Approximate controllability for a retarded semilinear stochastic evolution system

Approximate controllability for a retarded semilinear stochastic evolution system Abstract In this work, we study the approximate controllability for a class of semilinear stochastic evolution systems with finite delays in $$L_p$$ space. The main technique is the fundamental solution theory constructed through Laplace transformation. The approximate controllability result is obtained via the so-called resolvent condition. The nonlinear terms are only required to be partly uniformly bounded. An example is provided to illustrate the obtained results. 1. Introduction This article is concerned with the approximate controllability of the following stochastic partial differential equation (SPDE) with finite delay   \begin{equation}\label{eq41} \left\{\begin{aligned} &d y(t) =\big(-Ay(t)+L(y_t)+ f(t,y_t)+Bu(t)\big)dt+g(t,y_t)dW(t), \,\,\,\, 0\leq t\leq T,\\ &y_0=\phi \in ML(p,r,\alpha), \end{aligned}\right. \end{equation} (1) on a Hilbert space $$X$$. Here, $$y(t)$$ is the state variable and its histories $$y_{t}$$ are given in the usual way by $$y_{t}(\theta)=y(t+\theta)$$, for $$\theta\in[-r,0]$$, and belong to the phase space $$ML(p,r,\alpha)$$ to be defined later. The control function $$u(\cdot)$$ takes values in another Hilbert space $$U$$. The operator $$-A: D(-A)\subseteq X\to X$$ is the infinitesimal generator of an analytic semigroup $$\left(S(t)\right)_{t\geq 0}$$ while the operators $$L$$ and $$B$$ are linear bounded. The functions $$f(\cdot,\cdot)$$ and $$g(\cdot,\cdot)$$ are Lipschitz continuous and uniformly bounded given below, and $$W(t)$$ is a $$Q-$$Wiener process. System (1) is an abstract form of SPDEs in infinite dimensional space. This kind of equations arises naturally in the mathematical models of various phenomena in natural and social science; see DA Prato & Zabbczyk (1992), Grecksch & Tudor (1995) and Sobczyk (1991), for some practical background of this kind of equations. Recently, quantitative and qualitative properties (existence, stability, invariant measures, etc.) of SPDEs with finite delays have gained much attention of many mathematicians, see Bezandry & Diagana (2012), Caraballo (1990), Lei et al. (2007), Liu (2014), Taniguchi (1998), Taniguchi et al. (2002) and You & Yuan (2014) among others. In particular, the controllability problems of finite-delayed SPDEs have also received the attention of many authors, see Dauer & Mahmudov (2004), Ganthi & Muthukumar (2012), Muthukumar & Balasubramaniam (2009) and Sakthivel et al. (2012) for instance. In reference Dauer & Mahmudov (2004), the authors have investigated the approximate controllability of the following stochastic evolution equation with finite delay   \begin{equation}\label{eq52} \left\{\begin{aligned} &d y(t) =\big(-Ay(t)+ f(t,y_t)+Bu(t)\big)dt+g(t,y_t)dW(t),\,\,\,\, 0\leq t\leq T,\\ &y_0=\phi, \end{aligned}\right. \end{equation} (2) by using the $$C_0-$$semigroup theory and the resolvent condition. The so-called resolvent condition (see condition $$(H_5)$$ in Section 3), originating from Bashirov & Mahmudov (1999), is equivalent to the approximate controllability of the corresponding linear system, and it is an important approach to study the approximate controllability of (stochastic) semilinear evolution systems. In recent years, this condition has been extensively adopted to study the controllability problems for various deterministic and stochastic semilinear evolution systems. For instance, in references Balasubramaniam & Tamilalagan (2015), Tamilalagan & Balasubramaniam (2017), Ren et al. (2013), Sakthivel et al. (2011, 2012, 2013, 2016) and Shen & Sun (2012), the authors have successfully employed this resolvent condition to explore the approximate controllability respectively for deterministic and stochastic (fractional) differential systems with nonlocal conditions or with finite (infinite) delays. The outstanding feature of this approach is that the nonlinear terms in the considered systems should satisfy the uniform boundedness condition, see Balasubramaniam & Tamilalagan (2015), Dauer & Mahmudov (2004), Dauer et al. (2006), Fu et al. (2014), Ganthi & Muthukumar (2012), Mokkedem & Fu (2016), Mokkedem & Fu (2017), Sakthivel et al. (2011, 2012, 2013, 2016) and Shen & Sun (2012) and the references therein. We also point out that another method to discuss the approximate controllability for semilinear evolution systems is to apply the range condition initiated in Naito (1987). In Muthukumar & Balasubramaniam (2009) and Wang (2009), for example, the authors obtained the controllability results by using exactly this technique. However, one can see that generally it is difficult to verify such a range condition in practice, especially for (stochastic) functional differential equations (FDEs) with time delay (including the system (22) to be discussed in Section 5 since the initial function $$\phi$$ is not null). The motivation of this article is the approximate controllability of stochastic partial functional equations which can be rewritten as the abstract form (1), like the retarded stochastic heat conduction system (22). We will study this problem through investigating the approximate controllability for the system (1). It should be observed that, there is an extra linear term $$L(y_t)$$ in System (1) which is not uniformly bounded and hence it can not be regarded as a special case of System (2). As a result, the existing results in Dauer & Mahmudov (2004) and Dauer et al. (2006) on approximate controllability become invalid for System (1). Moreover, the theory of $$C_0-$$semigroup is not enough to solve such problem, since the semigroup generated by $$A+L$$ loses the compactness property. For this reason, instead of using the $$C_0-$$semigroup theory we study the approximate controllability of System (1) by adopting the fundamental solution one on the associated deterministic linear system with finite delay which will be founded in a similar way as that in Fu et al. (2014). By the construction of the fundamental solution $$G(t)$$, the linear term $$L\left(y_t\right)$$ is involved into $$G(t)$$ and it does not appear in the expression of mild solutions of System (1). In this manner, we overcome the obstacle of the non-uniformly boundedness of $$L\left(y_t\right)$$. Note the fundamental solution theory has been constructed for some linear (neutral) FDEs and has already been utilized to discuss the control problems for corresponding semilinear (neutral) FDEs in these years, and many interesting results are obtained in various topics, see Fu et al. (2014), Jeong et al. (1999), Jeong & Roh (2006), Liu (2009), Mokkedem & Fu (2016), Mokkedem & Fu (2017) and Wang (2009) among others. In particular, in article Mokkedem & Fu (2017) we have studied the approximate controllability for a semiliear stochastic FDEs with infinite delay via this technique. In this work, the analyticity of the fundamental solution $$G(t)$$ is discussed which allows us to employ the $$\alpha-$$norm and the fractional power operators approaches. In addition, results on compactness and uniform continuity of $$G(t)$$ and $$A^\alpha G(t)$$, for $$t>0$$ and $$\alpha \in (0,1)$$, are achieved as well. By these properties, the regularity of solutions of System (1) is improved and our obtained results become more meaningful. Clearly, the obtained results extend and develop many existing works such as Dauer & Mahmudov (2004) and Muthukumar & Balasubramaniam (2009). Moreover, the fundamental solution theory founded here can also be applied to discuss other important issues such as qualitative properties and optimal controls for semilinear FDEs with finite delay. This article is organized as follows. The notations and terminologies used in this article are presented in Section 2. In Section 3, we first discuss the fundamental solution of the corresponding deterministic linear system. Then, we apply it to define the mild solutions of System (1). After that, we present some lemmas to be used in the proofs of the main results. In Section 4, we start by showing the existence and uniqueness of mild solution of System (1) by making use of the Banach fixed point theorem. Then, we explore its approximate controllability on $$[0,T]$$ which is the main result of this article. Finally, in Section 5, an example is given to show the applications of the obtained results. 2. Preliminaries In this section, we collect the notations and the terminologies to be used in the whole article. Let $$X$$ be a separable Hilbert space with inner product $$<\cdot,\cdot>$$ and norm $$\|\cdot\|$$ and let $$K$$ be another separable Hilbert space with inner product $$<\cdot,\cdot>_K$$ and norm $$\|\cdot\|_K$$. We employ the same notation $$\|\cdot\|$$ for the norm of $${\mathscr{L}}(K;X)$$, where $${\mathscr{L}}(K;X)$$ denotes the space of all bounded linear operators from $$K$$ into $$X$$. Particularly, $${\mathscr{L}}(X)$$ will denote $${\mathscr{L}}(X;X)$$. Let $$-A: D(-A)\subseteq X\to X$$ be a closed, linear and densely defined operator generating an analytic semigroup $$\left(S(t)\right)_{t\ge 0}$$ on $$X$$. Assume that $$0 \in \rho(A)$$, where $$\rho(A)$$ denotes the resolvent set of $$A$$. Then, for $$\alpha \in(0,1]$$, it is possible to define the fractional power operator $$A^{\alpha}$$ as a closed linear operator on its domain $$D(A^{\alpha})$$. Furthermore, the subspace $$D(A^{\alpha})$$ is dense in $$X$$ and the expression   \[ \|x\|_{\alpha} =\|A^{\alpha}x\|,\qquad x \in D(A^{\alpha}), \] defines a norm on $$D(A^{\alpha})$$. Hereafter, we represent by $$X_{\alpha}$$ the space $$D(A^{\alpha})$$ endowed with the norm $$\|\cdot\|_{\alpha}$$. Then the following properties are well-known, see Engel & Nagel (2000) and Pazy (1983). Lemma 2.1 For the analytic semigroup $$\left(S(t)\right)_{t\geq 0}$$ generated by the operator $$(-A,D(-A))$$, there hold $$(i)$$ Let $$\alpha \in(0,1]$$, then $$X_{\alpha}$$ is a Banach space. $$(ii)$$ If $$0<\beta<\alpha\leq 1$$, then $$X_{\alpha} \mapsto X_{\beta}$$ and the imbedding is compact whenever $$(\lambda I+A)^{-1}$$, the resolvent operator of $$-A$$, is compact. $$(iii)$$ For every $$\alpha \in(0,1]$$, there exist a constant $$M_{\alpha}>0$$ and a real number $$a>0$$ such that   $$\left\|A^{\alpha}S(t)h\right\| \leq M_{\alpha} e^{-at} t^{-\alpha} \|h\|, \,\,\,\, t>0,$$ for any $$h\in X.$$ Particularly,   $$\|A^{\alpha}S(t)\|\leq \frac{M_\alpha}{t^\alpha},\,\,\,\, t\in (0,T].$$ Let $${\it{\Omega}} :=({\it{\Omega}},\mathscr{F},\left\{\mathscr{F}_t\right\}_{t\geq0},\mathbb{P})$$ be a filtered complete probability space satisfying the usual condition, which means that the filtration $$\left\{\mathscr{F}_t\right\}_{t\geq0}$$ is a right continuous increasing family and $$\mathscr{F}_0$$ contains all $$\mathbb{P}-$$null sets. We assume that $$\mathscr{F}_t=\sigma\left(W(s):0\leq s\leq t\right)$$ is the $$\sigma-$$algebra generated by $$W$$ and $$\mathscr{F}_T=\mathscr{F}$$, where $$W(t)$$ is a $$K-$$valued Wiener process defined on $$({\it{\Omega}},\mathscr{F},\left\{\mathscr{F}_t\right\}_{t\geq0},\mathbb{P})$$ with a finite trace nuclear covariance operator $$Q$$. Let $$\beta_n(t), (n=1,2,\cdots),$$ be a sequence of real-valued one-dimensional standard Brownian motions mutually independent over $$({\it{\Omega}},\mathscr{F},\left\{\mathscr{F}_t\right\}_{t\geq0},\mathbb{P})$$. Set   $$W(t)=\sum_{n=1}^{+\infty} \sqrt{\lambda_n} \beta_n(t) \xi_n, \,\,\,\, t\geq 0,$$ where $$\lambda_n\geq 0, (n=1,2,\cdots),$$ are non-negative real numbers and $$\xi_n, (n=1,2,\cdots),$$ is a complete orthonormal basis in $$K$$. Let $$Q \in {\mathscr{L}}(K)$$ be an operator defined by $$Q\xi_n=\lambda_n\xi_n$$ with finite trace $$Tr(Q)=\sum_{n=1}^{+\infty} \lambda_n<\infty$$. Then the above $$K-$$valued stochastic process $$W(t)$$ is called a $$Q-$$Wiener process. Definition 2.1 Let $$\sigma\in {\mathscr L}(K; X)$$ and define   \begin{equation}\label{eq43} \|\sigma\|^{2}_{{\mathscr L}_2^0}:={\mbox Tr} (\sigma Q\sigma^{*})=\sum\limits_{ n=1}^{+\infty}\|\sqrt{\lambda_{n}}\sigma \xi_{n}\|^2. \end{equation} (3) If $$\|\sigma\|_{{\mathscr L}_2^0}<\infty$$, then $$\sigma$$ is called a $$Q$$-Hilbert–Schmidt operator and let $${\mathscr L}_2^0 (K; X)$$ denote the space of all $$Q$$-Hilbert–Schmidt operators $$\sigma: K\to X$$. We recall that $$f$$ is said to be $$\mathscr{F}_t$$-adapted if $$f(t,\cdot):{\it{\Omega}}\to X$$ is $$\mathscr{F}_t$$-measurable, $$a.e. t\in[0.T]$$. Suppose that $$y(t):{\it{\Omega}}\to X_{\alpha}$$, $$t\ge 0$$, is a continuous, $${\mathscr F}_{t}$$-adapted, $$X_{\alpha}$$-valued stochastic process, we can associate it with another process $$y_t : {\it{\Omega}}\to {\mathscr{B}_\alpha}$$, $$t\ge 0$$, by setting $$y_t(s)(\omega)=y(t+s)(\omega)$$, $$s\in[-r, 0]$$. Then we say that the process $$y_t$$ is generated by the process $$y(t)$$. Finally, we introduce some function spaces to be used in the sequel. For any fixed $$0<r<\infty$$, let $$\mathcal{B}([-r,0];X)$$ be the Banach space of all measurable and bounded functions from $$[-r,0]$$ into $$X$$ endowed with the sup-norm, i.e.,   $$\mathcal{B}([-r,0];X):=\left\{x(\cdot):[-r,0]\to X \Big | x(\cdot)\,\, \mbox{is measurable and}\,\, \|x\|_{\mathcal{B}}=\sup\limits_{\theta \in [-r,0]}\|x(\theta)\|<+\infty\right\}\!.