarrow0^{+}, \xi\rightarrow y}\frac{h(\xi+\lambda z)-h(\xi)}{\lambda}.$$ Recall also that the generalized Clarke subdifferential of $$h$$ at $$y$$, denoted by $$\partial h(y)$$, is a subset of $$I^{*}$$ given by \[ \partial h(y):=\{y^{*}\in I^{*}:h^{0}(y,z)\geq \langle y^{*},z\rangle, \forall z\in I\}. \] Lemma 2.5 (Proposition 2.1.2 of Clarke (1983)) Let $$h$$ be locally Lipschitz of rank $$K$$ near $$y$$. Then (a) $$\partial h(y)$$ is a non-empty, convex, weak$$^{*}$$-compact subset of $$I^{*}$$ and $$\|y^{*}\|_{I^{*}}\leq K$$ for every $$y^{*}$$ in $$\partial h(y)$$; (b) for every $$z\in I$$, one has $$h^{0}(y,z)=\max\{\langle y^{*},z\rangle: \textrm{for all } y^{*}\in \partial h(y)\}.$$ The following is based on Lemma 3.1 in Fečkan et al. (2013) and Definition 2.5 in Li et al. (2016). Definition 2.6 For each $$u\in L^{2}(J,U)$$, a function $$x\in C(J,H)$$ is a mild solution of (1.1) if $$x(0)=x_{0}$$ and there exists $$f\in L^{2}(J,H)$$ such that $$f(t)\in\partial F(t,x(t))$$ a.e. on $$t\in J$$ and \begin{eqnarray}\label{2.4} x(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f(s)ds\nonumber\\ &&+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds, \quad t\in J. \end{eqnarray} (2.1) where $$\mathscr{T}_E(t):=\int_{0}^{\infty}E^{-1}\xi_{\alpha}(\theta)T(t^{\alpha}\theta)d\theta, \quad \mathscr{L}_E(t):=\alpha\int_{0}^{\infty}E^{-1}\theta\xi_{\alpha}(\theta)T(t^{\alpha}\theta)d\theta,$$ and $$\xi_{\alpha}(\theta)=\frac{1}{\alpha}\theta^{-1-\frac{1}{\alpha}}\varpi_{\alpha}(\theta^{-\frac{1}{\alpha}})\geq0,$$ $\varpi_{\alpha}(\theta)$$=\frac{1}{\pi}\sum_{n=1}^{\infty}(-1)^{n-1}\theta^{-n\alpha-1}\frac{{\it{\Gamma}}(n\alpha+1)}{n!}\sin(n\pi \alpha)$ ,$$\theta\in(0,\infty);$$ here $$\xi_{\alpha}$$ is a probability density function defined on $$(0,\infty),$$ that is $$\xi_{\alpha}(\theta)\geq 0, \theta\in(0,\infty)$$, and $$\int_{0}^{\infty}\xi_{\alpha}(\theta)d\theta=1.$$ Let $$K_{b}(F)=\{x(b)\in H: x(\cdot)$$ is a mild solution of system (1.1) corresponding to a control $$u\in L^{2}(J,U)$$ with initial value $$x_{0}\in H\}$$, which is called the reachable set of (1.1). If $$F\equiv0$$, then this system is called the corresponding linear system of (1.1). In this case, $$K_{b}(0)$$ denotes the reachable set of the linear system. Next recall (see Definitions 2.4 and 2.5 in Kumar & Sukavanam (2012)): Definition 2.7 The system (1.1) is said to be approximately controllable on $$J=[0,b]$$, if $$\overline{K_{b}(F)}=H$$, where $$\overline{K_{b}(F)}$$ denotes the closure of $$K_{b}(F)$$. Clearly, the corresponding linear system is approximately controllable on $$J$$, if $$\overline{K_{b}(0)}=H$$. Lemma 2.8 (Lemma 3.2 of Wang et al. (2014)) Under $$(S_1)$$-$$(S_3)$$, the operators $$\mathscr{T}_E(t)$$ and $$\mathscr{L}_E(t)$$ have the following properties: $${\rm (i)}$$ For any fixed $$t\geq 0, \mathscr{T}_E(t)$$ and $$\mathscr{L}_E(t)$$ are linear operators and bounded operators, i.e. for any $$x \in X$$, $$\big\|\mathscr{T}_E(t)x\big\|\leq M\big\|E^{-1}\big\|\|x\| \textrm{and} \|\mathscr{L}_E(t)x\|\leq \frac{M\|E^{-1}\big\|}{{\it{\Gamma}}(\alpha)}\|x\|.$$ $${\rm (ii)}$$$$\{\mathscr{T}_E(t),t\geq 0\}$$ and $$\{\mathscr{L}_E(t),t\geq 0\}$$ are compact. Finally we recall the following well known fixed point theorem (see Ma (1972)). Theorem 2.9 Let $$I$$ be a Banach space and $$\digamma:I\rightarrow 2^{I} $$ be a compact convex valued, u.s.c. multi-valued map such that there exists a closed neighbourhood $$V$$ of $$0$$ for which $$\digamma(V)$$ is a relatively compact set. If the set $${\it{\Omega}}=\{x\in I:\lambda x\in \digamma(x)\,\,\textrm{for some}\,\,\lambda>1\}$$ is bounded, then $$\digamma$$ has a fixed point. 