$$ Similarly, for $$\alpha\in (0,1]$$, $$\mathcal{B}([-r,0];X_\alpha)$$ is the Banach space of all measurable and bounded functions from $$[-r,0]$$ into $$X_\alpha$$ with the norm   $$\|x\|_{\mathcal{B_\alpha}}:= \sup_{\theta \in [-r,0]}\left\|A^{\alpha}x(\theta)\right\|<+\infty, \,\,\,\,\mbox{for any}\,\,\,\, x\in \mathcal{B_\alpha}.$$ Denote by $$C_\alpha := C([-r,0];X_\alpha)$$ the space of all continuous functions from $$[-r,0]$$ into $$X_\alpha$$. Then, denote by $$ML({p,r,\alpha}), 2<p<\infty,$$ the space of all $${\mathscr{F}_0}-$$measurable functions that belong to $$L_p({\it{\Omega}};C_{\alpha})$$, that is, $$ML({p,r,\alpha}), 2<p<\infty,$$is the space of $${\mathscr{F}_0}-$$measurable $$C_{\alpha}-$$valued functions $$\phi:{\it{\Omega}}\to C_{\alpha}$$ with the norm   $$\mathbb{E}\|\phi\|^p_{C_{\alpha}}=\mathbb{E}\left(\sup_{\theta\in[-r,0]}\left\|A^{\alpha}\phi(\theta)\right\|^p \right)<+\infty.$$ Additionally, let $$L_p^{\mathscr{F}}([0,T];X)$$ be the closed subspace of $$L_p([0,T]\times{\it{\Omega}};X)$$ consisting of $${\mathscr{F}_t}-$$adapted processes and let $$C([-r,T];L_p ({\it{\Omega}};X))$$ denote the Banach space of all continuous maps from $$[-r,T]$$ into $$L_p({\it{\Omega}};X)$$ satisfying the condition $$\sup\limits_{t\in [-r,T]}\mathbb{E}\|x(t)\|^p<\infty$$. 3. Fundamental solution In this section, we first discuss the fundamental solution for the linear deterministic system corresponding to System (1). Then, we express the mild solutions of System (1) via the fundamental solution. Finally, we introduce the concept of approximate controllability and collect some basic lemmas to be used in Section 4. To this end, we make the following assumptions on the terms of System (1). $$(H_1)$$ The operator $$(-A,D(-A))$$ generates an analytic semigroup $$\left(S(t)\right)_{t\geq 0}$$ on the Hilbert space $$X$$. Then there exist constants $$\mu \in \mathbb{R}$$ and $$M_{\mu}\geq 1$$ such that (see Pazy, 1983)   $$\|S(t)\|\leq M_{\mu} e^{\mu t}, \,\,\,\,\text{for all}\,\,\,\, t\geq 0,$$ Particularly,   $$\|S(t)\|\leq M,\,\,\,\, \text{for some}\,\,\,\, M\geq 1, \,\,\,\,\text{ and all }\,\,\,\, t\in [0,T],$$ and for any $$\alpha\in(0,1]$$, the fractional power operator $$A^\alpha$$ exists and Lemma 2.1 holds. $$(H_2)$$$$B$$ is a bounded linear operator from $$U$$ to $$X$$. $$(H_3)$$ The operator $$L: \mathcal{B}([-r,0];X)\to X$$ is a bounded linear operator with $$\|L\|=l$$ for some $$l>0$$, and it maps $$\mathcal{B}([-r,0];X_\alpha)$$ into $$D(A^\alpha)$$. We also assume that $$A^{\alpha}L=LA^{\alpha}$$ for all $$\alpha \in[0,1]$$. Here $$A^{\alpha}L=L A^{\alpha}$$ is understood as: for any $$\varphi\in \mathcal{B}([-r,0],X_\alpha),A^{\alpha}L(\varphi)=L(A^{\alpha}\varphi)$$. It is readily seen that this commuting property is verified for many systems, see Fu et al. (2014). $$(H_4)$$ $$(i)$$ The nonlinear functions $$f:[0,+\infty)\times C_{\alpha}\to X$$ and $$g:[0,+\infty)\times C_{\alpha}\to {{\mathscr L}_2^0}(K;X)$$ are two measurable mappings satisfying that $$f(t,0)$$ and $$g(t,0)$$ are bounded in $$X-$$norm and $${{\mathscr L}_2^0}(K;X)-$$norm, respectively. $$(ii)$$ For arbitrary $$\gamma, \xi\in C_{\alpha}$$ and $$t\in[0,T]$$, there exists a positive real constant $$N_1>0$$ such that   \begin{gather*} \|f(t,\gamma)-f(t,\xi)\|^p+\|g(t,\gamma)-g(t,\xi)\|^p_{{\mathscr L}_2^0} \leq N_1 \|\gamma-\xi\|^p_{C_{\alpha}},\\ \|f(t,\xi)\|^p+\|g(t,\xi)\|^p_{{\mathscr L}_2^0} \leq N_1\left(1+\|\xi\|^p_{C_{\alpha}}\right)\!. \end{gather*} $$(H'_4)$$ $$(i)$$ The nonlinear functions $$f:[0,+\infty)\times C_{\alpha}\to X$$ and $$g:[0,+\infty)\times C_{\alpha}\to {{\mathscr L}_2^0}(K;X)$$ are two measurable mappings satisfying that $$f(t,0)$$ and $$g(t,0)$$ are bounded in $$X-$$norm and $${{\mathscr L}_2^0}(K;X)-$$norm, respectively. $$(ii)$$ For arbitrary $$\gamma, \xi\in C_\alpha$$ and $$t\in[0,T]$$, there exists positive real constant $$N_1>0$$ such that   \begin{gather*} \|f(t,\gamma)-f(t,\xi)\|^p+\|g(t,\gamma)-g(t,\xi)\|^p_{{\mathscr L}_2^0}\leq N_1 \|\gamma-\xi\|^p_{C_{\alpha}},\\ \|f(t,\xi)\|^p+\|g(t,\xi)\|^p_{{\mathscr L}_2^0} \leq N_1. \end{gather*} We now establish the fundamental solution theory. To this end, denote by $$y(t,\phi)$$ the mild solution of the following linear retarded deterministic FDE on space $$X$$ associated to System (1):   \begin{equation}\label{eq44} \left\{\begin{aligned} &\frac{d}{dt} y(t) =-Ay(t)+L\left(y_t\right), \,\,\,\,t\in [0,T],\\ &y(t)=\phi(t)\in \mathcal{B}([-r,0];X), \,\,\,\,t\in[-r,0], \end{aligned}\right. \end{equation} (4) where $$-A:D(-A)\subseteq X\to X$$ and $$L: \mathcal{B}([-r,0];X)\to X$$ are operators given above. Then the mild solutions of System (4) can be expressed by $$C_0-$$semigroup $$(S(t))_{t\geq 0}$$ as   \begin{equation}\label{eq45} y(t,\phi)=\left\{ \begin{aligned} &S(t) \phi(0)+\int_0^tS(t-s)L(y_s(\cdot,\phi))ds,& t\in [0,T],\\ &\phi(t), &t\in[-r,0]. \end{aligned}\right. \end{equation} (5) For the solutions of System (4) satisfying (5), we can establish the following result. Theorem 3.1 For arbitrary $$\phi\in \mathcal{B}([-r,0];X)\cap L^1([-r,0];X)$$ and $$T>0$$, if $$(H_1)$$ and $$(H_3)$$ are satisfied, then there exists a unique solution $$y(t,\phi)$$, $$t\in[-r,T]$$, of System (4) satisfying (5) such that: $$(i)$$ The restriction of $$y(t,\phi)$$ on $$[0,T]$$ is continuous, i.e., $$y|_{[0,T]}\in C\left([0,T];X\right)$$, and $$(ii)$$$$\|y(t,\phi)\|\leq Ce^{\gamma t}\|\phi\|_{\mathcal{B}}$$, for all $$t\geq 0$$, where $$\gamma \in \mathbb{R}$$ and $$C>1$$ are constants. Proof. Let $$\delta\in (0,T]$$ such that   \begin{equation}\label{Q} Ml \delta<1, \end{equation} (6) and put   \begin{equation}\label{r} \rho=\frac{M+Mrl}{1-Ml\delta}\|\phi\|_{\mathcal{B}}. \end{equation} (7) Define the set $$E(\delta,\rho)$$ by   \[ E(\delta,\rho)=\left\{y:[0,\delta]\to \mathcal{B}([0,\delta];X) \big{|} y(0)=\phi (0)\,\,\,\, \text{and}\,\,\,\, \|y\|_{\delta}:=\sup_{ 0\leq t \leq \delta }\|y(t)\| \leq \rho \right\}\! . \] Clearly, $$E(\delta,\rho)$$ is a closed, bounded and convex subset of the Banach space $$\mathcal{B}([0,\delta];X)$$. Next, we define an operator $$Q$$ on $$E(\delta,\rho)$$ as follows:   $$(Qy)(t,\phi)=S(t) \phi(0) +\int_0^t S(t-s)L(\tilde{y}_s(\cdot,\phi))ds,$$ for any $$y \in E(\delta,\rho)$$ and all $$t \in [0,\delta]$$, where   \begin{equation*} \tilde{y}(t,\phi)=\left\{ \begin{aligned} &y(t,\phi),&\text{if}\,\,\,\, &t \in [0,\delta],&\\ &\phi(t), &\text{if}\,\,\,\, &t \in [-r,0].& \end{aligned} \right. \end{equation*} Obviously, $$(Qy)(0,\phi)=\phi(0)$$. In addition, from $$(H_1)$$ and $$(H_3)$$ we have, for any $$t\in [0,\delta]$$ and $$y \in E(\delta,\rho)$$,   \[ \begin{aligned} \|(Qy)(t,\phi)\| &\leq \|S(t)\| \|\phi(0)\|+l\int_0^t\|S(t-s)\| \|\tilde{y}_s(\cdot,\phi)\|_{\mathcal{B}}ds\\ & \leq M\|\phi\|_{\mathcal{B}}+Ml\int_{-r}^0\sup_{\theta\in[-r,0]}\|\phi(\theta)\| ds+Ml\int_{0}^t\sup_{\theta\in[0,\delta]}\|y(\theta,\phi)\| ds\\ & \leq \left(M+Mrl\right)\|\phi\|_{\mathcal{B}}+M\delta l\|y\|_\delta\\ & \leq \rho, \,\,\,\,(\text{by the definition} (7) \,\,\,\,\text{of}\,\,\,\, \rho ), \end{aligned} \] which shows that $$Q$$ maps $$E(\delta,\rho)$$ into itself. On the other hand, for any $$y^1$$ and $$y^2$$ in $$E(\delta,\rho)$$ and any $$t\in [0,\delta]$$, the assumptions $$(H_1)$$ and $$(H_3)$$ yield that   \[ \begin{aligned} \|(Qy^1)(t,\phi)-(Qy^2)(t,\phi)\|& \leq Ml\int_0^t\|\tilde{y}^1_s(\cdot,\phi)-\tilde{y}^2_s(\cdot,\phi)\|_{\mathcal{B}} ds\\ & \leq Ml\delta \|y^1-y^2\|_\delta, \end{aligned} \] which from (6) implies that $$Q$$ is a contractive mapping. Therefore, there exists a unique fixed point $$y(\cdot,\phi)$$ for the operator $$Q$$ in $$E(\delta,\rho)$$ which is a solution of system (4) defined on $$[0,\delta]$$ and satisfies $$y(0)=\phi(0)$$. Now, we extend the solution $$y(\cdot,\phi)$$ on $$[-r,0]$$ such that we get a solution of system (4) satisfying (5) on $$[-r,\delta]$$. Similarly, we can prove the existence of a solution of the system (4) defined on $$[\delta,2\delta]$$, $$[2\delta,3\delta]$$, $$\cdots$$ with $$y_0=\phi$$. In finite steps, we get the existence of a solution of system (4) on the whole interval $$[-r,T]$$ such that $$y(\cdot,\phi)$$ is given by (5). Moreover, it is easy to see that on the interval $$[0,T]$$, $$(Qy)(t,\phi)$$ is continuous. Next, we use the well-known Gronwall’s inequality to prove the uniqueness of the solution of system (4). Let $$y^1(t,\phi)$$ and $$y^2(t,\phi)$$ be any two solutions of system (4) satisfying (5). Then, using $$(H_1)$$ and $$(H_3)$$ we get   \[ \begin{aligned} \|y^1(t,\phi)-y^2(t,\phi)\| & \leq Ml \int_0^t\|y^1_s(\cdot,\phi)-y^2_s(\cdot,\phi)\|_{\mathcal{B}} ds\\ & \leq Ml \int_0^t \sup_{0\leq \tau\leq s}\|y^1(\tau,\phi)-y^2(\tau,\phi)\| ds.\\ \end{aligned} \] So   \[ \sup_{0\leq \tau\leq t}\|y^1(\tau,\phi)-y^2(\tau,\phi)\|\leq Ml \int_0^t \sup_{0\leq \tau\leq s}\|y^1(\tau,\phi)-y^2(\tau,\phi)\| ds. \] The Gronwall’s inequality implies that $$\sup\limits_{0\leq \tau\leq t}\|y^1(\tau,\phi)-y^2(\tau,\phi)\|\equiv 0$$, thus $$y^1(t,\phi)=y^2(t,\phi)$$ for all $$t\in[0,T]$$ and consequently $$y^1(\cdot,\phi)=y^2(\cdot,\phi)$$ on $$[-r,T]$$. The claim $$(ii)$$ can be shown also by the Gronwall’s inequality. Indeed, from (5), $$(H_1)$$ and $$(H_3)$$, the solution $$y(t,\phi)$$ satisfies that, for $$t\in [0,\delta]$$ with $$\delta>0$$,   \[ \begin{aligned} \|y(t,\phi)\| & \leq M \|\phi\|_{\mathcal{B}}+Ml \int_0^t\|y_s(\cdot,\phi)\|_{\mathcal{B}}ds\\ & \leq \left(M+Ml r\right) \|\phi\|_{\mathcal{B}}+Ml\int_{0}^t\sup_{0\leq \tau\leq s}\|y(\tau,\phi)\|ds.\\ \end{aligned} \] Hence,   \[ \sup_{0\leq \tau\leq t}\|y(t,\phi)\|\leq \left(M+Ml r\right) \|\phi\|_{\mathcal{B}}+Ml\int_{0}^t\sup_{0\leq \tau\leq s}\|y(\tau,\phi)\|ds, \] which by Gronwall’s inequality implies that   $$\sup_{0\leq \tau\leq t}\|y(\tau,\phi)\|\leq (M+Ml r)\|\phi\|_{\mathcal{B}} e^{Mlt}:=C(\delta).$$ Now, for any $$t>0$$, let $$t \in ((n -1)\delta,n\delta]$$$$(n\in \mathbb{N})$$, then proceeding inductively as above, we obtain easily that   $$\|y(t,\phi)\|\leq C^n(\delta),$$ which, letting $$\gamma= \frac{\ln C(\delta)}{\delta}$$, immediately yields that, for any $$t>0$$, $$\|y(t,\phi)\|\leq Ce^{\gamma t}$$ with $$C\ge 1$$ and $$\gamma\in \mathbb{R}$$. The claim $$(ii)$$ is proved. □ The solution $$y(t,\phi)$$ given by (5) is called a mild solution of System (4). Next, for any $$x\in X$$, define the function $$\phi^0_x$$ by   \begin{equation}\label{eq46} \phi^0_x(\theta)=\left\{ \begin{aligned} & x, &\theta=0,\\ &0, &\theta\in[-r,0). \end{aligned} \right. \end{equation} (8) Then, we define the fundamental solution $$G(t)\in {\mathscr{L}}(X)$$ of (4) with the initial datum $$\phi^0_x$$ by, for any $$x\in X$$,   \begin{equation}\label{eq47} G(t)x=\left\{ \begin{aligned} & y(t,\phi^0_x), &t\geq 0,\\ &0, &t<0. \end{aligned} \right. \end{equation} (9) According to Theorem 3.1, the fundamental solution $$G(t)$$ is well defined. Moreover, System (9) implies that $$G(t)$$ is the unique solution of the operator equation   \begin{equation}\label{eq48} G(t)=\left\{ \begin{aligned} &S(t) +\int_0^t S(t-s) L\left(G_s\right)ds, &t\geq 0,\\ &0,&t< 0, \end{aligned} \right. \end{equation} (10) where $$G_t(\theta):=G(t+\theta), \theta\in [-r,0]$$. Furthermore, for the fundamental solution $$G(t)$$ defined above, one has that. Theorem 3.2 For $$G(t)$$, $$t\in \mathbb{R}$$, there hold: $$(i)$$$$G(t)$$ is a strongly continuous one-parameter family of bounded linear operators on $$X$$ and satisfies that   $$\|G(t)\| \leq Ce^{\gamma t},\,\,\,\, t\geq 0,$$ where $$C>1$$, $$\gamma\in\mathbb{R}$$ are constants. Particularly, we have that   $$\|G(t)\| \leq \overline{M},\,\,\,\, t\in[0,T],$$ for some $$\overline{M}\geq 1$$. $$(ii)$$ If the semigroup $$\big(S(t)\big)_{t\ge 0}$$ is compact, then $$G(t)$$ is also compact for all $$t>0$$. $$(iii)$$$$G(t)$$ and $$A^{\alpha}G(t)$$ are uniformly continuous on $$(0,T]$$ for all $$\alpha \in (0,1)$$. $$(iv)$$ For each $$\alpha\in(0,1)$$, there exists a constant $$\overline{M}_{\alpha}>0$$ such that   $$\|A^\alpha G(t)\|\leq \frac{\overline{M}_\alpha}{t^\alpha}, \,\,\,\,\text{for all}\,\,\,\, t\in(0,T].$$ $$(v)$$ For all $$t\in[0,T]$$, $$G(t)$$ commutes with the operator $$A^{\alpha}$$, that is, $$A^{\alpha}G(t)=G(t)A^{\alpha}$$ for each $$\alpha \in [0,1]$$. Proof. Assertion $$(i)$$ is obvious by Theorem 3.1 and the definition (9) of $$G(\cdot)$$. For Assertion $$(ii)$$, let $$\epsilon>0$$ very small, then the operator   $$\int_0^{t-\epsilon} S(t-s)L \left(G_s\right)ds=S(\epsilon)\int_0^{t-\epsilon} S(t-s-\epsilon)L \left(G_s\right)ds,$$ is clearly compact and, since $$\left(S(t)\right)_{t\geq 0}$$ is assumed to be compact, it converges to $$\int_0^{t} S(t-s)L \left(G_s\right)ds$$ uniformly as $$\epsilon\to 0^+$$, which shows that $$\int_0^{t} S(t-s)L \left(G_s\right)ds$$ is compact too. Consequently, from (10) we deduce that $$G(t)$$ is a compact operator on $$X$$ for any $$t> 0$$. $$(iii)$$ Let $$0<\epsilon< t_1< t_2\leq T$$, then from (10) we have   $$ \begin{aligned} \left\|G(t_2)-G(t_1)\right\| & \leq \left\| S(t_2)-S(t_1) \right\| + \int^{t_1-\epsilon}_0 \left\| S(t_2-s)-S(t_1-s) \right\| \left\|L(G_s)\right\|ds \\ &\quad{} + \int_{t_1-\epsilon}^{t_1} \left\|\left(S(t_2-s)-S(t_1-s)\right) \right\| \left\|L(G_s)\right\|ds +\int^{t_2}_{t_1}\left\|S(t_2-s)\right\|\left\|L(G_s)\right\|ds. \end{aligned} $$ By the analyticity, $$S(t)$$ is uniformly continuous for $$t>0$$, then the right hand side tends to zero for $$t_2 \to t_1$$ and $$\epsilon$$ small enough. Hence $$G(t)$$ is uniformly continuous for $$t\in (0,T]$$. Similarly, using the fact that, for all $$\alpha \in (0,1)$$, $$A^{\alpha}S(t)$$ is uniformly continuous for $$t>0$$, one can get the uniform continuity of $$A^{\alpha}G(t)$$ on $$(0,T]$$. Hence, assertion $$(iii)$$ holds. $$(iv)$$ First we note from (10) that $$\mathscr{R}(G)\subset D(A)$$ since the semigroup $$S(t)$$ is analytic. Then, from $$(i)$$, (10), $$(H_3)$$ and Lemma 2.1 $$(iii)$$ it follows that, for any $$\alpha \in (0,1)$$ and $$0<t\leq T$$,   \[ \begin{aligned} \left\|A^\alpha G(t)\right\|& \leq \left\|A^\alpha S(t)\right\| +\left\|\int_0^t A^\alpha S(t-s)L \left(G_s\right)ds\right\|\\ & \leq \left\|A^\alpha S(t)\right\| + l \overline{M} \int_0^t \frac{M_\alpha}{(t-s)^\alpha}ds\\ & \leq \frac{M_\alpha}{t^\alpha}+ l \overline{M} \frac{M_\alpha}{1-\alpha} \frac{T}{t^\alpha}\\ & = \left(M_\alpha + \frac{l \overline{M} M_\alpha T}{1-\alpha}\right) \frac{1}{t^\alpha}. \end{aligned} \] Let $$\overline{M}_\alpha= M_\alpha + \frac{l \overline{M} M_\alpha T}{1-\alpha}$$, then $$(iv)$$ is proved. $$(v)$$ Since the semigroup $$\left(S(t)\right)_{t\geq 0}$$ is analytic, we know that $$A^{\alpha} S(t)=S(t)A^{\alpha}, \text{for all} \alpha\in [0,1]$$ and $$t\in [0,T]$$. Now, from (10), we have that   \begin{align*} A^{\alpha} G(t) & = A^{\alpha}S(t)+\int_0^t S(t-s)L\left(A^{\alpha}G_s(\theta)\right)ds,\\ G(t)A^{\alpha} & = S(t)A^{\alpha}+\int_0^tS(t-s)L\left(G_s(\theta)A^{\alpha}\right) ds. \end{align*} Combined with $$(H_1)$$ and $$(H_3)$$ this yields   $$ \begin{aligned} \left\|A^{\alpha}G(t)-G(t)A^{\alpha} \right\| & \leq \int_0^t\left\|S(t-s)L \left(A^{\alpha}G_s(\theta)-G_s(\theta)A^{\alpha}\right)\right\|ds\\ & \leq M l \int_0^t \left\|A^{\alpha} G_s(\theta)-G_s(\theta)A^{\alpha}\right\|ds\\ & \leq M l \int_0^t\sup_{\nu\in[0,s]}\left\|A^{\alpha}G(\nu)-G(\nu)A^{\alpha}\right\|ds, \end{aligned} $$ hence   $$ \begin{aligned} \sup_{\tau\in[0,t]}\left\|A^{\alpha}G(\tau)-G(\tau)A^{\alpha}\right\|& \leq M l \int_0^t\sup_{\tau\in[0,s]}\left\|A^{\alpha}G(\tau)-G(\tau)A^{\alpha}\right\|ds. \end{aligned}$$ The Gronwall’s inequality implies that $$\sup\limits_{t\in[0,T]}\|A^{\alpha}G(t)-G(t)A^{\alpha}\|\equiv0$$, thus $$A^{\alpha} G(t)=G(t)A^{\alpha}$$ for all $$\alpha\in [0,1]$$ and $$t\in[0,T]$$. Then we get the assertion $$(v)$$. □ Now we consider the following linear deterministic inhomogeneous FDEs on $$X$$:   \begin{equation}\label{eq49}\left\{\begin{aligned} &\frac{d}{dt} y(t) =-Ay(t)+L \left(y_t\right)+f(t), \,\,\,\,t\geq 0,\\ &y(\theta)=\phi(\theta)\in C([-r,0];X), \,\,\,\,\theta\in [-r,0], \end{aligned}\right. \end{equation} (11) where the operators $$A$$ and $$L$$ are as described above and the function $$f(t)$$ belongs to $$L^1(\mathbb{R}^+;X)$$. The mild solutions of System (11) are represented through the $$C_0$$-semigroup $$(S(t))_{t\geq 0}$$ by   \begin{equation}\label{eq410} y(t,\phi)=\left\{ \begin{aligned} &S(t) \phi(0)+\int_0^tS(t-s)\Big(L \left(y_s(\cdot,\phi)\right)+f(s)\Big)ds, &t\geq 0,\\ &\phi(t), & t \in [-r,0]. \end{aligned} \right. \end{equation} (12) For the subsequent discussions, we need to represent the mild solutions (12) as an ‘explicit formula’ via the fundamental solution $$G(t)$$ defined above, that is Theorem 3.3 For $$\phi\in C([-r,0];X)$$, the mild solutions (12) of System (11) can be expressed equivalently by   \begin{equation*} y(t,\phi)=\left\{ \begin{aligned} &G(t)\phi(0)+\int_0^tG(t-s)\Big(L \left({\tilde{\phi}_s}\right)+f(s)\Big)ds, \,\,\,\,t\geq 0,\\ &\phi(t), \quad\quad t \in [-r,0], \end{aligned}\right. \end{equation*} where the function $$\tilde{\phi}(\cdot)$$ is defined as   \begin{equation}\label{eq411} \tilde{\phi}(t)=\left\{ \begin{aligned} & \phi(t), & t \in [-r,0],\\ &0, &t>0. \end{aligned} \right. \end{equation} (13) Proof. The proof is very similar to that of (Mokkedem & Fu (2017), Theorem 3.3), so we omit it here. □ Next, we define the mild solutions of the stochastic System (1) via the fundamental solution $$G(t)$$ as follows. Definition 3.1 A stochastic process $$y(\cdot)$$ defined on $$[-r,T], 0<r, T<\infty$$ is said to be a mild solution of System (1) if the following conditions are satisfied: (1) $$y(t,\omega)$$ is measurable as a function from $$[0,T]\times{\it{\Omega}}$$ to $$X$$ and $$y(t)$$ is $${\mathscr{F}_t}-$$adapted; (2) $$\mathbb{E}\|y(t)\|^p<\infty$$ for each $$t\in[-r,T]$$; (3) For each $$u\in L_p^{\mathscr{F}}([0,T];U)$$, the process $$y(\cdot)$$ satisfies the following integral equation:   \begin{equation*} y(t)=\left\{ \begin{aligned} &G(t)\phi(0)+\int_0^t G(t-s)\Big(L \left({\tilde{\phi}_s}\right)+f(s,y_s)+Bu(s)\Big)ds\\ & +\int_0^t G(t-s)g(s,y_s)dW(s), &t\in[0,T],\\ &\phi(t) \in ML(p,r,\alpha) ,& t \in [-r,0], \end{aligned} \right.\end{equation*} where the function $$\tilde{\phi}(\cdot)$$ is given by (13). Next, we turn to present the concept of approximate controllability. Definition 3.2 The System (1) is said to be approximately controllable on $$[0,T]$$ if $$\mathscr{R}(T,\phi)$$ is dense in $$L_p^{\mathscr{F}}\left({\it{\Omega}};X\right)$$, i.e.,   $$\overline{\mathscr{R}(T,\phi)}= L_p^{\mathscr{F}}\left({\it{\Omega}};X\right)\!,$$ where $$\mathscr{R}(T,\phi)=\left\{y(T,\phi,u): u(\cdot)\in L_p^{\mathscr{F}}([0,T];U)\right\}.$$ In the sequel, we shall study the approximate controllability for System (1) by assuming the approximate controllability of the linear deterministic system corresponding to (1). For this purpose, we introduce the following resolvent operator. Let   $${\it{\Gamma}}^t_s=\int_s^t G(t-\tau)BB^\ast G^\ast(t-\tau)d\tau,\,\,\,\, \text{for} \,\,\,\,0\leq s<t\leq T,$$ where $$B^*$$ and $$G^*$$ denote respectively the adjoint operators of $$B$$ and $$G$$, then the resolvent operator $$R(\lambda,-{\it{\Gamma}}^\top_s)\in {\mathscr{L}}(X)$$ for $$\lambda>0$$ is given by   $$R(\lambda,-{\it{\Gamma}}^\top_s)=(\lambda I+{\it{\Gamma}}^\top_s)^{-1}.$$ Since the operator $${\it{\Gamma}}^\top_s$$ is clearly positive, $$R(\lambda,-{\it{\Gamma}}^\top_s)$$ is well defined. We will always assume that $$(H_5)$$$$\lambda R(\lambda,-{\it{\Gamma}}^\top_0)\to 0$$ as $$\lambda\to 0^+$$ in the strong operator topology. The above condition $$(H_5)$$ is equivalent to the approximate controllability of the following deterministic linear system corresponding to (1):   \begin{equation}\label{eq413} \left\{\begin{aligned} &\frac{d}{dt}y(t) =-Ay(t)+L \left(y_t\right)+ Bu(t),\,\,\,\, t\in [0,T],\\ &y(\theta)=0, \,\,\,\,\theta\in[-r,0]. \end{aligned}\right. \end{equation} (14) More precisely, we have that Theorem 3.4 The following statements are equivalent: $$(i)$$ The control system (14) is approximately controllable on $$[0,T]$$. $$(ii)$$ If $$B^*G^*(t)y=0$$ for all $$t\in [0,T]$$, then $$y=0$$. $$(iii)$$ The condition ($$H_5$$) holds. Proof. The proof of this theorem is similar to that of (Curtain & Zwart, 1995, Theorem 4.4.17) and (Bashirov & Mahmudov, 1999, Theorem 2), so we omit it here. □ By this theorem the condition $$(H_5)$$ implies that the deterministic linear system (14) is approximately controllable on interval $$[0,T]$$ and hence on $$[s,T]$$ for all $$0\leq s <T$$, we thus infer that   $$\lambda R(\lambda,-{\it{\Gamma}}^\top_s)\to 0,\,\,\,\, \text{strongly as} \,\,\,\,\lambda \to 0^+, \text{ for any } s\in[0,T),$$ from which we can assume that   $$\left\|R(\lambda,-{\it{\Gamma}}^\top_s)\right\|\leq \frac{1}{\lambda}, \qquad\text{ for any } s\in[0,T), \,\,\,\,\lambda\in(0,1).$$ Finally, to study the existence of mild solutions of System (1), we need the following lemmas: Lemma 3.1 (DA Prato & Zabbczyk, 1992, Proposition 4.15) If $$\phi\in L_2^{\mathscr{F}}(0,T;{{\mathscr L}_2^0}(K;X))$$, $$A^\alpha \phi\in L_2^{\mathscr{F}}(0,T;{{\mathscr L}_2^0}(K;X))$$ and $$\phi(t)k\in X_\alpha, t\geq 0,$$ for arbitrary $$k\in K$$, then   $$A^\alpha \int_0^t \phi(s)dW(s)=\int_0^t A^\alpha \phi(s)dW(s).$$ Lemma 3.2 (DA Prato & Zabbczyk, 1992, Lemma 7.2) For any $$p>2$$ and $$\phi\in L_p^{\mathscr{F}}({\it{\Omega}};L_2(0,T;{{\mathscr L}_2^0}(K;X))),$$ we have   \begin{eqnarray*} \mathbb{E}\left( \sup_{s\in[0,t]} \left\| \int_0^s \phi(r)dW(r) \right\|^p \right) &\leq& c_p \sup_{s\in[0,t]} \mathbb{E}\left\| \int_0^s \phi(r)dW(r) \right\|^p\\ &\leq& C_p \mathbb{E}\left(\int_0^t \left\| \phi(r) \right\|^2_{{\mathscr L}_2^0} dr \right)^{\frac{p}{2}}, t\in [0,T], \end{eqnarray*} where $$c_p=\left(\frac{p}{p-1}\right)^p, C_p=\left(\frac{p}{2}(p-1)\right)^{\frac{p}{2}}\left(\frac{p}{p-1}\right)^{\frac{p^2}{2}}.$$ Lemma 3.3 (Dauer & Mahmudov, 2004, Lemma 6) Let $$p\geq 2$$ and $$h\in L_p\left({\it{\Omega}};X\right)$$ be fixed, then there exists a function $$\varphi$$ in space $$L_p^{\mathscr{F}}({\it{\Omega}};L_2(0,T;{{\mathscr L}_2^0}(K;X)))$$ such that   $$h=\mathbb{E}h+\int_0^\top \varphi(s) dW(s).$$ Moreover, we can prove that Lemma 3.4 Let $$p>2$$, $$\frac{1}{p}+\alpha<\frac12$$ and $$g\in L_p^{\mathscr{F}}(0,T;{{\mathscr L}_2^0}(K;X)).$$ Then there exists a constant $$N_2>0$$ such that   $$ \mathbb{E}\sup_{\theta\in[-r,0]} \left\| \int_0^{t+\theta} A^\alpha G(t+\theta-\tau) g(\tau)dW(\tau) \right\|^p \leq N_2 \int_0^t \mathbb{E}\left\|g(\tau)\right\|^p_{{\mathscr L}_2^0} d\tau, $$ where   \begin{eqnarray*} N_2 &=&2^{p-1}M^p_\alpha \Bigg({\it{\Gamma}}(1+q(\beta-1-\alpha))(aq)^{q(1+\alpha-\beta)}\Bigg)^{\frac{p}{q}} C_{p} \frac{T^{\frac{p(1-2\beta)}{2}}}{(1-2\beta)^{\frac{p}{2}}}\left(1+T^{\frac{p}{q}+1}\left(l \overline{M}\right)^p\right)\!, \end{eqnarray*} with $$\frac{1}{p}+\alpha<\beta<\frac12, \frac{1}{p}+\frac{1}{q}=1$$ and $${\it{\Gamma}}(\cdot)$$ is the well known Gamma function. Proof. From (10) we have   \begin{align*} & \sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-\tau)g(\tau)dW(\tau)\right\|^p\\ &\quad{} = \sup_{\theta\in[-r,0]} \Bigg\|\int_0^{t+\theta} A^\alpha S(t+\theta-\tau)g(\tau)dW(\tau)\\ &\qquad{} +\int_0^{t+\theta}\int_0^{t+\theta-\tau}A^\alpha S(t+\theta-\tau-u)L\left(G_u\right)dug(\tau)dW(\tau)\Bigg\|^p\\ & \quad{} = \sup_{\theta\in[-r,0]}\Bigg\| \int_0^{t+\theta} A^\alpha S(t+\theta-\tau)g(\tau)dW(\tau)\\ &\qquad{} +\int_0^{t+\theta}\int_{\tau}^{t+\theta} A^\alpha S(\nu-\tau) L\left(G_{t+\theta-\nu}\right)d\nu g(\tau)dW(\tau)\Bigg\|^p\\ &\quad{} = \sup_{\theta\in[-r,0]}\Bigg\| \int_0^{t+\theta} A^\alpha S(t+\theta-\tau)g(\tau)dW(\tau)\\ &\qquad +\int_0^{t+\theta}\int_{0}^{\nu} A^\alpha S(\nu-\tau)L\left(G_{t+\theta-\nu}\right)g(\tau)dW(\tau)d\nu\Bigg\|^p, \end{align*} then   \begin{align*} & \mathbb{E}\sup_{\theta\in[-r,0]} \left\| \int_0^{t+\theta} A^\alpha G(t+\theta-\tau) g(\tau)dW(\tau) \right\|^p \\ &\quad{} \leq 2^{p-1} \mathbb{E} \sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha S(t+\theta-\tau)g(\tau)dW(\tau) \right\|^p \\ &\qquad{} +2^{p-1} \mathbb{E}\sup_{\theta\in[-r,0]} \left\|\int_0^{t+\theta}\int_{0}^{\nu} A^\alpha S(\nu-\tau)L\left(G_{t+\theta-\nu}\right)g(\tau)dW(\tau)d\nu\right\|^p\\ & = 2^{p-1}(I_1+I_2), \end{align*} where from (Dauer & Mahmudov, 2004, Lemma 7)   \begin{align}\label{eq414} I_1 \leq N \int_0^t \mathbb{E}\left\|g(\tau)\right\|^{p}_Q d\tau, \end{align} (15) with   $$N = M^p_\alpha \Bigg({\it{\Gamma}}(1+q(\beta-1-\alpha))(aq)^{q(1+\alpha-\beta)}\Bigg)^{\frac{p}{q}} C_{p} \frac{T^{\frac{p(1-2\beta)}{2}}}{(1-2\beta)^{\frac{p}{2}}}, \,\,\,\,\frac{1}{p}+\alpha<\beta<\frac12,$$ where $$\frac{1}{p}+\frac{1}{q}=1.