3. Existence of mild solutions In this section, we study the existence of mild solutions for (1.1). We assume the following: $$H(F):$$$$F:J\times H\rightarrow R$$ is a function such that: (i) the function $$t\mapsto F(t,x)$$ is measurable for all $$x\in H$$; (ii) the function $$x\mapsto F(t,x)$$ is locally Lipschitz for a.e. $$t\in J$$; (iii) there exist a function $$a\in L^{2}(J,R^{+})$$ and a constant $$c>0$$ such that $$\|\partial F(t,x)\|_H =\textrm{sup}\{\|f\|_H :f\in\partial F(t,x)\}\leq a(t)+c\|x\|_{H},$$ for a.e. $$t\in J\,\,\textrm{and all}\,\,x\in H.$$ Now, we define an operator $$\mathcal{N}: L^{2}(J,H)\rightarrow 2^{L^{2}(J,H)}$$ as follows $$\mathcal{N}(x)=\{w\in L^{2}(J,H): w(t)\in \partial F(t;x(t)) \textrm{a.e.} t \in J\}, \textrm{ for } x\in L^{2}(J,H).$$ Lemma 3.1 (Lemma 5.3 of Migórski et al. (2013)) If $$H(F)$$ holds, then for $$x\in L^{2}(J,H),$$ the set $$\mathcal{N}(x)$$ has non-empty, convex and weakly compact values. Lemma 3.2 (Lemma 11 of Migórski & Ochal (2009)) If $$H(F)$$ holds, the operator $$\mathcal{N}$$ satisfies: if $$x_{n}\rightarrow x$$ in $$L^{2}(J,H)$$, $$w_{n}\rightarrow w $$ weakly in $$L^{2}(J,H)$$ and $$ w_{n}\in \mathcal{N}(x_{n})$$, then we have $$w\in \mathcal{N}(x)$$. Theorem 3.3 Assume $$(S_1)$$-$$(S_3)$$ are satisfied. For each $$u\in L^2(J,U)$$, if $$H(F)$$ holds, then (1.1) has a mild solution on $$J$$. Proof. For any $$x\in C(J,H)\subset L^2(J,H)$$, from Lemma 3.1 we can consider the map $$\digamma:C(J,H)\rightarrow 2^{C(J,H)} $$ as follows \begin{eqnarray} \digamma(x)&=&\bigg{\{} h\in C(J,H): h(t)= \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f(s)ds\nonumber\\& & +\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds, f\in \mathcal{N}(x)\bigg{\}}, \textrm{ for } x\in C(J,H). \nonumber\end{eqnarray} We will show $$\digamma$$ has a fixed point using Theorem 2.9. Note $$\digamma(x)$$ is convex from the convexity of $$\mathcal{N}(x)$$. We divide the proof into five steps. Step 1: $$\digamma$$ maps bounded sets into bounded sets in $$C(J,H)$$. For any $$ x\in B_{r}=\{x\in C(J,H):\|x\|_C\leq r\}, r>0$$ and $$\varphi\in\digamma(x)$$, we choose a $$f\in \mathcal{N}(x)$$ with \begin{eqnarray*} \varphi(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f(s)ds\\&& +\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds, \quad t\in J. \nonumber \nonumber\end{eqnarray*} From $$H(F)(iii)$$, Lemma 2.8(i) and the Hölder inequality, we have \begin{eqnarray*} \|\varphi(t)\|_H&\leq& M\|E^{-1}\|\|x_{0}\|_H+\frac{M\|E^{-1}\|cb^{\alpha}r}{{\it{\Gamma}}(1+\alpha)}\\&& +\frac{M\|E^{-1}\|b^{\alpha-\frac{1}{2}}}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)}(\|a\|_{L^{2}(J,R^+)} +\|B\|\|u\|_{L^{2}(J,U)}). \nonumber\end{eqnarray*} Thus $$\digamma(B_{r})$$ is bounded in $$C(J,H)$$. Step 2. $$\{\digamma(x):x\in B_{r}\}$$ is equicontinuous (for all $$r>0$$). For any $$x\in B_{r}, \varphi\in\digamma(x)$$, there exists $$f\in \mathcal{N}(x)$$ such that \begin{eqnarray} \varphi(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)[f(s)+Bu(s)]ds, \quad t\in J. \nonumber \nonumber\end{eqnarray} From $$H(F)(iii)$$ and Lemma 2.8(i), for $$\forall t\in J$$, we have \begin{eqnarray*} &&\|\varphi(t)-\varphi(0)\|_H \\ &\leq &\|\mathscr{T}_E(t)Ex_{0}-x_{0}\|_H+\frac{M\|E^{-1}\|t^{\alpha-\frac{1}{2}}}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)} (\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^2(J,U)})\nonumber \\&&+\frac{M\|E^{-1}\|crt^\alpha}{{\it{\Gamma}}(1+\alpha)}. \nonumber\end{eqnarray*} Thus, for $$\forall\varepsilon>0$$ and for sufficiently small $$\delta_1>0$$, with $$0<t\leq\delta_1$$, we have $$\|\varphi(t)-\varphi(0)\|_H<\frac{\varepsilon}{2}$$. Hence, for $$\forall\varepsilon>0, \forall \tau_1, \tau_2\in [0, \delta_1]$$ and $$\forall\varphi\in\digamma(B_r)$$, we have $$\|\varphi(\tau_2)-\varphi(\tau_1)\|_H<\varepsilon$$. For any $$x\in B_{r}$$ and $$\frac{\delta_1}{2}\leq\tau_{1}<\tau_{2}\leq b$$, we obtain \begin{eqnarray*} & &\|\varphi(\tau_{2})-\varphi(\tau_{1})\|_H\nonumber\\&\leq& \|\mathscr{T}_E(\tau_{2})Ex_{0}-\mathscr{T}_E(\tau_{1})Ex_{0}\|_H\nonumber \\& &+\|\int_{0}^{\tau_{1}}[(\tau_{2}-s)^{\alpha-1}-(\tau_{1}-s)^{\alpha-1}] \mathscr{L}_E(\tau_{2}-s)[f(s)+Bu(s)]ds\|_H\nonumber \\& & +\|\int_{0}^{\tau_{1}}(\tau_{1}-s)^{\alpha-1}[\mathscr{L}_E(\tau_{2}-s)-\mathscr{L}_E(\tau_{1}-s)][f(s)+Bu(s)]ds\|_H\nonumber \\& & +\|\int_{\tau_{1}}^{\tau_{2}}(\tau_{2}-s)^{\alpha-1}\mathscr{L}_E(\tau_{2}-s)[f(s)+Bu(s)]ds\|_H\nonumber \\& & := Q_{1}+Q_{2}+Q_{3}+Q_{4}. \nonumber\end{eqnarray*} Clearly, \begin{eqnarray*} Q_1\leq\|Ex_{0}\|_H\|\mathscr{T}_E(\tau_{2})-\mathscr{T}_E(\tau_{1})\|_H. \end{eqnarray*} From $$H(F)(iii)$$, Lemma 2.8(i) and the Hölder’s inequality, we obtain \begin{eqnarray*} Q_{2}&\leq &\frac{M\|E^{-1}\|(\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^{2}(J,U)})}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)} [\tau_{1}^{2\alpha-1}-\tau_{2}^{2\alpha-1}-(\tau_{2}-\tau_{1})^{2\alpha-1}]^{\frac{1}{2}}\nonumber \\& &+ \frac{M\|E^{-1}\|cr}{{\it{\Gamma}}(1+\alpha)}[\tau_{2}^{\alpha}-\tau_{1}^{\alpha}+(\tau_{2}-\tau_{1})^{\alpha}]. \nonumber\end{eqnarray*} Taking $$\delta_2>0$$ small enough, we have \begin{eqnarray*} &&Q_{3}\\&\leq&\textrm{sup}_{s\in[0,\tau_{1}-\delta_2]}\|\mathscr{L}_E(\tau_{2}-s)-\mathscr{L}_E(\tau_{1}-s)\|\\&&\times \bigg{[}\frac{\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^{2}(J,U)}}{\sqrt{2\alpha-1}} (\tau_{1}^{\alpha-\frac{1}{2}}-\delta_2^{\alpha-\frac{1}{2}})\nonumber +\frac{cr}{\alpha}(\tau_{1}^{\alpha}-\delta_2^{\alpha})\bigg{]} \\ &&+\frac{2M\|E^{-1}\|(\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^{2}(J,U)})}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)} \delta_2^{\alpha-\frac{1}{2}}+\frac{2M\|E^{-1}\|cr}{{\it{\Gamma}}(1+\alpha)}\delta_2^{\alpha}. \nonumber\end{eqnarray*} Also, we have \begin{eqnarray*} Q_{4}&\leq&\frac{M\|E^{-1}\|(\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^{2}(J,U)})}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)} (\tau_{2}-\tau_{1})^{\alpha-\frac{1}{2}}\\&&+\frac{M\|E^{-1}\|cr}{{\it{\Gamma}}(1+\alpha)}(\tau_{2}-\tau_{1})^{\alpha}. \nonumber\end{eqnarray*} From the compactness of $$T(t)(t>0)$$ and Definition 2.6, we see that the operator $$\mathscr{L}_E(t)(t>0)$$ is continuous in the uniform operator topology, and thus, for any $$x\in B_{r}$$, $$Q_{3}$$ tends to zero as $$\tau_{2}\rightarrow \tau_{1}, \delta_2\rightarrow0$$. Also note $$Q_{i}(i=1,2,4)$$ tends to zero as $$\tau_{2}\rightarrow\tau_{1}$$. Thus we get that $$\|\varphi(\tau_{2})-\varphi(\tau_{1})\|_H$$ tends to zero as $$\tau_{2}\rightarrow \tau_{1}$$ and $$\delta_2\rightarrow0$$. Let $$\delta=\min\{\delta_1, \delta_2\}$$. For $$\forall \varepsilon>0, \forall \tau_1, \tau_2\in [0, b], |\tau_1-\tau_2|<\delta,\forall \varphi\in\digamma(B_r)$$, one sees that $$\|\varphi(\tau_{2})-\varphi(\tau_{1})\|_H<\varepsilon$$ independently of $$x\in B_r$$. Therefore, we deduce that $$\{\digamma(x):x\in B_{r}\}$$ is an equicontinuous family of functions in $$C(J,H)$$. Step 3: $$\digamma$$ is completely continuous. We prove that for $$\forall t\in J$$, $$r>0$$, the set $${\it{\Pi}}(t)=\{\varphi(t):\varphi\in \digamma(B_{r})\}$$ is relatively compact in $$H$$. Obviously, $${\it{\Pi}}(0)=\{x_{0}\}$$ is compact, so we only need to consider $$t>0$$. Let $$0<t\leq b$$ be fixed. For any $$x\in B_{r}, \varphi\in\digamma(x)$$, we choose $$f\in \mathcal{N}(x)$$ with \begin{eqnarray} \varphi(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)[f(s)+Bu(s)]ds, \quad t\in J. \nonumber \nonumber\end{eqnarray} For each $$\epsilon\in(0,t), t\in(0,b], x\in B_{r}$$ and any $$\delta>0$$, we define \begin{eqnarray*} \varphi^{\epsilon,\delta}(t)&=&\mathscr{T}_E(t)Ex_{0}+\alpha \int_{0}^{t-\epsilon}\int_{\delta}^{\infty} E^{-1}\theta(t-s)^{\alpha-1}\xi_{\alpha}(\theta)\\&&\times T((t-s)^{\alpha}\theta)[f(s)+Bu(s)]d\theta ds. \nonumber\\ &=&\mathscr{T}_E(t)Ex_{0}+\alpha T(\epsilon^{\alpha}\delta)\int_{0}^{t-\epsilon}\int_{\delta}^{\infty} E^{-1}\theta(t-s)^{\alpha-1}\xi_{\alpha}(\theta)\\&&\times T((t-s)^{\alpha}\theta-\epsilon^{\alpha}\delta)[f(s)+Bu(s)]d\theta ds. \nonumber \nonumber\end{eqnarray*} From the compactness of $$\mathscr{T}_E(t)(t>0)$$ (see Lemma 2.8(ii)) we see that the set $${\it{\Pi}}_{\epsilon,\delta}(t)=\{\varphi^{\epsilon,\delta}(t):\varphi\in \digamma(B_{r})\},$$ is relatively compact in $$H$$ for each $$\epsilon\in(0,t)$$ and $$\delta>0$$. Moreover, we have \begin{eqnarray} & &\|\varphi(t)-\varphi^{\epsilon,\delta}(t)\|_H\nonumber\\ &\leq&\frac{\alpha M\|E^{-1}\|}{\sqrt{2\alpha-1}}\bigg{(}\|a\|_{L^{2}(J,R^+)}+\|B\|\|u\|_{L^{2}(J,U)}\bigg{)} \bigg{[}b^{\alpha-\frac{1}{2}}\int_{0}^{\delta}\theta\xi_{\alpha}(\theta)d\theta\nonumber\\& & +\frac{1}{{\it{\Gamma}}(1+\alpha)}\epsilon^{\alpha-\frac{1}{2}}\bigg{]} +M\|E^{-1}\|cr\bigg{[}\frac{1}{{\it{\Gamma}}(1+\alpha)}\epsilon^{\alpha} +b^{\alpha}\int_{0}^{\delta}\theta\xi_{\alpha}(\theta)d\theta\bigg{]}. \nonumber\end{eqnarray} Now since $$0\leq\int_{0}^{\delta}\theta\xi_{\alpha}(\theta)d\theta\leq \int_{0}^{\infty}\theta\xi_{\alpha}(\theta)d\theta=\frac{1}{{\it{\Gamma}}(1+\alpha)}$$, the inequality above tends to zero when $$\epsilon\rightarrow 0$$ and $$\delta\rightarrow 0.$$ Therefore, the set $${\it{\Pi}}(t) (t>0)$$ is totally bounded, i.e., relatively compact in $$H$$. From above (and Step 2) and the Ascoli–Arzela Theorem, we see that $$\digamma$$ is completely continuous. Step 4: $$\digamma$$ has a closed graph. Let $$x_{n}\rightarrow x_{*}$$ as $$n\rightarrow \infty $$ in $$C(J,H)$$, $$\varphi_{n}\in \digamma(x_{n})$$ and $$\varphi_{n}\rightarrow \varphi_{*}$$ as $$n\rightarrow \infty $$ in $$C(J,H)$$. We prove that $$\varphi_{*}\in \digamma(x_{*}).$$ Now, $$\varphi_{n}\in \digamma(x_{n})$$ so there exists $$f_{n}\in \mathcal{N}(x_{n})$$ with \begin{eqnarray} \label{3.4} \varphi_{n}(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f_{n}(s)ds\nonumber \\&&+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds. \end{eqnarray} (3.1) From $$H(F)(iii)$$, $$\{f_{n}\}_{n\geq1}\subseteq L^{2}(J,H)$$ is bounded. Hence we assume that \begin{eqnarray}\label{3.5} f_{n}\rightarrow f_{*}, \textrm{ weakly in } L^{2}(J,H), \end{eqnarray} (3.2) From (3.1), (3.2) and the compactness of the operator $$\mathscr{L}_E(t)$$, we have that \begin{eqnarray}\label{3.6} \varphi_{n}(t)&\rightarrow & \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f_{*}(s)ds\nonumber\\&& +\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds. \end{eqnarray} (3.3) Note that $$\varphi_{n}\rightarrow \varphi_{*}$$ in $$C(J,H)$$ and $$f_{n}\in \mathcal{N}(x_{n})$$. From Lemma 3.2, we obtain $$f_{*}\in \mathcal{N}(x_{*})$$. Hence, $$\varphi_{*}\in \digamma(x_{*}),$$ which implies $$\digamma$$ has closed graph. From Proposition 3.3.12(2) of Migórski et al. (2013), $$\digamma$$ is u.s.c. Step 5: Apriori estimate. From Steps 1–4, we have that $$\digamma$$ is u.s.c. and is compact convex valued and $$\digamma(B_r)$$ is a relatively compact set (here $$r>0$$). We now prove that the set $${\it{\Omega}}=\{x\in C(J,H): \lambda x\in \digamma (x), \lambda>1\}$$ is bounded. For $$\forall x\in{\it{\Omega}}$$, there exists $$f\in \mathcal{N}(x)$$ with \begin{eqnarray*} x(t)&=& \lambda^{-1}\mathscr{T}_E(t)Ex_{0}+\lambda^{-1}\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f(s)ds\\&& +\lambda^{-1}\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu(s)ds. \nonumber\end{eqnarray*} Then from assumption $$H(F)(iii)$$, we derive \begin{eqnarray}\label{3.7} \|x(t)\|_H\leq \rho+\frac{M\|E^{-1}\|c}{{\it{\Gamma}}(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\|x(s)\|_Hds, \nonumber\end{eqnarray} where \begin{eqnarray} \rho=M\|E^{-1}\|\|x_0\|_H +\frac{M\|E^{-1}\|b^{\alpha-\frac{1}{2}}}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)}\bigg{(}\|a\|_{L^2(J,R^+)}+\|B\|\|u\|_{L^2(J,U)}\bigg{)}. \nonumber\end{eqnarray} It follows from (3.4) and Corollary 2 in Ye et al. (2007) that $$\|x(t)\|_H\leq \rho E_\alpha(M\|E^{-1}\|ct^\alpha).$$ Hence, $$\|x\|_C=\sup_{t\in J}\|x(t)\|_H\leq \rho E_\alpha(M\|E^{-1}\|cb^\alpha)$$, which implies the set $${\it{\Omega}}$$ is bounded. From Theorem 2.9, $$\digamma$$ has a fixed point, i.e., (1.1) has a mild solution. The proof is complete. □ 4. Approximate controllability of a linear system In this section, we consider the approximate controllability of the following linear fractional control system of Sobolev type: \begin{eqnarray}\label{4.1} \left\{\begin{array}{@{}llll} ^{C}D^{\alpha}_{t}(Ex(t))+ Ax(t)=Bu(t), t\in J=[0,b], \frac{1}{2}<\alpha<1,\\ x(0)=x_{0}. \end{array}\right. \end{eqnarray} (4.1) Define a bounded linear operator $$\mathcal {K}:L^{2}(J,H)\rightarrow H$$ as follows: $$\mathcal {K}h=\int_{0}^{b}(b-s)^{\alpha-1}\mathscr{L}_E(b-s)h(s)ds, \quad h(\cdot)\in L^{2}(J,H).$$ We assume the following: $$H(B):$$ For each $$h(\cdot)\in L^{2}(J,H)$$, there exists a function $$g(\cdot)\in \overline{R(B)}$$, such that $$\mathcal{K}h=\mathcal{K}g,$$ where $$R(B)$$ denotes the range of operator $$B$$ and $$\overline{R(B)}$$ is the closure of $$R(B)$$. Theorem 4.1 Suppose $$H(B)$$ is satisfied. Then (4.1) is approximately controllable on $$J$$ provided that $$T(t)$$ is a differentiable semi-group. Proof. The idea comes from Kumar & Sukavanam (2012). Since the domain $$D(A)$$ of the operator $$A$$ is dense in $$H$$, it is sufficient to show that $$D(A)\subset \overline{K_{b}(0)},$$ i.e. for any $$\epsilon>0$$ and $$\eta\in D(A)$$, there exists a control function $$u\in L^{2}(J,U)$$, such that \begin{eqnarray}\label{4.2} \|\eta-\mathscr{T}_E(b)Ex_{0}-\mathcal{K}Bu\|_{H}<\epsilon. \end{eqnarray} (4.2) For any $$x_{0}\in H$$, $$T(t)$$ is a differentiable semi-group, so $$T(t)Ex_{0}\in D(A)$$, such that $$\mathscr{T}_E(b)Ex_{0}\in D(A)$$. For a given $$ \eta\in D(A)$$, we see that there exists a function $$h(\cdot)\in L^{2}(J,H)$$, such that $$\mathcal{K}h=\eta-\mathscr{T}_E(b)Ex_{0}$$. for example, one can choose $$h(t)=\frac{E^2[{\it{\Gamma}}(\alpha)]^{2}(b-t)^{1-\alpha}}{b}\bigg{[}\mathscr{L}_E(b-t)+2t \frac{d \mathscr{L}_E(b-t)}{dt}\bigg{]}[\eta-\mathscr{T}_E(b)Ex_{0}];$$ note that, $$\mathcal{K}h=\frac{E^2[{\it{\Gamma}}(\alpha)]^{2}}{b}\big[\eta-\mathscr{T}_E(b)Ex_{0}\big](b\mathscr{L}_E^{2}(0)).\nonumber$$ From Definition 2.6, $$\mathscr{L}_E(0)=\frac{E^{-1}}{{\it{\Gamma}}(\alpha)}$$. Further, $$\mathcal{K}h=\eta-\mathscr{T}_E(b)Ex_{0}.$$ Next, we prove there is a control function $$u_\varepsilon\in L^{2}(J,U)$$ such that (4.2) holds. From assumption $$H(B)$$, for $$h(\cdot)\in L^{2}(J,H)$$, there exists a function $$g\in \overline{R(B)}$$ with $$\mathcal{K}h=\int_{0}^{b}(b-s)^{\alpha-1}\mathscr{L}_E(b-s)h(s)ds=\int_{0}^{b}(b-s)^{\alpha-1}\mathscr{L}_E(b-s)g(s)ds.$$ Since $$g\in \overline{R(B)}$$, for a given $$\varepsilon>0$$, there exists a control function $$u_\varepsilon\in L^{2}(J,U)$$ such that $$\|Bu_{\varepsilon}-g\|_{L^{2}(J,U)}<\frac{{\it{\Gamma}}(\alpha)}{M\|E^{-1}\|}\sqrt{2\alpha-1} b^{\frac{1}{2}-\alpha}\varepsilon.$$ Then for $$\varepsilon>0, u_\varepsilon\in L^{2}(J,U)$$, from the above arguments, we have \begin{eqnarray*} \|\eta-\mathscr{T}_E(b)Ex_{0}-\mathcal{K}Bu_\varepsilon\|_{H} &=&\|\mathcal{K}h-\mathcal{K}Bu_\varepsilon\|_{H} \nonumber\\ &\leq& \frac{M\|E^{-1}\| b^{\alpha-\frac{1}{2}}}{\sqrt{2\alpha-1}{\it{\Gamma}}(\alpha)}\|Bu_{\varepsilon}-g\|_{L^2(J,U)}<\varepsilon. \nonumber\end{eqnarray*} Since $$\varepsilon$$ is arbitrary, we deduce that $$D(A)\subset \overline{K_{b}(0)}.$$ The denseness of the domain $$D(A)$$ in $$H$$ implies the approximate controllability of (4.1) on $$J$$. This completes the proof. □ For Section 5, we consider two relevant operators associated with (4.1): $${\it{\Gamma}}_{0}^{b}=\int_{0}^{b}(b-s)^{\alpha-1}\mathscr{L}_E(b-s)BB^{*}\mathscr{L}_E^{*}(b-s)ds$$ and $$R(\varepsilon,{\it{\Gamma}}_{0}^{b})=(\varepsilon {\mathbb I}+{\it{\Gamma}}_{0}^{b})^{-1}, \varepsilon>0,$$ respectively, where $$B^{*}$$ denotes the adjoint of $$B$$ and $$\mathscr{L}_E^{*}(t)$$ is the adjoint of $$\mathscr{L}_E(t)$$. Lemma 4.2 (Lemma 2.10 of Liu & Li (2015)). The linear fractional control system (4.1) is approximately controllable on $$J$$ if and only if $$\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})\rightarrow 0$$ as $$\varepsilon\rightarrow 0^{+}$$ in the strong operator topology. 5. Approximate controllability for the semi-linear case In this section, we consider the approximate controllability of (1.1). For any $$x\in C(J,H)\subset L^2(J,H)$$, from Lemma 3.1, we know that $$\mathcal{N}(x)\not=\emptyset$$. Now, for any $$\varepsilon>0$$ and $$x_{1}\in H$$ is fixed, we consider the map $$\digamma_{\varepsilon}:C(J,H)\rightarrow 2^{C(J,H)} $$ given by: \begin{eqnarray} \digamma_{\varepsilon}(x)&=&\bigg{\{} h\in C(J,H): h(t)= \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f(s)ds\nonumber\\& & +\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)Bu_{\varepsilon}(s)ds, f\in \mathcal{N}(x)\bigg{\}}, \nonumber\end{eqnarray} where $$ u_{\varepsilon}(t)=B^{*}\mathscr{L}_E^{*}(b-t) R(\varepsilon,{\it{\Gamma}}_{0}^{b})\bigg{(}x_{1}-\mathscr{T}_E(b)Ex_{0} -\int_{0}^{b}(b-\tau)^{\alpha-1}\mathscr{L}_E(b-\tau)f(\tau)d\tau\bigg{)}. $$ Theorem 5.1 Suppose $$H(F)$$ (i), (ii) are satisfied and there exists a function $$\psi\in L^{2}(J,R^{+})$$ such that $$\|\partial F(t,x)\|_{H}\leq \psi(t),\,\,\textrm{for a.e.}\,\,t\in J,\,\,\textrm{all}\,\,x\in H.$$ Then $$\digamma_\varepsilon$$ has a fixed point on $$J$$. The proof is similar to that in Theorem 3.3 so we omit it. Theorem 5.2 Assume that the assumptions of Theorem 5.1 hold, and in addition, suppose (4.1) is approximately controllable. Then the system (1.1) is approximately controllable on $$J$$. Proof. Let $$x_{\varepsilon}$$ be a fixed point of $$\digamma_\varepsilon(x)$$ in $$C(J,H)$$. Then, there exists $$f_{\varepsilon}\in \mathcal{N}(x_{\varepsilon})$$ such that for each $$t\in J,$$ \begin{eqnarray*} x_{\varepsilon}(t)&=& \mathscr{T}_E(t)Ex_{0}+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)f_{\varepsilon}(s)ds \\&&+\int_{0}^{t}(t-s)^{\alpha-1}\mathscr{L}_E(t-s)BB^{*}\mathscr{L}_E^{*}(b-s)\nonumber\\& &\times R(\varepsilon,{\it{\Gamma}}_{0}^{b})\bigg{(}x_{1}-\mathscr{T}_E(b)Ex_{0} -\int_{0}^{b}(b-\tau)^{\alpha-1}\mathscr{L}_E(b-\tau)f_{\varepsilon}(\tau)d\tau\bigg{)}ds. \nonumber \nonumber\end{eqnarray*} Since $${\mathbb I}-{\it{\Gamma}}_{0}^{b}R(\varepsilon,{\it{\Gamma}}_{0}^{b})=\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b}),$$ we get $$x_\varepsilon(b)=x_{1}-\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})G(f_{\varepsilon}),$$ where $$G(f_{\varepsilon})=x_{1}-\mathscr{T}_E(b)Ex_{0} -\int_{0}^{b}(b-\tau)^{\alpha-1}\mathscr{L}_E(b-\tau)f_{\varepsilon}(\tau)d\tau.