$$ On the other hand, using $$(H_3)$$, Theorem 3.2 $$(i)$$ and H$$\ddot{o}$$lder inequality, we have   \begin{align}\label{eq415} I_2 & \leq T^{\frac{p}{q}}\mathbb{E}\sup_{\theta\in[-r,0]} \left(\int_0^{t+\theta}\left\|\int_{0}^{\nu} A^\alpha S(\nu-\tau)L\left(G_{t+\theta-\nu}\right)g(\tau)dW(\tau)\right\|^p d\nu\right) \nonumber\\ & \leq T^{\frac{p}{q}}\left(l \overline{M}\right)^p \mathbb{E} \sup_{\theta\in[-r,0]} \left(\int_0^{t+\theta}\left\|\int_{0}^{\nu} A^\alpha S(\nu-\tau)g(\tau)dW(\tau)\right\|^p d\nu\right)\nonumber\\ & \leq T^{\frac{p}{q}}\left(l \overline{M}\right)^p \mathbb{E}\left(\int_0^{t}\left\|\int_{0}^{\nu} A^\alpha S(\nu-\tau)g(\tau)dW(\tau)\right\|^p d\nu\right)\!, \end{align} (16) or from (Dauer & Mahmudov, 2004, Lemma 7) it yields that   \begin{align}\label{eq41*} \mathbb{E} \left\|\int_0^t A^\alpha S(t-\tau)g(\tau)dW(\tau) \right\|^p & \leq N \int_0^t \mathbb{E} \left\|g(\tau)\right\|^{p}_Q d\tau, \end{align} (17) substituting (17) into (16), we get   \begin{align}\label{eq416} I_2 & \leq T^{\frac{p}{q}}\left(l \overline{M}\right)^p N \int_0^t\int_{0}^{\nu} \mathbb{E}\left\|g(\tau)\right\|^{p}_{Q} d\tau d\nu\nonumber\\ & \leq T^{\frac{p}{q}}\left(l \overline{M}\right)^p N \int_0^t (t-\tau) \mathbb{E}\left\|g(\tau)\right\|^{p}_{Q} d\tau \nonumber\\ & \leq T^{\frac{p}{q}+1}\left(l \overline{M}\right)^p N \int_0^t \mathbb{E}\left\|g(\tau)\right\|^{p}_{Q} d\tau. \end{align} (18) Combining (15) and (18) gives that   $$\mathbb{E} \left\| \int_0^t A^\alpha G(t-\tau) g(\tau)dW(\tau) \right\|^p \leq N_2 \int_0^t\mathbb{E}\left\|g(\tau)\right\|^{p}_{Q} d\tau,$$ with   $$N_2=2^{p-1}N \left(1+T^{\frac{p}{q}+1}\left(l \overline{M}\right)^p\right)\!.$$ Hence Lemma 3.4 is proved. □ 4. Approximate controllability In this section, we present the main result of this article on the approximate controllability of System (1). That is, for any $$h\in L_p^{\mathscr{F}}\left({\it{\Omega}};X\right)$$, by selecting proper control $$u^\lambda$$ in $$L_p^{\mathscr{F}}([0,T];U)$$ (for any given $$\lambda \in (0,1)$$), there exists a mild solution $$y^\lambda(\cdot,\phi,u^\lambda)\in C([-r,T];L_p({\it{\Omega}};X_\alpha))$$ for System (1), such that $$y^\lambda(T,\phi,u^\lambda)\to h$$ in $$L_p({\it{\Omega}};X)$$ as $$\lambda\to 0^+$$. For this purpose, besides the previous assumptions $$(H_1)-(H_5)$$, we will assume that the analytic semigroup $$\left(S(t)\right)_{t\ge 0}$$ is compact for all $$t>0$$. First, we define $$D_p$$ to be the closed subspace of $\(C([-r,T];L_p({\it{\Omega}};X))\)$ consisting of measurable and $${\mathscr{F}_t}-$$adapted processes $$Z$$ with $$\|Z\|_{D_p}<\infty$$, where   $$\|Z\|_{D_p}:= \left(\sup_{t\in[0,T]}\mathbb{E}\left\|Z_t\right\|^p_{C}\right)^{\frac{1}{p}} =\left(\sup_{t\in[0,T]}\mathbb{E} \sup_{\theta\in[-r,0]}\left\|Z_t(\theta)\right\|^p\right)^{\frac{1}{p}},$$ and put   $$D_{p,\alpha}:= \left\{z\in D(A^{\alpha}): A^{\alpha}z\in D_p\right\}\!.$$ Then, let $$\lambda\in(0,1)$$, $$\phi\in ML(p,r,\alpha)$$ and $$h\in L_p\left({\it{\Omega}};X\right)$$ be fixed. For any $$t\in [0,T]$$ and $$y\in D_{p,\alpha}$$, we define the control function $$u^{\lambda}(t,y)$$ as   \begin{align}\label{eq417} u^{\lambda}(t,y):=& B^*G^*(T-t)R(\lambda,-{\it{\Gamma}}^\top_0)\Big(\mathbb{E}h-G(T)\phi(0)\Big)\nonumber\\ &- B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^\top_s)G(T-s)\left(L\left({\tilde{\phi}_s}\right)+f(s,y_s)\right)ds\nonumber\\ &- B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^\top_s)\Big(G(T-s)g(s,y_s)-\varphi(s)\Big)dW(s), \end{align} (19) where $$\tilde{\phi}$$ is given by (13), $${\it{\Gamma}}^\top_s=\int_s^\top G(T-\tau)BB^\ast G^\ast(T-\tau)d\tau$$ is the controllability Grammian and $$h=\mathbb{E} h+\int_0^\top \varphi(s)dW(s)$$ by Lemma 3.3. Then, we can obtain the following estimates. Lemma 4.1 There exists a positive real constant $$N_3>0$$ such that, for all $$(x, y) \in D_{p,\alpha}^2$$,   \begin{align*} \mathbb{E}\left\|u^{\lambda}(t,x)-u^{\lambda}(t,y)\right\|^p & \leq \frac{1}{\lambda^p} N_3 \mathbb{E}\int_0^t\|x_s-y_s\|^p_{C_{\alpha}} ds,\\ \mathbb{E}\left\|u^{\lambda}(t,x)\right\|^p & \leq \frac{1}{\lambda^p} N_3\left(1+\mathbb{E} \int_0^t\|x_s\|^p_{C_{\alpha}} ds\right)\!. \end{align*} Proof. We will only prove the first inequality since the proof of the second is similar. Let $$x$$ and $$y$$ be two fixed functions in $$D_{p,\alpha}$$. Then, from (19), we have   \begin{eqnarray*} \mathbb{E}\left\|u^{\lambda}(t,x)-u^{\lambda}(t,y)\right\|^p &\leq& 2^{p-1}\Bigg(\mathbb{E}\left\|B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^\top_s)G(T-s)\Big(f(s,x_s)-f(s,y_s)\Big)ds\right\|^p\\ &&+\mathbb{E}\left\|B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^\top_s)G(T-s)\Big(g(s,x_s)-g(s,y_s)\Big)dW(s)\right\|^p\Bigg). \end{eqnarray*} According to $$(H_2)-(H_5)$$, Theorem 3.2 $$(i)$$, Lemma 3.2 and H$$\ddot{o}$$lder inequality we have   \begin{eqnarray*} \mathbb{E}\left\|u^{\lambda}(t,x)-u^{\lambda}(t,y)\right\|^p &\leq& 2^{p-1}\|B\|^p \Bigg[\frac{1}{\lambda^p}\overline{M}^{2p} T^{\frac{p}{q}} N_1 \mathbb{E}\int_0^t\|x_s-y_s\|^p_{C_{\alpha}} ds \\ &&+\overline{M}^{p}C_p\mathbb{E}\left(\int_0^t \left\|R(\lambda,-{\it{\Gamma}}^\top_s)G(T-s)\right\|^2\left\|g(s,x_s)-g(s,y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}}\Bigg]\\ &\leq&\frac{1}{\lambda^p} 2^{p-1}\|B\|^p {\overline{M}}^{2p} T^{\frac{p}{q}} N_1 \mathbb{E}\int_0^t\|x_s-y_s\|^p_{C_{\alpha}} ds \\ &&+\frac{1}{\lambda^p}2^{p-1}\|B\|^p \overline{M}^{2p} C_p N_1\mathbb{E}\left(\int_0^t\|x_s-y_s\|^2_{C_{\alpha}} ds\right)^{\frac{p}{2}} \\ &\leq& \frac{1}{\lambda^p} N_3 \mathbb{E}\int_0^t\|x_s-y_s\|^p_{C_{\alpha}} ds, \end{eqnarray*} for some constant $$N_3>0$$. Hence the lemma is proved. □ Next, for any $$\phi\in ML(p,r,\alpha)$$, we define the following mapping $$P^{\lambda}$$ on $$D_p$$ as   \begin{equation}\label{eq418} (P^{\lambda} y)(t) =\left\{ \begin{aligned} &G(t)A^\alpha\phi(0)+\int_0^t A^\alpha G(t-s)\Big(L \left({\tilde{\phi}_s}\right)+f(s,A^{-\alpha}y_s)+Bu^{\lambda}(s,A^{-\alpha}y)\Big)ds \\ & +\int_0^t A^\alpha G(t-s)g(s,A^{-\alpha}y_s)dW(s), \,\,\,\,t\in[0,T],\\ &A^\alpha \phi(t), \,\,\,\,\,\,\,\,t\in [-r,0], \end{aligned} \right. \end{equation} (20) with   \begin{align}\label{eq419} u^{\lambda}(t,A^{-\alpha}y):=& B^*G^*(T-t)R(\lambda,-{\it{\Gamma}}^\top_0)\Big(\mathbb{E}h-G(T)\phi(0)\Big)\nonumber\\ &- B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^\top_s)G(T-s)\Big(L\left({\tilde{\phi}_s}\right)+f(s,A^{-\alpha}y_s)\Big)ds \nonumber\\ &- B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^T_s)\Big(G(T-s)g(s,A^{-\alpha}y_s)-\varphi(s)\Big)dW(s). \end{align} (21) In the sequel, we need to prove the existence of mild solutions of System (1). Clearly, if the operator $$P^{\lambda}$$ has a fixed point $$y(\cdot,\phi)$$ on $$D_{p}$$, then $$x(\cdot,\phi)=A^{-\alpha}y(\cdot,\phi)$$ is a mild solution of System (1) belonging to $\(C([-r,T];L_p({\it{\Omega}};X_\alpha))\)$. To make our arguments clear we first prove several auxiliary lemmas before presenting the existence theorem. Lemma 4.2 Assume that $$(H_1)-(H_4)$$ hold. Let $$2<p<\infty$$ and $$0<\alpha<\frac{p-2}{2p}$$. Then, for any $$\phi\in ML(p,r,\alpha)$$ and $$y\in D_p, (P^{\lambda} y)(\cdot)$$ is continuous on the interval $$[0,T]$$ in the $$L_p$$ sense. Proof. Let $$0 \leq t_1 < t_2 < T$$. Then for any fixed $$y\in D_p$$, define   $$J_1 := \mathbb{E}\left\|\left(P^{\lambda}y\right)(t_1)-\left(P^{\lambda}y\right)(t_2)\right\|^p.$$ We will show that $$J_1$$ tends to zero when $$t_2\to t_1$$ which implies the right continuity of $$(P^{\lambda} y)(\cdot)$$ on $$[0,T)$$ in the $$L_p$$ sense. By definition (20) of $$P^{\lambda} y$$ we have   \begin{eqnarray*} J_1 &\leq&5^{p-1}\mathbb{E} \left\|\left(G(t_1)-G(t_2)\right)A^\alpha\phi(0)\right\|^p \\ &&+ 5^{p-1}\mathbb{E} \left\|\int_0^{t_2} A^\alpha G(t_2-s)L \left({\tilde{\phi}_s}\right) ds-\int_0^{t_1} A^\alpha G(t_1-s)L \left({\tilde{\phi}_s}\right)ds\right\|^p \\ &&+ 5^{p-1}\mathbb{E} \left\|\int_0^{t_2} A^\alpha G(t_2-s)f(s,A^{-\alpha}y_s)ds-\int_0^{t_1} A^\alpha G(t_1-s)f(s,A^{-\alpha}y_s)ds\right\|^p \\ &&+ 5^{p-1}\mathbb{E} \left\|\int_0^{t_2} A^\alpha G(t_2-s)Bu^{\lambda}(s,A^{-\alpha}y)ds-\int_0^{t_1} A^\alpha G(t_1-s)Bu^{\lambda}(s,A^{-\alpha}y)ds \right\|^p \\ &&+ 5^{p-1}\mathbb{E} \left\|\int_0^{t_2} A^\alpha G(t_2-s)g(s,A^{-\alpha}y_s)dW(s)-\int_0^{t_1} A^\alpha G(t_1-s)g(s,A^{-\alpha}y_s)dW(s)\right\|^p, \end{eqnarray*} hence   \begin{eqnarray*} J_1 &\leq&5^{p-1}\mathbb{E} \left\|\left(G(t_1)-G(t_2)\right)A^\alpha\phi(0)\right\|^p +10^{p-1}\Bigg(\mathbb{E}\left\|\int_{t_1}^{t_2} A^\alpha G(t_2-s)L \left({\tilde{\phi}_s}\right)ds\right\|^p \\ &&+\mathbb{E}\left\|\int_{t_1}^{t_2} A^\alpha G(t_2-s)f(s,A^{-\alpha}y_s)ds\right\|^p +\mathbb{E}\left\|\int_{t_1}^{t_2} A^\alpha G(t_2-s)Bu^{\lambda}(s,A^{-\alpha}y)ds\right\|^p\\ &&+\mathbb{E}\left\|\int_{t_1}^{t_2} A^\alpha G(t_2-s)g(s,A^{-\alpha}y_s)dW(s)\right\|^p +\mathbb{E}\left\|\int_0^{t_1} A^\alpha \left(G(t_2-s)-G(t_1-s)\right)L \left({\tilde{\phi}_s}\right) ds\right\|^p\\ &&+\mathbb{E}\left\|\int_0^{t_1} A^\alpha \left(G(t_2-s)-G(t_1-s)\right)f(s,A^{-\alpha}y_s)ds\right\|^p \\ &&+\mathbb{E}\left\|\int_0^{t_1} A^\alpha \left(G(t_2-s)-G(t_1-s)\right)Bu^{\lambda}(s,A^{-\alpha}y)ds\right\|^p \\ &&+\mathbb{E} \left\|\int_0^{t_1} A^\alpha \left(G(t_2-s)-G(t_1-s)\right)g(s,A^{-\alpha}y_s)dW(s)\right\|^p \Bigg)\\ &=&\sum_{\imath=1}^9 I_{\imath}. \end{eqnarray*} According to Theorem 3.2 $$(iv)$$ and $$(H_3)$$ together with H$$\ddot{o}$$lder inequality we can find a positive constant $$l_{2}>0$$ such that   \begin{align*} I_{2} &\leq 10^{p-1}\overline{M}^p_\alpha\mathbb{E}\left(\int_{t_1}^{t_2}(t_2-s)^{-\alpha} \left\|L \left({\tilde{\phi}_s}\right)\right\|ds\right)^p \\ &\leq 10^{p-1} \overline{M}^p_\alpha \mathbb{E}\left[\left(\int_{t_1}^{t_2}(t_2-s)^{-\alpha q}ds\right)^{\frac{p}{q}}\left(\int_{t_1}^{t_2}\left\|L \left({\tilde{\phi}_s}\right)\right\|^p ds\right)\right]\\ &\leq 10^{p-1} \overline{M}^p_\alpha \mathbb{E}\left[\left(\frac{(t_2-t_1)^{1-\alpha q}}{1-\alpha q}\right)^{\frac{p}{q}}l^p \left\|A^{-\alpha}\right\|^p \int_{t_1}^{t_2}\|{\tilde{\phi}_s}\|^p_{{\mathcal{B}}_{\alpha}}ds\right]\\ &\leq l_{2} \left(\frac{1}{1-\alpha q}\right)^{\frac{p}{q}}(t_2-t_1)^{p(1-\alpha)}\mathbb{E}\|\phi\|^p_{C_{\alpha}}, \end{align*} where $$\frac{1}{p}+\frac{1}{q}=1$$. Similarly, using Theorem 3.2 $$(iv)$$, Lemma 4.1, $$(H_2)$$, $$(H_4)$$ and H$$\ddot{o}$$lder inequality, one has that   \begin{eqnarray*} I_{3} &\leq& l_{3}\left(\frac{1}{1-\alpha q}\right)^{\frac{p}{q}} (t_2-t_1)^{p(1-\alpha)}\left(1+\|y\|^p_{D_p}\right)\!, \end{eqnarray*} and   \[ I_{4} \leq l_{4} \left(\frac{1}{1-\alpha q}\right)^{\frac{p}{q}}(t_2-t_1)^{p(1-\alpha)}\left(1+\|y\|^p_{D_p}\right)\!, \] for some constants $$l_{3}, l_4 >0$$. On the other hand, by using $$(H_4)$$, Theorem 3.2 $$(iv)$$ and Lemma 3.2, we obtain that for some $$C'_p$$,   \begin{eqnarray*} I_5 &\leq & 10^{p-1}C'_p\mathbb{E}\left( \int_{t_1}^{t_2}\left\| A^\alpha G(t_2-s)g(s,A^{-\alpha}y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}}\\ &\leq& 10^{p-1}C'_p \overline{M}^p_\alpha \mathbb{E}\left(\int_{t_1}^{t_2}(t_2-s)^{-2\alpha} \left\|g(s,A^{-\alpha}y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}} \\ &\leq& 10^{p-1}C'_p \overline{M}^p_\alpha \mathbb{E}\left[\left(\int_{t_1}^{t_2}(t_2-s)^{-2\alpha {\frac{p}{p-2}}}ds\right)^{\frac{p-2}{2}} \left(\int_{t_1}^{t_2}\left\|g(s,A^{-\alpha}y_s)\right\|^p_{{\mathscr L}_2^0} ds\right)\right] \\ &\leq& 10^{p-1}C'_p N_1 \overline{M}^p_\alpha \left(\frac{p-2}{p-2-2\alpha p}\right)^{\frac{p-2}{2}}(t_2-t_1)^{\frac{p-2-2\alpha p}{2}} \mathbb{E}\int_{t_1}^{t_2}\left(1+\left\|y_s\right\|^p_{C}\right)ds \\ &\leq& l_{5}\left(\frac{p-2}{p-2-2\alpha p}\right)^{\frac{p-2}{2}}(t_2-t_1)^{\frac{p-2\alpha p}{2}}\left(1+\|y\|^p_{D_p}\right)\!, \end{eqnarray*} for $$l_{5}>0$$ and $$p-2-2\alpha p>0.