$$ Since $$\|\partial F(t,x)\|_H\leq\psi(t)$$, we have $$\int_{0}^{b}\|f_{\varepsilon}(s)\|ds\leq \|\psi\|_{L^{2}(J,R^+)}\sqrt{b}.$$ Thus, the sequence $$\{f_{\varepsilon}\}$$ is bounded in $$L^{2}(J,H)$$, so there is a subsequence, still denoted by $$\{f_{\varepsilon}\}$$, which converges weakly to $$f$$, that is $$f_{\varepsilon}\xrightarrow{w} f$$. Let $$h=x_{1}-\mathscr{T}_E(b)Ex_{0}-\int_{0}^{b}(b-\tau)^{\alpha-1}\mathscr{L}_E(b-\tau)f(\tau)d\tau.$$ The linear system (4.1) is approximately controllable, so from Lemma 4.2, as $$\varepsilon \to 0^{+}$$, we have $$\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})\to 0.$$ It follows that \begin{eqnarray} \|G(f_{\varepsilon})-h\| & \leq&\sup_{0\leq t\leq b}\|\int_{0}^{t}(t-\tau)^{\alpha-1}\mathscr{L}_E(t-\tau)\big[f_{\varepsilon}(\tau)-f(\tau)\big]d\tau\|. \nonumber\end{eqnarray} The compactness of the operator $$\mathscr{L}_E(t),t>0$$ guarantees that $$\|G(f_{\varepsilon})-h\|\to 0$$ as $$\varepsilon\rightarrow 0^{+}$$. From the above arguments, we also get \begin{eqnarray} \|x_{\varepsilon}(b)-x_{1}\| &\leq&\|\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})(h)\|+\|\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})[G(f_{\varepsilon})-h]\|\nonumber\\ &\leq&\|\varepsilon R(\varepsilon,{\it{\Gamma}}_{0}^{b})(h)\|+\|G(f_{\varepsilon})-h\|\rightarrow0,\,\,\textrm{as}\,\,\varepsilon\rightarrow 0^{+}, \nonumber\end{eqnarray} and this proves the approximately controllability of (1.1) on $$J$$. □ Remark 5.3 From the above conclusion, we find that it is generalizes the results in Li et al. (2016). In fact, if $$E={\mathbb I}$$, the results are similar to Li et al. (2016). 6. Examples Example 6.1 Motivated by an example in Fečkan et al. (2013), we consider the following fractional heat equation: \begin{eqnarray}\label{6.1} \left\{\begin{array}{@{}llll} ^CD^{3/4}_{t}(x(t,y)-x_{yy}(t,y))=x_{yy}(t,y)+Bu(t,y)+\phi(t,y),\\ 0<t<1, 0<y<\pi, x(t,0)=x(t,\pi)=0, 0<t<1,\\ x(0,y)=x_0(y), 0<y<\pi, \end{array}\right. \end{eqnarray} (6.1) where $$x(t,y)$$ represents the temperature at the point $$y\in (0,\pi)$$ and time $$t\in (0,1)$$. Now, set $$H=L^2[0,\pi]$$ and $$e_n(y)=\sqrt{2/\pi}\sin(ny), n=1,2,\cdots.$$ Then $$\{e_n(y)\}$$ is an orthonormal base for $$H$$. Define $$A:D(A)\subset X\rightarrow H$$ by $$Ax=-x_{yy}$$ and $$E:D(E)\subset X\rightarrow H$$ by $$Ex:=x-x_{yy}$$ with domain \begin{eqnarray}\label{6.1-dom} \{x\in H: x, x'\textrm{ are absolutely continuous}, x''\in H, x(0)=x(\pi)=0\}. \end{eqnarray} (6.2) Then $$Ax=\sum_{n=1}^\infty n^2\langle x,e_n\rangle e_n, x\in D(A)$$ and $$Ex=\sum_{n=1}^\infty(1+n^2)\langle x,e_n\rangle e_n, x\in D(A).$$ Furthermore, for $$x\in X$$, we have $$E^{-1}x=\sum_{n=1}^\infty\frac{1}{1+n^2}\langle x,e_n\rangle e_n, \quad -AE^{-1}x=\sum_{n=1}^\infty\frac{-n^2}{1+n^2}\langle x,e_n\rangle e_n x\in D(A).$$ It is well known that $$-AE^{-1}$$ generates a compact semi-group $$T(t)(t>0)$$ on $$H$$ and \begin{eqnarray}\label{6.1-T} T(t)x=\sum_{n=1}^\infty e^{\frac{-n^2}{1+n^2}t}\langle x,e_n\rangle e_n, x\in H. \end{eqnarray} (6.3) It is easy to see that $$T(t)$$ is compact with $$\|T(t)\|\leq e^{-t}\leq 1$$, $$E^{-1}$$ is also compact and bounded with $$\|E^{-1}\|\leq1$$. Moreover, the two operators $$\mathscr{T}_E(t)$$ and $$\mathscr{L}_E(t)$$ can be defined by $$\mathscr{T}_E(t)=\int_{0}^{\infty}E^{-1}\xi_{\frac{3}{4}}(\theta)T(t^{\frac{3}{4}}\theta)d\theta, \quad \mathscr{L}_E(t)=\frac{3}{4}\int_{0}^{\infty}E^{-1}\theta\xi_{\frac{3}{4}}(\theta)T(t^{\frac{3}{4}}\theta)d\theta.$$ We easily get, $$\|\mathscr{T}_E(t)\|\leq1, \quad \|\mathscr{L}_E(t)\|\leq\frac{1}{{\it{\Gamma}}(\frac{3}{4})}.