$$ Also, there holds   \begin{align*} I_{6} &\leq 20^{p-1} \mathbb{E}\left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right) L \left({\tilde{\phi}_s}\right)\right\| ds\right)^p\\ &\quad{} +20^{p-1}\mathbb{E}\left(\int_{t_1-\epsilon}^{t_1}\left\|A^\alpha \left(G(t_2-s)-G(t_1-s)\right)L \left({\tilde{\phi}_s}\right)\right\|ds\right)^p\\ &\leq 20^{p-1}\mathbb{E}\left[\left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}} \left(\int_0^{t_1-\epsilon}\left\| L \left({\tilde{\phi}_s}\right)\right\|^p ds\right)\right]\\ &\quad{} +20^{p-1}\mathbb{E}\left[\left(\int_{t_1-\epsilon}^{t_1}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}} \left(\int_{t_1-\epsilon}^{t_1}\left\| L \left({\tilde{\phi}_s}\right)\right\|^p ds\right)\right]\\ &\leq 20^{p-1}\left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}}l^p \left\|A^{-\alpha}\right\|^p \mathbb{E}\int_{0}^{t_1-\epsilon}\|\tilde{\phi}_s\|^p_{{\mathcal{B}}_{\alpha}}ds\\ &\quad{} +20^{p-1}\left[ \int_{t_1-\epsilon}^{t_1}\left(\frac{\overline{M}_\alpha}{(t_2-s)^\alpha}+\frac{\overline{M}_\alpha}{(t_1-s)^\alpha}\right)^q ds\right]^{\frac{p}{q}}l^p \left\|A^{-\alpha}\right\|^p \mathbb{E}\int_{t_1-\epsilon}^{t_1}\|\tilde{\phi}_s\|^p_{{\mathcal{B}}_{\alpha}}ds\\ &\leq l_{61}\left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}}\\ &\quad{} +l_{62}\frac{\overline{M}_\alpha^p}{(1-\alpha q)^{\frac{p}{q}}}\left[\left(t_2-t_1\right)^{1-\alpha q}-\left(t_2-t_1+\epsilon\right)^{1-\alpha q}-\epsilon^{1-\alpha q}\right]^{\frac{p}{q}}\!, \end{align*} where $$l_{61},l_{62}>0$$. Similarly, there exist $$l_{71}, l_{72}, l_{81}, l_{82}>0$$ such that   \begin{eqnarray*} I_7 &\leq & l_{71} \left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}}\\ &&+l_{72}\frac{\overline{M}_\alpha^p}{(1-\alpha q)^{\frac{p}{q}}}\left[\left(t_2-t_1\right)^{1-\alpha q}-\left(t_2-t_1+\epsilon\right)^{1-\alpha q}-\epsilon^{1-\alpha q}\right]^{\frac{p}{q}}\!, \end{eqnarray*} and   \begin{eqnarray*} I_8 &\leq & l_{81} \left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}}\\ &&+l_{82}\frac{\overline{M}_\alpha^p}{(1-\alpha q)^{\frac{p}{q}}}\left[\left(t_2-t_1\right)^{1-\alpha q}-\left(t_2-t_1+\epsilon\right)^{1-\alpha q}-\epsilon^{1-\alpha q}\right]^{\frac{p}{q}}\!. \end{eqnarray*} Finally, from Lemma 3.2, it follows immediately that, for some constant $$C''_p\geq 0$$,   \begin{eqnarray*} I_9 &\leq & 10^{p-1}C''_p\mathbb{E} \left(\int_0^{t_1}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)g(s,A^{-\alpha}y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}} \\ &\leq& 10^{p-1}C''_p \mathbb{E}\left(\int_{0}^{t_1-\epsilon} \left\|A^\alpha \left(G(t_2-s)-G(t_1-s)\right)g(s,A^{-\alpha}y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}} \\ &&+ 10^{p-1}C''_p \mathbb{E}\left(\int_{t_1-\epsilon}^{t_1} \left\|A^\alpha G(t_2-s)-G(t_1-s)g(s,A^{-\alpha}y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}} \\ &=& I_{91}+I_{92}. \end{eqnarray*} By virtue of (3), we see easily that there exists a positive constant $$l_{91}>0$$ such that   \begin{eqnarray*} I_{91} &\leq& 10^{p-1} C''_p \left(\int_{0}^{t_1-\epsilon}\left\|A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^{2\frac{p}{p-2}} ds\right)^{\frac{p-2}{p}}\mathbb{E}\int_{0}^{t_1-\epsilon} \left\|g(s,A^{-\alpha}y_s)\right\|^p_{{\mathscr L}_2^0} ds\\ &\leq& l_{91} \left(\int_{0}^{t_1-\epsilon}\left\|A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^{2\frac{p}{p-2}} ds\right)^{\frac{p-2}{p}} \left(1+\|y\|^p_{D_p}\right)\!. \end{eqnarray*} Similarly, we have, for some constant $$l_{92}>0$$,   \begin{eqnarray*} I_{92} &\leq & l_{91} \left(\int_{t_1-\epsilon}^{t_1}\left\|A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^{2\frac{p}{p-2}} ds\right)^{\frac{p-2}{p}} \left(1+\|y\|^p_{D_p}\right)\\ &\leq & l_{91} \left[\int_{t_1-\epsilon}^{t_1}\left(\frac{\overline{M}_\alpha}{(t_2-s)^\alpha}+\frac{\overline{M}_\alpha}{(t_1-s)^\alpha}\right)^{\frac{2p}{p-2}} ds\right]^{\frac{p-2}{p}} \left(1+\|y\|^p_{D_p}\right)\\ &\leq & l_{92} \overline{M}_\alpha^2 \left(\frac{p-2}{p-2-2\alpha p}\right)^{\frac{p-2}{p}}\left[\left(t_2-t_1\right)^{\frac{p-2-2\alpha p}{p-2}}-\left(t_2-t_1+\epsilon\right)^{\frac{p-2-2\alpha p}{p-2}}-\epsilon^{\frac{p-2-2\alpha p}{p-2}}\right]^{\frac{p-2}{p}}\!. \end{eqnarray*} By Theorem 3.2 $$(i)$$, $$G(t)$$ is strongly continuous for all $$t\geq 0$$. Hence $$I_1$$ tends to zero when $$t_2\to t_1$$. On the other hand, since from Theorem 3.2 $$(iii)$$, $$A^\alpha G(t)$$ is uniformly continuous for $$t\in (0,T]$$, it follows that $$I_\imath, \imath=2,\cdots,9,$$ tend to zero as $$t_2\to t_1$$ and $$\epsilon$$ small enough. Hence, for $$2<p<\infty$$ and $$0<\alpha<\frac{p-2}{2p} (<\frac{p-1}{p}<1)$$, $$(P^\lambda y)(\cdot)$$ is continuous from the right in $$[0,T)$$. A similar reasoning shows that it is also continuous from the left in $$(0,T]$$. Therefore, the proof of the lemma is complete. □ Lemma 4.3 Assume that all the hypotheses of Lemma 4.2 hold. Then the operator $$P^\lambda$$ sends $$D_p$$ into itself, i.e., $$P^\lambda(D_p) \subset D_p$$. Proof. Let $$y\in D_p$$ and let $$t\in[0,T]$$. Then   \begin{align*} \mathbb{E} \left\|\left(P^\lambda y\right)_t \right\|^p_C &\leq 5^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|G(t+\theta)A^\alpha\phi(0)\right\|^p\\ &\quad{} +5^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s) L \left({\tilde{\phi}_s}\right)ds\right\|^p\\ &\quad{} +5^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s) f(s,A^{-\alpha}y_s)ds\right\|^p\\ &\quad{} +5^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s) Bu^{\lambda}(s,A^{-\alpha}y)ds\right\|^p\\ &\quad{} +5^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s) g(s,A^{-\alpha}y_s)dW(s)\right\|^p\\ &= \sum_{\imath=10}^{14} I_{\imath}, \end{align*} where, from Theorem 3.2 $$(i)$$,   $$ I_{10}\leq 5^{p-1}\overline{M}^p \mathbb{E} \|\phi\|^p_{C_\alpha}. $$ Moreover, using $$(H_3)$$, Theorem 3.2 $$(iv)$$ and H$$\ddot{o}$$lder inequality we get   \begin{eqnarray*} I_{11} &\leq& 5^{p-1}\overline{M}^p_\alpha\mathbb{E}\sup_{\theta\in[-r,0]}\left(\int_0^{t+\theta} (t+\theta-s)^{-\alpha}\left\|L \left({\tilde{\phi}_s}\right)\right\|ds\right)^p\\ &\leq& 5^{p-1}\overline{M}^p_\alpha \mathbb{E}\sup_{\theta\in[-r,0]}\left[\left(\int_0^{t+\theta}(t+\theta-s)^{-\alpha q}ds \right)^{\frac{p}{q}}\left(\int_0^{t+\theta}\left\|L \left({\tilde{\phi}_s}\right)\right\|^p ds\right)\right]\\ &\leq& l_{11}\mathbb{E}\left\|\phi\right\|^p_{C_{\alpha}}, \end{eqnarray*} for a positive constant $$l_{11}$$. Similarly, due to $$(H_4)$$, Lemma 4.1, Theorem 3.2 $$(iv)$$ and H$$\ddot{o}$$lder inequality, there exist $$l_{12}, l_{13}>0$$ such that   \begin{eqnarray*} I_{12} &\leq& l_{12}\left(1+\|y\|^p_{D_p}\right)\!,\\ I_{13} &\leq& l_{13}\left(1+\|y\|^p_{D_p}\right)\!, \end{eqnarray*} and from $$(H_4)$$ and Lemma 3.4 it follows that   \begin{eqnarray*} I_{14} &\leq& 5^{p-1}N_2 \int_0^t \mathbb{E}\left\|g(s,A^{-\alpha}y_s)\right\|^p_{{\mathscr L}_2^0} ds\\ &\leq&l_{14}\left(1+\|y\|^p_{D_p}\right)\!, \end{eqnarray*} for a positive constant $$l_{14}$$. The above computations show well that   $$\left\|P^\lambda y\right\|^p_{D_p}=\sup_{t\in[0,T]}\mathbb{E} \left\|\left(P^\lambda y\right)_t \right\|^p_C <+\infty.$$ Therefore, we obtain that $$P^\lambda(D_p)\subset D_p$$ and this completes the proof. □ Now, we are on the position to prove the existence and uniqueness of mild solution of (1). Theorem 4.1 Let $$2<p<\infty$$ and $$0<\alpha<\frac{p-2}{2p}$$. Suppose that the assumptions $$(H_1)-(H_5)$$ hold. Then, for any $$\phi\in ML(p,r,\alpha)$$, the operator $$P^\lambda$$ has a unique fixed point in $$D_p$$. Proof. We use the classical Banach fixed point theorem to prove this theorem. By Lemmas 4.2 and 4.3, $$P^\lambda$$ is a continuous operator on $$[0,T]$$ that maps $$D_p$$ into itself. It remains to show that there exists an $$n\in \mathbb{N}$$ such that $${P^\lambda}^n$$ is a contraction. Let $$(x, y) \in D_p^2$$, then, for any fixed $$t\in[0,T]$$, we have   \begin{eqnarray*} J_2&:=& \mathbb{E} \left\| \left(P^\lambda x\right)_t -\left(P^\lambda y\right)_t \right\|^p_C \\[1ex] &\leq& 3^{p-1} \mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s)\left(f(s,A^{-\alpha}x_s)-f(s,A^{-\alpha}y_s)\right)ds\right\|^p \\[1ex] &&+3^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s)B\left(u^{\lambda}(s,A^{-\alpha}x)-u^{\lambda}(s,A^{-\alpha}y)\right)ds\right\|^p \\[1ex] &&+3^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s)\left(g(s,A^{-\alpha}x_s)-g(s,A^{-\alpha}y_s)\right)dW(s)\right\|^p \\[1ex] &=&\sum_{\imath=15}^{17} I_{\imath}, \end{eqnarray*} which by using $$(H_4)$$, Theorem 3.2 $$(iv)$$ and H$$\ddot{o}$$lder inequality yields that   \begin{eqnarray*} I_{15} &\leq & 3^{p-1}\overline{M}_\alpha^p \mathbb{E}\sup_{\theta\in[-r,0]}\left(\int_0^{t+\theta}(t+\theta-s)^{-\alpha} \left\|f(s,A^{-\alpha}x_s)-f(s,A^{-\alpha}y_s)\right\|ds \right)^p\\[1ex] &\leq & 3^{p-1}\overline{M}_\alpha^p\mathbb{E}\sup_{\theta\in[-r,0]} \left[\left(\int_0^{t+\theta} (t+\theta-s)^{-\alpha q} ds\right)^{\frac{p}{q}}\left(\int_0^{t+\theta}\left\|f(s,A^{-\alpha}x_s)-f(s,A^{-\alpha}y_s)\right\|^p ds\right)\right] \\[1ex] &\leq & 3^{p-1}\overline{M}_\alpha^p N_1 \sup_{\theta\in[-r,0]}\left(\frac{(t+\theta)^{1-\alpha q}}{1-\alpha q}\right)^{\frac{p}{q}} \mathbb{E} \int_0^t \left\|x_s-y_s\right\|^p_{C}ds \\[1ex] &\leq & 3^{p-1}\overline{M}_\alpha^p N_1 \left(\frac{ T^{1-\alpha q}}{1-\alpha q}\right)^{\frac{p}{q}} \mathbb{E} \int_0^t \left\|x_s-y_s\right\|^p_{C}ds . \end{eqnarray*} Similarly, from $$(H_2)$$, Lemma 4.1, Theorem 3.2 $$(iv)$$ and H$$\ddot{o}$$lder inequality, we obtain   \begin{eqnarray*} I_{16} &\leq & \frac{1}{\lambda^p} 3^{p-1}\overline{M}_\alpha^p \|B\|^p N_3 \left(\frac{T^{1-\alpha q}}{1-\alpha q}\right)^{\frac{p}{q}}T \mathbb{E} \int_0^t \left\|x_s-y_s\right\|^p_{C}ds. \end{eqnarray*} Finally, $$(H_4)$$ and Lemma 3.4 yields to   \begin{eqnarray*} I_{17} &\leq & 3^{p-1}N_2 \int_0^t \mathbb{E} \left\|g(s,A^{-\alpha}x_s)-g(s,A^{-\alpha}y_s)\right\|^p_{{\mathscr L}_2^0} ds \\[1ex] &\leq & 3^{p-1}N_1 N_2\mathbb{E} \int_0^t \left\|x_s-y_s\right\|^p_{C}ds. \end{eqnarray*} Hence, there exists a $$B(\lambda)>0$$ such that   \begin{eqnarray*} \mathbb{E} \left\| \left(P^\lambda x\right)_t -\left(P^\lambda y\right)_t \right\|^p_C &\leq& B(\lambda)\mathbb{E}\int_0^t \left\|x_s-y_s\right\|^p_{C}ds\\ &\leq& T B(\lambda) \left\|x-y\right\|^p_{D_p}, \end{eqnarray*} for any $$t\in[0,T]$$. Then   $$\left\| P^\lambda x-P^\lambda y \right\|^p_{D_p} \leq T B(\lambda)\left\|x-y\right\|^p_{D_p},\,\,\,\,\,\,\,\, \mbox{for any}\,\,\,\, (x,y)\in D_p.$$ For any integer $$n\geq 1$$, by iteration, it follows that, for any $$x,y\in D_p$$,   $$\left\|{P^\lambda}^n x-{P^\lambda}^n y \right\|^p_{D_p} \leq \frac{(T B(\lambda))^n}{n!}\left\|x-y\right\|^p_{D_p}.$$ Since for sufficiently large $$n$$, $$\frac{(T B(\lambda))^n}{n!}< 1$$, $${P^\lambda}^n$$ is a contraction map on $$D_p$$ and therefore $${P^\lambda}$$ itself has a unique fixed point $$y(\cdot,\phi)$$ in $$D_p$$. The theorem is proved. □ Thus, by Theorem 4.1, for any $$\lambda\in(0,1)$$, the operator $${P^\lambda}$$ defined by (20) has a unique fixed point $$y^{\lambda}\in D_p$$, from which by setting $$x^{\lambda}(t,\phi)=A^{-\alpha}y^{\lambda}(t,\phi), t\in [-r,T],$$ we get a mild solution of System (1) which belongs to $\(C([-r,T];L_p({\it{\Omega}};X_\alpha))\)$. Using the Gronwall’s inequality, one can easily derive the uniqueness of the mild solution of System (1). Next we prove the main result of this work on the approximate controllability of System (1). That is Theorem 4.2 Let $$2<p<\infty$$ and $$0<\alpha<\frac{p-2}{2p}$$. Assume that the hypotheses $$(H_1)-(H_3), (H'_4)$$ and $$(H_5)$$ are all satisfied and suppose in addition that the semigroup $$\left(S(t)\right)_{t>0}$$ is compact. Then, for any $$\phi\in ML(p,r,\alpha)$$, System (1) is approximately controllable on $\([0,T]\)$. Proof. Let $$x^{\lambda}(\cdot,\phi)=A^{-\alpha}y^{\lambda}(\cdot,\phi)$$ be the mild solution of System (1) obtained in Theorem 4.1 under the control function $$u^{\lambda}(\cdot,A^{-\alpha}y)$$ given by (21). Then substituting (21) into (20) immediately yields that $$x^{\lambda}(\cdot,\phi)=\phi(\cdot)$$ on $$[-r,0]$$ and, for $$t\in[0,T]$$,   \begin{equation*} \begin{aligned} x^{\lambda}(t,\phi)&= G(t)\phi(0)+\int_0^t G(t-s)B B^* G^*(T-s)R(\lambda,-{\it{\Gamma}}_0^\top)\Big(\mathbb{E}h-G(T)\phi(0)\Big)\\ &\quad{} +\!\int_0^t\!\left(\!G(t-s)-R(\lambda,-{\it{\Gamma}}_s^\top)G(T-s)\int_s^t G(t-s)BB^*G^*(T-s)ds \!