$$ Let the infinite dimensional Hilbert space $$U$$ be defined by $$U:=\{u:u=\sum_{n=2}^\infty u_ne_n, \sum_{n=2}^\infty u_{n}^2<\infty\}.$$ The norm in $$U$$ is defined by $$\|u\|_U=(\sum_{n=2}^\infty u_n^2)^{1/2}$$. Define a mapping $$B\in \mathcal{L}(U,H)$$ as follows: $$ Bu=2u_2e_1+\sum_{n=2}^\infty u_ne_n \quad \textrm{ for } u=\sum_{n=2}^\infty u_ne_n\in U, $$ and for $$v=\sum_{n=1}^\infty v_ne_n\in H$$, inner product $$\langle Bu,v\rangle=\langle u,B^*v\rangle$$, thus $$ B^*v=(2v_1+v_2)e_2+\sum_{n=3}^\infty v_ne_n$$ and $$ B^*T^*(t)x=(2x_1e^{-t}+x_2e^{-4t})e_2+\sum_{n=3}^\infty e^{\frac{-n^2}{1+n^2}}x_ne_n. $$ It follows that $$\|B^*T^*(t)x\|=0$$ for some $$t\in J$$, which implies that $$x=0$$, and thus the linear part of (6.1) is approximate controllable on $$J$$ (see Theorem 4.1.7 of Curtain & Zwart (1995)). Then all hypotheses in Theorem 5.2 are satisfied, so (6.1) is approximate controllable. Example 6.2 Consider the partial differential system of the form \begin{eqnarray}\label{6.2} \left\{\begin{array}{@{}llll} ^CD^{4/5}_{t}(x(t,\theta)-x_{\theta\theta}(t,\theta))=x_{\theta\theta}(t,y)+\bar{b}(\theta)u(t)+\phi(t,\theta),\\ 0<t<1, 0<\theta<\pi, x(t,0)=x(t,\pi)=0, 0<t<1,\\ x(0,\theta)=x_0(\theta), \quad 0<\theta<\pi, \end{array}\right. \end{eqnarray} (6.4) where $$u\in H=L^2[0,\pi]$$. Let $$e_n(\theta)=\sqrt{2/\pi}\sin(n\theta), n=1,2,\cdots.$$ Then $$\{e_n(\theta)\}$$ is an orthonormal base for $$H$$. Let $$B$$ can be define as $$(Bu)(\theta)=\bar{b}(\theta)u,$$$$(B^*v)(\theta)=\sum_{n=0}^{\infty}\langle \bar{b},e_n\rangle \langle v,e_n\rangle.$$ Define $$A:D(A)\subset X\rightarrow H$$ by $$Ax=-x_{\theta\theta}$$ and $$E:D(E)\subset X\rightarrow H$$ by $$Ex:=x-x_{\theta\theta}$$ with the domain (6.2). Like Example 6.1, $$-AE^{-1}$$ generates a compact semi-group $$T(t)(t>0)$$ on $$H$$ given in (6.3) and $$\mathscr{L}_E(t)$$ can be defined by $$\mathscr{L}_E(t)x=\frac{4}{5}\sum_{n=1}^\infty \int_{0}^{\infty}E^{-1}\theta\xi_{\frac{4}{5}}(\theta)e^{\frac{-n^2}{1+n^2}t^{\frac{4}{5}}\theta}d\theta\langle x,e_n\rangle e_n.$$ To prove the system (6.4) is approximately controllable, we will show that $$(b-s)^{\alpha-1}B^*\mathscr{L}_E(b-s)x=0\Rightarrow x=0$$. In fact, \begin{eqnarray} &&(b-s)^{\alpha-1}B^*\mathscr{L}_E(b-s)x\nonumber\\&=&(b-s)^{\alpha-1}\frac{4}{5}\sum_{n=1}^\infty \int_{0}^{\infty}E^{-1}\theta\xi_{\frac{4}{5}}(\theta)e^{\frac{-n^2}{1+n^2}(b-s)^{\frac{4}{5}}\theta}d\theta\langle \bar{b},e_n\rangle \langle x,e_n\rangle=0. \nonumber\end{eqnarray} Thus, provided that $$\langle \bar{b},e_n\rangle= \int_0^{\pi}\bar{b}e_n(\theta)d\theta \neq 0$$, so $$\langle x,e_n\rangle=0\Rightarrow x=0$$. Therefore, the system (6.4) is approximately controllable (Curtain & Zwart, 1995) provided that $$\langle \bar{b},e_n\rangle \neq 0$$ for $$n=1,2,3\cdots$$. 7. Conclusion Approximate controllability for fractional evolution inclusions of Sobolev and Clarke subdifferential type has been investigated. Utilizing two characteristic solution operators and their fundamental properties via a fixed point theorem for multi-valued maps, sufficient conditions for the mild solution of (6.1) are proved, and using two related operators, we studied the approximate controllability. In the future, we will study the controllability of relative problems by using the techniques of the measure of non-compactness. For example, one can consider approximation controllability of impulsive fractional problems (Wang et al., 2016), fractional damped equations (Wang et al., 2017), fractional delay problems (Li & Wang, 2017b) and fractional Navier-Stokes equations (Zhou & Peng, 2017a,b). 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