\right)\left(L \left({\tilde{\phi}_s}\right)+f(s,x^{\lambda}_s)\right)\!ds\\ &\quad{} +\int_0^t\left(G(t-s)-R(\lambda,-{\it{\Gamma}}_s^\top)G(T-s)\int_s^t G(t-s)BB^*G^*(T-s)ds \right)g(s,x^{\lambda}_s)dW(s)\\ &\quad{} +\int_0^t\int_s^tG(t-\nu)BB^*G^*(T-\nu) d\nu R(\lambda,-{\it{\Gamma}}_s^\top)\varphi(s)dW(s), \end{aligned} \end{equation*} which at $$t=T$$ gives   \begin{align*} x^{\lambda}(T,\phi)& = G(T)\phi(0)+{\it{\Gamma}}_0^\top R(\lambda,-{\it{\Gamma}}_0^\top)\Big(\mathbb{E} h-G(T)\phi(0)\Big)\\ &\quad{} +\int_0^\top\left(I-{\it{\Gamma}}_s^\top R(\lambda,-{\it{\Gamma}}_s^\top) \right)G(T-s)\left(L \left({\tilde{\phi}_s}\right)+f(s,x^{\lambda}_s)\right)ds\\ &\quad{} +\int_0^\top\left(I-{\it{\Gamma}}_s^\top R(\lambda,-{\it{\Gamma}}_s^\top) \right)G(T-s)g(s,x^{\lambda}_s)dW(s)\\ &\quad{} +\int_0^\top {\it{\Gamma}}_s^\top R(\lambda,-{\it{\Gamma}}_s^\top)\varphi(s)dW(s), \end{align*} hence, using the fact that $$I-{\it{\Gamma}}_s^\top R(\lambda,-{\it{\Gamma}}_s^\top)=\lambda R(\lambda,-{\it{\Gamma}}_s^\top)$$, we get   $$ \begin{aligned} x^{\lambda}(T,\phi) & = h-\lambda R(\lambda,-{\it{\Gamma}}_0^\top)\Big(\mathbb{E} h-G(T)\phi(0)\Big)\\ &\quad{} +\int_0^\top\lambda R(\lambda,-{\it{\Gamma}}_s^\top)G(T-s)\left(L \left({\tilde{\phi}_s}\right)+f(s,x^{\lambda}_s)\right)ds\\ &\quad{} +\int_0^\top\lambda R(\lambda,-{\it{\Gamma}}_s^\top)\left(G(T-s)g(s,x^{\lambda}_s)-\varphi(s)\right)dW(s). \end{aligned} $$ By assumption $$(H'_4)$$, we know that $$\left\{f(s,x^{\lambda}_s):\lambda\in(0,1)\right\}$$ and $$\left\{g(s,x^{\lambda}_s):\lambda\in(0,1)\right\}$$ are uniformly bounded in $$\lambda\in(0,1)$$ in $$X$$ and $${{\mathscr L}_2^0}(K;X)$$ respectively, from which it follows that there exist subsequences, still denoted by $$f(s,x^{\lambda}_s)$$ and $$g(s,x^{\lambda}_s)$$, that converge weakly to, say, $$f(s)$$ and $$g(s)$$ in $$X$$ and $${{\mathscr L}_2^0}(K;X)$$, respectively, for each $$s\in[0,T]$$. Then the compactness of $$G(t), t > 0,$$ which is guaranteed by Theorem 3.2 $$(ii)$$, implies that   \begin{align*} G(T-s)f(s,x^{\lambda}_s)&\to G(T-s)f(s),\\ G(T-s)g(s,x^{\lambda}_s)&\to G(T-s)g(s), \mbox{in} (0,T]\times {\it{\Omega}}. \end{align*} Hence   \begin{align*} \mathbb{E}\left\|x^{\lambda}(T,\phi)-h\right\|^p &\leq 7^{p-1}\left\|\lambda R(\lambda,-{\it{\Gamma}}_0^\top)\left(\mathbb{E} h-G(T)\phi(0)\right)\right\|^p\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top\left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top)G(T-s) L \left({\tilde{\phi}_s}\right)\right\|ds\right)^p\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top\left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top)\right\|\left\|G(T-s)\left(f(s,x^{\lambda}_s)-f(s)\right)\right\|ds\right)^p\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top\left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top)G(T-s)f(s)\right\|ds\right)^p\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top\left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top)\right\|^2 \left\|G(T-s)\left(g(s,x^{\lambda}_s)-g(s)\right)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}}\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top \left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top) G(T-s)g(s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}}\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top\left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top)\varphi(s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}}\\ &\to 0 \,\,\,\,\mbox{as} \,\,\,\, \lambda\to 0^+. \end{align*} Consequently, System (1) is approximately controllable on $$[0,T]$$. □ 5. Example To illustrate the obtained results, we present in this section an example of controllable SPDE. Consider the following stochastic boundary value problem.   \begin{equation}\label{eq420}\left\{ \begin{aligned} &d z(t,x)= \left[-\frac{\partial^2}{\partial x^2}z(t,x)+z(t-1,x)+F_1(t,z(t-r_1(t),x))+Bu(t,x)\right]dt\\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +F_2(t,z(t-r_2(t),x)) d\beta(t), \,\,\,\,0< t\leq 2, \,\,\,\,0\leq x\leq\pi,\\ &z(t,0)=z(t,\pi)=0, \,\,\,\,0\leq t\leq 2,\\ &z(\theta,x)=\phi_0(\theta,x), \,\,\,\,-1\leq \theta\leq 0, \,\,\,\,0\leq x\leq\pi, \end{aligned}\right. \end{equation} (22) where $$r_1, r_2$$ are continuous functions with $$0<r_1(\cdot), r_2(\cdot)\le r$$, and $$\phi_0$$ is $$\mathscr{F}_0-$$measurable. The functions $$F_1$$ and $$F_2$$ will be described below. $$\beta(t)$$ denotes a one-dimensional standard Brownian motion. System (22) arises in the study of stochastic heat flow in materials of the so-called retarded type. Here, $$z(t,x)$$ represents the temperature of the point $$x$$ at time $$t$$. For this stochastic equation without the second term $$z(t-1,x)$$ on the right hand side, reference Dauer & Mahmudov (2004) has already established the sufficient conditions for its approximate controllability. Clearly, the results in Dauer & Mahmudov (2004) can not be applied to the system (22) as $$z(t-1,x)$$ is not uniformly bounded. However, we may explore here the approximate controllability of the stochastic system (22) by using the results achieved in the last section. Let $$X=L_2([0,\pi]), K=\mathbb{R}$$ and $$U=L_2([0,2])$$. Let $$A: D(A)\to X$$ be the operator given by   $$A\xi=-\xi'',$$ with the domain   \begin{align*} D(A)= \Bigg\{&\xi\in X \,\,\,\,\mbox{such that}\,\,\,\, \;\xi(0)=\xi(\pi)=0 \,\,\,\,\mbox{with}\,\,\,\, \frac{\partial\xi}{\partial x},\;\frac{\partial^2\xi }{\partial x^2} \,\,\,\,\mbox{all in}\,\,\,\, X, \\ & \xi\,\,\,\, \mbox{and}\,\,\,\, \frac{\partial\xi}{\partial x}\,\,\,\, \mbox{are both absolutely continuous on} \,\,\,\,[0,\pi]\;\Bigg\}. \end{align*} Then $$-A$$ generates a strongly continuous semigroup $$(S(t))_{t\ge 0}$$ which is analytic, compact and self-adjoint. Furthermore, $$-A$$ has a discrete spectrum, the eigenvalues are $$-n^2,\;n\in \mathbb{N^+}$$, with the corresponding normalized eigenvectors $$e_n(x)= \sqrt{\frac{2}{\pi}}\sin(nx)$$, $$n=1,2,\cdots$$. Then the following properties hold: (i) If $$\xi\in D(A)$$, then   $$A\xi= \sum_{n=1}^{\infty} n^2 \langle \xi,e_n \rangle e_n.$$ (ii) For every $$\xi\in X$$,   \begin{equation*} S(t)\xi=\sum_{n=1}^{\infty} e^{-n^2t}\langle \xi,e_n \rangle e_n. \end{equation*} Define $$A^\alpha$$ for self-adjoint operator $$A$$ by the classical spectral theorem and it is easy to deduce that   $$|A^{\alpha}|S(t)\xi=\sum\limits_{n=1}^{\infty}\left(n^2\right)^{\alpha} e^{-n^2 t}\langle \xi,e_n \rangle e_n ,$$ which immediately implies   \begin{align*} \left\|A^{\alpha}S(t)\xi\right\|^2&=\sum\limits_{n=1}^{\infty} n^{4\alpha} e^{-2 n^2 t} \left|\langle \xi,e_n \rangle \right|^2\\ &=e^{-2 a t} t^{-2\alpha}\sum\limits_{n=1}^{\infty} (n^2 t)^{2\alpha} e^{-\left(2 n^2-2a\right)t} \left|\langle \xi,e_n \rangle \right|^2\!. \end{align*} Then $$(H_1)$$ holds. Let $$2<p<\infty$$ and $$0<\alpha<\frac{p-2}{2p}$$ such that there exists some number $$\beta$$ satisfies $$\frac{1}{p}+\alpha<\beta<\frac12$$. Suppose that the following conditions hold for System (22): (i) The functions $$F_i:[0,2]\times \mathbb{R}\to \mathbb{R}$$ are Lipschitz continuous in the second variable and uniformly bounded. i.e., there exist positive constants $$c_1$$ and $$c_2$$ such that   $$|F_{i}(t,x_1)-F_{i}(t,x_2)| \leq c_i |x_1-x_2|, \,\,\,\,i=1,2,$$ for any $$t\in [0,2], x_i\in \mathbb{R}$$. Moreover, there exists a constant $$c_0> 0$$ such that, for any $$t\in [0,2], x\in \mathbb{R}$$,   $$|F_{i}(t,x,y)|\leq c_0.$$ (ii) The function $$\phi_0(\theta,x)$$ belongs to $$ML(p,r,\alpha)$$. To represent problem (22) as the abstract form of System (1), we define $$Z(t)(\cdot):= z(t,\cdot)$$ and $$\phi(t)(\cdot):= \phi(t,\cdot)$$. Now define the operator $$L: \mathcal{B}([-1,0];X) \to X$$, the maps $$f(\cdot,\cdot):[0,2]\times C_\alpha\to X$$ and $$g(\cdot,\cdot):[0,2]\times C_\alpha \to {{\mathscr L}_2^0}(\mathbb{R};X)$$, respectively, as   \begin{align*} L(\phi)(x) & =L(\phi(\cdot,x))=\phi(-1,x),\\ f(t,\psi)(x) & = f(t,\psi(\cdot,x))=F_1\left(t,\psi(-r_1(t),x)\right)\!,\\ g(t,\psi)(x) & =g(t,\psi(\cdot,x))=F_2\left(t,\psi(-r_2(t),x)\right)\!, \end{align*} for any $$t\in [0,2]$$, $$\phi\in \mathcal{B}([-1,0];X)$$ and $$\psi\in C_\alpha$$. Then under these notations System (22) can be rewritten into the form of System (1). Clearly, $$L$$ is a linear bounded operator satisfying $$(H_3)$$. On the other hand, from the computations in Taniguchi et al. (2002) we see that the functions $$f(\cdot,\cdot)$$ and $$g(\cdot,\cdot)$$ are Lipschitz continuous with respect to the second variable. Moreover, their uniform boundedness follows directly from the uniform boundedness of $$F_{\imath}(\cdot,\cdot), \imath=1,2$$. Therefore, hypotheses $$(H_4')$$ is satisfied. Here, as usual, we take   $$ U=\left\{u=\sum\limits_{n=2}^{\infty}u_ne_n: \sum\limits_{n=2}^{\infty}u_n^2< +\infty\right\}\!, $$ with the norm   $$ \|u\|= \bigg(\sum\limits_{n=2}^{\infty}u_n^2\bigg)^{\frac{1}{2}}. $$ Then $$U$$ is a Hilbert space. Now define the linear continuous operator $$B$$ from $$U$$ into $$X$$ as   $$ Bu=2u_2e_1(x)+\sum\limits_{n=2}^{\infty}u_ne_n(x), \,\,\,\,\text{for}\,\,\,\, u=\sum\limits_{n=2}^{\infty}u_ne_n\in U. $$ It is easy to compute that   \begin{equation}\label{eq421} B^*v=(2v_1+v_2)e_2(x)+\sum\limits_{n=3}^{\infty}v_ne_n(x), \end{equation} (23) with $$v=\sum\limits_{n=1}^{\infty}v_ne_n(x)\in X.$$ So $$(H_2)$$ is verified too. According to Theorem 4.2, to obtain the approximate controllability for System (22), it remains to us to verify the resolvent condition ($$H_5$$). As one can see, it is difficult for us to obtain the explicit expression of the fundamental solution $$G(t)$$ associated to the linear system. Fortunately, however, we are able to calculate the expression of $$G(t)$$ on the interval $$[0,1]$$, and this is enough to guarantee that the condition ($$H_5$$) holds in this situation. Indeed, the solution on the interval $$[0,1]$$ of the corresponding linear deterministic equation   $$\left\{ \begin{aligned} &\frac{d}{dt}Z(t)=-AZ(t)+L(Z_t)+ f(t),\,\,\,\, t\in [0,2],\\ &Z_0=0, \end{aligned} \right.$$ is given by the $$C_0-$$semigroup $$S(t)$$ as follows:   $$Z(t)=\int_0^t S(t-s)f(s)ds,\,\,\,\,\,\,\,\, t\in [0,1].$$ This indicates that $$G(t)=S(t)$$ for $$t\in [0,1]$$. Thus we have   $$G^*(t)=S^*(t)=S(t)=G(t),\,\,\,\,\text{for}\,\,\,\, t\in [0,1].$$ Hence combining (23) we calculate directly that   $$B^*G^*(t)\xi =(2\xi_1e^{-t}+\xi_2e^{-4t})e_2(x)+\sum\limits_{n=3}^{+\infty}\xi_ne^{-n^2t}e_n(x),$$ for $$\xi=\sum\limits_{n=1}^{+\infty}\xi_ne_n(x)\in X$$ and $$t\in [0,1]$$. Now let $$\|B^*G^*(t)\xi\|=0,\text{for all} t\in [0,2]$$, then   $$ \|B^*G^*(t)\xi\|=0,\,\,\,\, t\in [0,1], $$ it follows that   $$\|2\xi_1e^{-t}+\xi_2e^{-4t}\|^2+\sum\limits_{n=3}^{+\infty}\|\xi_ne^{-n^2t}\|^2=0, \,\,\,\,t\in [0,1],$$ which implies $$\xi_n=0, n=1,2,\cdots.$$ and hence $$\xi =0$$. So, by virtue of Theorem 3.4, $$(H_5)$$ holds. Therefore by Theorem 4.2 we infer that System (22) is approximately controllable on the interval [0,2]. 6. Conclusion In this article, we studied the approximate controllability of the stochastic system (1) with finite delay in $$L_p$$ spaces ($$2<p<\infty$$). The resolvent condition was used to get the desired results by equivalently assuming that the linear deterministic part of the system (1) is approximately controllable. Since this approach requires the nonlinear terms of the system be uniformly bounded and since system (1) involves an additional linear term which is not uniformly bounded, we established the fundamental solution theory corresponding to the associated deterministic linear system and we have used it to describe in an explicit way the mild solutions of the considered stochastic system. Thus we could discuss the approximate controllability for System (1) with partially non-uniformly bounded term under the resolvent condition and obtained sufficient conditions for it. Therefore, this work extends somewhat the existing related results in the literature. It is worth pointing out that, as the fundamental solution $$G(t)$$ has the same regular properties as the semigroup and the expression of mild solutions becomes much simpler due to the use of $$G(t)$$, it is convenient for us to apply the fundamental solution theory to study qualitative issues and other topics on control theory such as stabilization, optimal control for deterministic and stochastic partial functional differential equations (PFDEs) with finite delay, which are our subsequent works in the recent future. There are some direct problems for our further discussion. First we may extend the results of this article to neutral stochastic PFDEs with finite delay by establishing the fundamental solution theory for the associated neutral linear system. We will also explore the optimal control problems for stochastic PFDEs with finite delay by utilizing the fundamental solution theory founded here. 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Approximate controllability for a retarded semilinear stochastic evolution system

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Oxford University Press
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© The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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0265-0754
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1471-6887
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10.1093/imamci/dnx045
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Abstract

Abstract In this work, we study the approximate controllability for a class of semilinear stochastic evolution systems with finite delays in $$L_p$$ space. The main technique is the fundamental solution theory constructed through Laplace transformation. The approximate controllability result is obtained via the so-called resolvent condition. The nonlinear terms are only required to be partly uniformly bounded. An example is provided to illustrate the obtained results. 1. Introduction This article is concerned with the approximate controllability of the following stochastic partial differential equation (SPDE) with finite delay   \begin{equation}\label{eq41} \left\{\begin{aligned} &d y(t) =\big(-Ay(t)+L(y_t)+ f(t,y_t)+Bu(t)\big)dt+g(t,y_t)dW(t), \,\,\,\, 0\leq t\leq T,\\ &y_0=\phi \in ML(p,r,\alpha), \end{aligned}\right. \end{equation} (1) on a Hilbert space $$X$$. Here, $$y(t)$$ is the state variable and its histories $$y_{t}$$ are given in the usual way by $$y_{t}(\theta)=y(t+\theta)$$, for $$\theta\in[-r,0]$$, and belong to the phase space $$ML(p,r,\alpha)$$ to be defined later. The control function $$u(\cdot)$$ takes values in another Hilbert space $$U$$. The operator $$-A: D(-A)\subseteq X\to X$$ is the infinitesimal generator of an analytic semigroup $$\left(S(t)\right)_{t\geq 0}$$ while the operators $$L$$ and $$B$$ are linear bounded. The functions $$f(\cdot,\cdot)$$ and $$g(\cdot,\cdot)$$ are Lipschitz continuous and uniformly bounded given below, and $$W(t)$$ is a $$Q-$$Wiener process. System (1) is an abstract form of SPDEs in infinite dimensional space. This kind of equations arises naturally in the mathematical models of various phenomena in natural and social science; see DA Prato & Zabbczyk (1992), Grecksch & Tudor (1995) and Sobczyk (1991), for some practical background of this kind of equations. Recently, quantitative and qualitative properties (existence, stability, invariant measures, etc.) of SPDEs with finite delays have gained much attention of many mathematicians, see Bezandry & Diagana (2012), Caraballo (1990), Lei et al. (2007), Liu (2014), Taniguchi (1998), Taniguchi et al. (2002) and You & Yuan (2014) among others. In particular, the controllability problems of finite-delayed SPDEs have also received the attention of many authors, see Dauer & Mahmudov (2004), Ganthi & Muthukumar (2012), Muthukumar & Balasubramaniam (2009) and Sakthivel et al. (2012) for instance. In reference Dauer & Mahmudov (2004), the authors have investigated the approximate controllability of the following stochastic evolution equation with finite delay   \begin{equation}\label{eq52} \left\{\begin{aligned} &d y(t) =\big(-Ay(t)+ f(t,y_t)+Bu(t)\big)dt+g(t,y_t)dW(t),\,\,\,\, 0\leq t\leq T,\\ &y_0=\phi, \end{aligned}\right. \end{equation} (2) by using the $$C_0-$$semigroup theory and the resolvent condition. The so-called resolvent condition (see condition $$(H_5)$$ in Section 3), originating from Bashirov & Mahmudov (1999), is equivalent to the approximate controllability of the corresponding linear system, and it is an important approach to study the approximate controllability of (stochastic) semilinear evolution systems. In recent years, this condition has been extensively adopted to study the controllability problems for various deterministic and stochastic semilinear evolution systems. For instance, in references Balasubramaniam & Tamilalagan (2015), Tamilalagan & Balasubramaniam (2017), Ren et al. (2013), Sakthivel et al. (2011, 2012, 2013, 2016) and Shen & Sun (2012), the authors have successfully employed this resolvent condition to explore the approximate controllability respectively for deterministic and stochastic (fractional) differential systems with nonlocal conditions or with finite (infinite) delays. The outstanding feature of this approach is that the nonlinear terms in the considered systems should satisfy the uniform boundedness condition, see Balasubramaniam & Tamilalagan (2015), Dauer & Mahmudov (2004), Dauer et al. (2006), Fu et al. (2014), Ganthi & Muthukumar (2012), Mokkedem & Fu (2016), Mokkedem & Fu (2017), Sakthivel et al. (2011, 2012, 2013, 2016) and Shen & Sun (2012) and the references therein. We also point out that another method to discuss the approximate controllability for semilinear evolution systems is to apply the range condition initiated in Naito (1987). In Muthukumar & Balasubramaniam (2009) and Wang (2009), for example, the authors obtained the controllability results by using exactly this technique. However, one can see that generally it is difficult to verify such a range condition in practice, especially for (stochastic) functional differential equations (FDEs) with time delay (including the system (22) to be discussed in Section 5 since the initial function $$\phi$$ is not null). The motivation of this article is the approximate controllability of stochastic partial functional equations which can be rewritten as the abstract form (1), like the retarded stochastic heat conduction system (22). We will study this problem through investigating the approximate controllability for the system (1). It should be observed that, there is an extra linear term $$L(y_t)$$ in System (1) which is not uniformly bounded and hence it can not be regarded as a special case of System (2). As a result, the existing results in Dauer & Mahmudov (2004) and Dauer et al. (2006) on approximate controllability become invalid for System (1). Moreover, the theory of $$C_0-$$semigroup is not enough to solve such problem, since the semigroup generated by $$A+L$$ loses the compactness property. For this reason, instead of using the $$C_0-$$semigroup theory we study the approximate controllability of System (1) by adopting the fundamental solution one on the associated deterministic linear system with finite delay which will be founded in a similar way as that in Fu et al. (2014). By the construction of the fundamental solution $$G(t)$$, the linear term $$L\left(y_t\right)$$ is involved into $$G(t)$$ and it does not appear in the expression of mild solutions of System (1). In this manner, we overcome the obstacle of the non-uniformly boundedness of $$L\left(y_t\right)$$. Note the fundamental solution theory has been constructed for some linear (neutral) FDEs and has already been utilized to discuss the control problems for corresponding semilinear (neutral) FDEs in these years, and many interesting results are obtained in various topics, see Fu et al. (2014), Jeong et al. (1999), Jeong & Roh (2006), Liu (2009), Mokkedem & Fu (2016), Mokkedem & Fu (2017) and Wang (2009) among others. In particular, in article Mokkedem & Fu (2017) we have studied the approximate controllability for a semiliear stochastic FDEs with infinite delay via this technique. In this work, the analyticity of the fundamental solution $$G(t)$$ is discussed which allows us to employ the $$\alpha-$$norm and the fractional power operators approaches. In addition, results on compactness and uniform continuity of $$G(t)$$ and $$A^\alpha G(t)$$, for $$t>0$$ and $$\alpha \in (0,1)$$, are achieved as well. By these properties, the regularity of solutions of System (1) is improved and our obtained results become more meaningful. Clearly, the obtained results extend and develop many existing works such as Dauer & Mahmudov (2004) and Muthukumar & Balasubramaniam (2009). Moreover, the fundamental solution theory founded here can also be applied to discuss other important issues such as qualitative properties and optimal controls for semilinear FDEs with finite delay. This article is organized as follows. The notations and terminologies used in this article are presented in Section 2. In Section 3, we first discuss the fundamental solution of the corresponding deterministic linear system. Then, we apply it to define the mild solutions of System (1). After that, we present some lemmas to be used in the proofs of the main results. In Section 4, we start by showing the existence and uniqueness of mild solution of System (1) by making use of the Banach fixed point theorem. Then, we explore its approximate controllability on $$[0,T]$$ which is the main result of this article. Finally, in Section 5, an example is given to show the applications of the obtained results. 2. Preliminaries In this section, we collect the notations and the terminologies to be used in the whole article. Let $$X$$ be a separable Hilbert space with inner product $$<\cdot,\cdot>$$ and norm $$\|\cdot\|$$ and let $$K$$ be another separable Hilbert space with inner product $$<\cdot,\cdot>_K$$ and norm $$\|\cdot\|_K$$. We employ the same notation $$\|\cdot\|$$ for the norm of $${\mathscr{L}}(K;X)$$, where $${\mathscr{L}}(K;X)$$ denotes the space of all bounded linear operators from $$K$$ into $$X$$. Particularly, $${\mathscr{L}}(X)$$ will denote $${\mathscr{L}}(X;X)$$. Let $$-A: D(-A)\subseteq X\to X$$ be a closed, linear and densely defined operator generating an analytic semigroup $$\left(S(t)\right)_{t\ge 0}$$ on $$X$$. Assume that $$0 \in \rho(A)$$, where $$\rho(A)$$ denotes the resolvent set of $$A$$. Then, for $$\alpha \in(0,1]$$, it is possible to define the fractional power operator $$A^{\alpha}$$ as a closed linear operator on its domain $$D(A^{\alpha})$$. Furthermore, the subspace $$D(A^{\alpha})$$ is dense in $$X$$ and the expression   \[ \|x\|_{\alpha} =\|A^{\alpha}x\|,\qquad x \in D(A^{\alpha}), \] defines a norm on $$D(A^{\alpha})$$. Hereafter, we represent by $$X_{\alpha}$$ the space $$D(A^{\alpha})$$ endowed with the norm $$\|\cdot\|_{\alpha}$$. Then the following properties are well-known, see Engel & Nagel (2000) and Pazy (1983). Lemma 2.1 For the analytic semigroup $$\left(S(t)\right)_{t\geq 0}$$ generated by the operator $$(-A,D(-A))$$, there hold $$(i)$$ Let $$\alpha \in(0,1]$$, then $$X_{\alpha}$$ is a Banach space. $$(ii)$$ If $$0<\beta<\alpha\leq 1$$, then $$X_{\alpha} \mapsto X_{\beta}$$ and the imbedding is compact whenever $$(\lambda I+A)^{-1}$$, the resolvent operator of $$-A$$, is compact. $$(iii)$$ For every $$\alpha \in(0,1]$$, there exist a constant $$M_{\alpha}>0$$ and a real number $$a>0$$ such that   $$\left\|A^{\alpha}S(t)h\right\| \leq M_{\alpha} e^{-at} t^{-\alpha} \|h\|, \,\,\,\, t>0,$$ for any $$h\in X.$$ Particularly,   $$\|A^{\alpha}S(t)\|\leq \frac{M_\alpha}{t^\alpha},\,\,\,\, t\in (0,T].$$ Let $${\it{\Omega}} :=({\it{\Omega}},\mathscr{F},\left\{\mathscr{F}_t\right\}_{t\geq0},\mathbb{P})$$ be a filtered complete probability space satisfying the usual condition, which means that the filtration $$\left\{\mathscr{F}_t\right\}_{t\geq0}$$ is a right continuous increasing family and $$\mathscr{F}_0$$ contains all $$\mathbb{P}-$$null sets. We assume that $$\mathscr{F}_t=\sigma\left(W(s):0\leq s\leq t\right)$$ is the $$\sigma-$$algebra generated by $$W$$ and $$\mathscr{F}_T=\mathscr{F}$$, where $$W(t)$$ is a $$K-$$valued Wiener process defined on $$({\it{\Omega}},\mathscr{F},\left\{\mathscr{F}_t\right\}_{t\geq0},\mathbb{P})$$ with a finite trace nuclear covariance operator $$Q$$. Let $$\beta_n(t), (n=1,2,\cdots),$$ be a sequence of real-valued one-dimensional standard Brownian motions mutually independent over $$({\it{\Omega}},\mathscr{F},\left\{\mathscr{F}_t\right\}_{t\geq0},\mathbb{P})$$. Set   $$W(t)=\sum_{n=1}^{+\infty} \sqrt{\lambda_n} \beta_n(t) \xi_n, \,\,\,\, t\geq 0,$$ where $$\lambda_n\geq 0, (n=1,2,\cdots),$$ are non-negative real numbers and $$\xi_n, (n=1,2,\cdots),$$ is a complete orthonormal basis in $$K$$. Let $$Q \in {\mathscr{L}}(K)$$ be an operator defined by $$Q\xi_n=\lambda_n\xi_n$$ with finite trace $$Tr(Q)=\sum_{n=1}^{+\infty} \lambda_n<\infty$$. Then the above $$K-$$valued stochastic process $$W(t)$$ is called a $$Q-$$Wiener process. Definition 2.1 Let $$\sigma\in {\mathscr L}(K; X)$$ and define   \begin{equation}\label{eq43} \|\sigma\|^{2}_{{\mathscr L}_2^0}:={\mbox Tr} (\sigma Q\sigma^{*})=\sum\limits_{ n=1}^{+\infty}\|\sqrt{\lambda_{n}}\sigma \xi_{n}\|^2. \end{equation} (3) If $$\|\sigma\|_{{\mathscr L}_2^0}<\infty$$, then $$\sigma$$ is called a $$Q$$-Hilbert–Schmidt operator and let $${\mathscr L}_2^0 (K; X)$$ denote the space of all $$Q$$-Hilbert–Schmidt operators $$\sigma: K\to X$$. We recall that $$f$$ is said to be $$\mathscr{F}_t$$-adapted if $$f(t,\cdot):{\it{\Omega}}\to X$$ is $$\mathscr{F}_t$$-measurable, $$a.e. t\in[0.T]$$. Suppose that $$y(t):{\it{\Omega}}\to X_{\alpha}$$, $$t\ge 0$$, is a continuous, $${\mathscr F}_{t}$$-adapted, $$X_{\alpha}$$-valued stochastic process, we can associate it with another process $$y_t : {\it{\Omega}}\to {\mathscr{B}_\alpha}$$, $$t\ge 0$$, by setting $$y_t(s)(\omega)=y(t+s)(\omega)$$, $$s\in[-r, 0]$$. Then we say that the process $$y_t$$ is generated by the process $$y(t)$$. Finally, we introduce some function spaces to be used in the sequel. For any fixed $$0<r<\infty$$, let $$\mathcal{B}([-r,0];X)$$ be the Banach space of all measurable and bounded functions from $$[-r,0]$$ into $$X$$ endowed with the sup-norm, i.e.,   $$\mathcal{B}([-r,0];X):=\left\{x(\cdot):[-r,0]\to X \Big | x(\cdot)\,\, \mbox{is measurable and}\,\, \|x\|_{\mathcal{B}}=\sup\limits_{\theta \in [-r,0]}\|x(\theta)\|<+\infty\right\}\!.$$ Similarly, for $$\alpha\in (0,1]$$, $$\mathcal{B}([-r,0];X_\alpha)$$ is the Banach space of all measurable and bounded functions from $$[-r,0]$$ into $$X_\alpha$$ with the norm   $$\|x\|_{\mathcal{B_\alpha}}:= \sup_{\theta \in [-r,0]}\left\|A^{\alpha}x(\theta)\right\|<+\infty, \,\,\,\,\mbox{for any}\,\,\,\, x\in \mathcal{B_\alpha}.$$ Denote by $$C_\alpha := C([-r,0];X_\alpha)$$ the space of all continuous functions from $$[-r,0]$$ into $$X_\alpha$$. Then, denote by $$ML({p,r,\alpha}), 2<p<\infty,$$ the space of all $${\mathscr{F}_0}-$$measurable functions that belong to $$L_p({\it{\Omega}};C_{\alpha})$$, that is, $$ML({p,r,\alpha}), 2<p<\infty,$$is the space of $${\mathscr{F}_0}-$$measurable $$C_{\alpha}-$$valued functions $$\phi:{\it{\Omega}}\to C_{\alpha}$$ with the norm   $$\mathbb{E}\|\phi\|^p_{C_{\alpha}}=\mathbb{E}\left(\sup_{\theta\in[-r,0]}\left\|A^{\alpha}\phi(\theta)\right\|^p \right)<+\infty.$$ Additionally, let $$L_p^{\mathscr{F}}([0,T];X)$$ be the closed subspace of $$L_p([0,T]\times{\it{\Omega}};X)$$ consisting of $${\mathscr{F}_t}-$$adapted processes and let $$C([-r,T];L_p ({\it{\Omega}};X))$$ denote the Banach space of all continuous maps from $$[-r,T]$$ into $$L_p({\it{\Omega}};X)$$ satisfying the condition $$\sup\limits_{t\in [-r,T]}\mathbb{E}\|x(t)\|^p<\infty$$. 3. Fundamental solution In this section, we first discuss the fundamental solution for the linear deterministic system corresponding to System (1). Then, we express the mild solutions of System (1) via the fundamental solution. Finally, we introduce the concept of approximate controllability and collect some basic lemmas to be used in Section 4. To this end, we make the following assumptions on the terms of System (1). $$(H_1)$$ The operator $$(-A,D(-A))$$ generates an analytic semigroup $$\left(S(t)\right)_{t\geq 0}$$ on the Hilbert space $$X$$. Then there exist constants $$\mu \in \mathbb{R}$$ and $$M_{\mu}\geq 1$$ such that (see Pazy, 1983)   $$\|S(t)\|\leq M_{\mu} e^{\mu t}, \,\,\,\,\text{for all}\,\,\,\, t\geq 0,$$ Particularly,   $$\|S(t)\|\leq M,\,\,\,\, \text{for some}\,\,\,\, M\geq 1, \,\,\,\,\text{ and all }\,\,\,\, t\in [0,T],$$ and for any $$\alpha\in(0,1]$$, the fractional power operator $$A^\alpha$$ exists and Lemma 2.1 holds. $$(H_2)$$$$B$$ is a bounded linear operator from $$U$$ to $$X$$. $$(H_3)$$ The operator $$L: \mathcal{B}([-r,0];X)\to X$$ is a bounded linear operator with $$\|L\|=l$$ for some $$l>0$$, and it maps $$\mathcal{B}([-r,0];X_\alpha)$$ into $$D(A^\alpha)$$. We also assume that $$A^{\alpha}L=LA^{\alpha}$$ for all $$\alpha \in[0,1]$$. Here $$A^{\alpha}L=L A^{\alpha}$$ is understood as: for any $$\varphi\in \mathcal{B}([-r,0],X_\alpha),A^{\alpha}L(\varphi)=L(A^{\alpha}\varphi)$$. It is readily seen that this commuting property is verified for many systems, see Fu et al. (2014). $$(H_4)$$ $$(i)$$ The nonlinear functions $$f:[0,+\infty)\times C_{\alpha}\to X$$ and $$g:[0,+\infty)\times C_{\alpha}\to {{\mathscr L}_2^0}(K;X)$$ are two measurable mappings satisfying that $$f(t,0)$$ and $$g(t,0)$$ are bounded in $$X-$$norm and $${{\mathscr L}_2^0}(K;X)-$$norm, respectively. $$(ii)$$ For arbitrary $$\gamma, \xi\in C_{\alpha}$$ and $$t\in[0,T]$$, there exists a positive real constant $$N_1>0$$ such that   \begin{gather*} \|f(t,\gamma)-f(t,\xi)\|^p+\|g(t,\gamma)-g(t,\xi)\|^p_{{\mathscr L}_2^0} \leq N_1 \|\gamma-\xi\|^p_{C_{\alpha}},\\ \|f(t,\xi)\|^p+\|g(t,\xi)\|^p_{{\mathscr L}_2^0} \leq N_1\left(1+\|\xi\|^p_{C_{\alpha}}\right)\!. \end{gather*} $$(H'_4)$$ $$(i)$$ The nonlinear functions $$f:[0,+\infty)\times C_{\alpha}\to X$$ and $$g:[0,+\infty)\times C_{\alpha}\to {{\mathscr L}_2^0}(K;X)$$ are two measurable mappings satisfying that $$f(t,0)$$ and $$g(t,0)$$ are bounded in $$X-$$norm and $${{\mathscr L}_2^0}(K;X)-$$norm, respectively. $$(ii)$$ For arbitrary $$\gamma, \xi\in C_\alpha$$ and $$t\in[0,T]$$, there exists positive real constant $$N_1>0$$ such that   \begin{gather*} \|f(t,\gamma)-f(t,\xi)\|^p+\|g(t,\gamma)-g(t,\xi)\|^p_{{\mathscr L}_2^0}\leq N_1 \|\gamma-\xi\|^p_{C_{\alpha}},\\ \|f(t,\xi)\|^p+\|g(t,\xi)\|^p_{{\mathscr L}_2^0} \leq N_1. \end{gather*} We now establish the fundamental solution theory. To this end, denote by $$y(t,\phi)$$ the mild solution of the following linear retarded deterministic FDE on space $$X$$ associated to System (1):   \begin{equation}\label{eq44} \left\{\begin{aligned} &\frac{d}{dt} y(t) =-Ay(t)+L\left(y_t\right), \,\,\,\,t\in [0,T],\\ &y(t)=\phi(t)\in \mathcal{B}([-r,0];X), \,\,\,\,t\in[-r,0], \end{aligned}\right. \end{equation} (4) where $$-A:D(-A)\subseteq X\to X$$ and $$L: \mathcal{B}([-r,0];X)\to X$$ are operators given above. Then the mild solutions of System (4) can be expressed by $$C_0-$$semigroup $$(S(t))_{t\geq 0}$$ as   \begin{equation}\label{eq45} y(t,\phi)=\left\{ \begin{aligned} &S(t) \phi(0)+\int_0^tS(t-s)L(y_s(\cdot,\phi))ds,& t\in [0,T],\\ &\phi(t), &t\in[-r,0]. \end{aligned}\right. \end{equation} (5) For the solutions of System (4) satisfying (5), we can establish the following result. Theorem 3.1 For arbitrary $$\phi\in \mathcal{B}([-r,0];X)\cap L^1([-r,0];X)$$ and $$T>0$$, if $$(H_1)$$ and $$(H_3)$$ are satisfied, then there exists a unique solution $$y(t,\phi)$$, $$t\in[-r,T]$$, of System (4) satisfying (5) such that: $$(i)$$ The restriction of $$y(t,\phi)$$ on $$[0,T]$$ is continuous, i.e., $$y|_{[0,T]}\in C\left([0,T];X\right)$$, and $$(ii)$$$$\|y(t,\phi)\|\leq Ce^{\gamma t}\|\phi\|_{\mathcal{B}}$$, for all $$t\geq 0$$, where $$\gamma \in \mathbb{R}$$ and $$C>1$$ are constants. Proof. Let $$\delta\in (0,T]$$ such that   \begin{equation}\label{Q} Ml \delta<1, \end{equation} (6) and put   \begin{equation}\label{r} \rho=\frac{M+Mrl}{1-Ml\delta}\|\phi\|_{\mathcal{B}}. \end{equation} (7) Define the set $$E(\delta,\rho)$$ by   \[ E(\delta,\rho)=\left\{y:[0,\delta]\to \mathcal{B}([0,\delta];X) \big{|} y(0)=\phi (0)\,\,\,\, \text{and}\,\,\,\, \|y\|_{\delta}:=\sup_{ 0\leq t \leq \delta }\|y(t)\| \leq \rho \right\}\! . \] Clearly, $$E(\delta,\rho)$$ is a closed, bounded and convex subset of the Banach space $$\mathcal{B}([0,\delta];X)$$. Next, we define an operator $$Q$$ on $$E(\delta,\rho)$$ as follows:   $$(Qy)(t,\phi)=S(t) \phi(0) +\int_0^t S(t-s)L(\tilde{y}_s(\cdot,\phi))ds,$$ for any $$y \in E(\delta,\rho)$$ and all $$t \in [0,\delta]$$, where   \begin{equation*} \tilde{y}(t,\phi)=\left\{ \begin{aligned} &y(t,\phi),&\text{if}\,\,\,\, &t \in [0,\delta],&\\ &\phi(t), &\text{if}\,\,\,\, &t \in [-r,0].& \end{aligned} \right. \end{equation*} Obviously, $$(Qy)(0,\phi)=\phi(0)$$. In addition, from $$(H_1)$$ and $$(H_3)$$ we have, for any $$t\in [0,\delta]$$ and $$y \in E(\delta,\rho)$$,   \[ \begin{aligned} \|(Qy)(t,\phi)\| &\leq \|S(t)\| \|\phi(0)\|+l\int_0^t\|S(t-s)\| \|\tilde{y}_s(\cdot,\phi)\|_{\mathcal{B}}ds\\ & \leq M\|\phi\|_{\mathcal{B}}+Ml\int_{-r}^0\sup_{\theta\in[-r,0]}\|\phi(\theta)\| ds+Ml\int_{0}^t\sup_{\theta\in[0,\delta]}\|y(\theta,\phi)\| ds\\ & \leq \left(M+Mrl\right)\|\phi\|_{\mathcal{B}}+M\delta l\|y\|_\delta\\ & \leq \rho, \,\,\,\,(\text{by the definition} (7) \,\,\,\,\text{of}\,\,\,\, \rho ), \end{aligned} \] which shows that $$Q$$ maps $$E(\delta,\rho)$$ into itself. On the other hand, for any $$y^1$$ and $$y^2$$ in $$E(\delta,\rho)$$ and any $$t\in [0,\delta]$$, the assumptions $$(H_1)$$ and $$(H_3)$$ yield that   \[ \begin{aligned} \|(Qy^1)(t,\phi)-(Qy^2)(t,\phi)\|& \leq Ml\int_0^t\|\tilde{y}^1_s(\cdot,\phi)-\tilde{y}^2_s(\cdot,\phi)\|_{\mathcal{B}} ds\\ & \leq Ml\delta \|y^1-y^2\|_\delta, \end{aligned} \] which from (6) implies that $$Q$$ is a contractive mapping. Therefore, there exists a unique fixed point $$y(\cdot,\phi)$$ for the operator $$Q$$ in $$E(\delta,\rho)$$ which is a solution of system (4) defined on $$[0,\delta]$$ and satisfies $$y(0)=\phi(0)$$. Now, we extend the solution $$y(\cdot,\phi)$$ on $$[-r,0]$$ such that we get a solution of system (4) satisfying (5) on $$[-r,\delta]$$. Similarly, we can prove the existence of a solution of the system (4) defined on $$[\delta,2\delta]$$, $$[2\delta,3\delta]$$, $$\cdots$$ with $$y_0=\phi$$. In finite steps, we get the existence of a solution of system (4) on the whole interval $$[-r,T]$$ such that $$y(\cdot,\phi)$$ is given by (5). Moreover, it is easy to see that on the interval $$[0,T]$$, $$(Qy)(t,\phi)$$ is continuous. Next, we use the well-known Gronwall’s inequality to prove the uniqueness of the solution of system (4). Let $$y^1(t,\phi)$$ and $$y^2(t,\phi)$$ be any two solutions of system (4) satisfying (5). Then, using $$(H_1)$$ and $$(H_3)$$ we get   \[ \begin{aligned} \|y^1(t,\phi)-y^2(t,\phi)\| & \leq Ml \int_0^t\|y^1_s(\cdot,\phi)-y^2_s(\cdot,\phi)\|_{\mathcal{B}} ds\\ & \leq Ml \int_0^t \sup_{0\leq \tau\leq s}\|y^1(\tau,\phi)-y^2(\tau,\phi)\| ds.\\ \end{aligned} \] So   \[ \sup_{0\leq \tau\leq t}\|y^1(\tau,\phi)-y^2(\tau,\phi)\|\leq Ml \int_0^t \sup_{0\leq \tau\leq s}\|y^1(\tau,\phi)-y^2(\tau,\phi)\| ds. \] The Gronwall’s inequality implies that $$\sup\limits_{0\leq \tau\leq t}\|y^1(\tau,\phi)-y^2(\tau,\phi)\|\equiv 0$$, thus $$y^1(t,\phi)=y^2(t,\phi)$$ for all $$t\in[0,T]$$ and consequently $$y^1(\cdot,\phi)=y^2(\cdot,\phi)$$ on $$[-r,T]$$. The claim $$(ii)$$ can be shown also by the Gronwall’s inequality. Indeed, from (5), $$(H_1)$$ and $$(H_3)$$, the solution $$y(t,\phi)$$ satisfies that, for $$t\in [0,\delta]$$ with $$\delta>0$$,   \[ \begin{aligned} \|y(t,\phi)\| & \leq M \|\phi\|_{\mathcal{B}}+Ml \int_0^t\|y_s(\cdot,\phi)\|_{\mathcal{B}}ds\\ & \leq \left(M+Ml r\right) \|\phi\|_{\mathcal{B}}+Ml\int_{0}^t\sup_{0\leq \tau\leq s}\|y(\tau,\phi)\|ds.\\ \end{aligned} \] Hence,   \[ \sup_{0\leq \tau\leq t}\|y(t,\phi)\|\leq \left(M+Ml r\right) \|\phi\|_{\mathcal{B}}+Ml\int_{0}^t\sup_{0\leq \tau\leq s}\|y(\tau,\phi)\|ds, \] which by Gronwall’s inequality implies that   $$\sup_{0\leq \tau\leq t}\|y(\tau,\phi)\|\leq (M+Ml r)\|\phi\|_{\mathcal{B}} e^{Mlt}:=C(\delta).$$ Now, for any $$t>0$$, let $$t \in ((n -1)\delta,n\delta]$$$$(n\in \mathbb{N})$$, then proceeding inductively as above, we obtain easily that   $$\|y(t,\phi)\|\leq C^n(\delta),$$ which, letting $$\gamma= \frac{\ln C(\delta)}{\delta}$$, immediately yields that, for any $$t>0$$, $$\|y(t,\phi)\|\leq Ce^{\gamma t}$$ with $$C\ge 1$$ and $$\gamma\in \mathbb{R}$$. The claim $$(ii)$$ is proved. □ The solution $$y(t,\phi)$$ given by (5) is called a mild solution of System (4). Next, for any $$x\in X$$, define the function $$\phi^0_x$$ by   \begin{equation}\label{eq46} \phi^0_x(\theta)=\left\{ \begin{aligned} & x, &\theta=0,\\ &0, &\theta\in[-r,0). \end{aligned} \right. \end{equation} (8) Then, we define the fundamental solution $$G(t)\in {\mathscr{L}}(X)$$ of (4) with the initial datum $$\phi^0_x$$ by, for any $$x\in X$$,   \begin{equation}\label{eq47} G(t)x=\left\{ \begin{aligned} & y(t,\phi^0_x), &t\geq 0,\\ &0, &t<0. \end{aligned} \right. \end{equation} (9) According to Theorem 3.1, the fundamental solution $$G(t)$$ is well defined. Moreover, System (9) implies that $$G(t)$$ is the unique solution of the operator equation   \begin{equation}\label{eq48} G(t)=\left\{ \begin{aligned} &S(t) +\int_0^t S(t-s) L\left(G_s\right)ds, &t\geq 0,\\ &0,&t< 0, \end{aligned} \right. \end{equation} (10) where $$G_t(\theta):=G(t+\theta), \theta\in [-r,0]$$. Furthermore, for the fundamental solution $$G(t)$$ defined above, one has that. Theorem 3.2 For $$G(t)$$, $$t\in \mathbb{R}$$, there hold: $$(i)$$$$G(t)$$ is a strongly continuous one-parameter family of bounded linear operators on $$X$$ and satisfies that   $$\|G(t)\| \leq Ce^{\gamma t},\,\,\,\, t\geq 0,$$ where $$C>1$$, $$\gamma\in\mathbb{R}$$ are constants. Particularly, we have that   $$\|G(t)\| \leq \overline{M},\,\,\,\, t\in[0,T],$$ for some $$\overline{M}\geq 1$$. $$(ii)$$ If the semigroup $$\big(S(t)\big)_{t\ge 0}$$ is compact, then $$G(t)$$ is also compact for all $$t>0$$. $$(iii)$$$$G(t)$$ and $$A^{\alpha}G(t)$$ are uniformly continuous on $$(0,T]$$ for all $$\alpha \in (0,1)$$. $$(iv)$$ For each $$\alpha\in(0,1)$$, there exists a constant $$\overline{M}_{\alpha}>0$$ such that   $$\|A^\alpha G(t)\|\leq \frac{\overline{M}_\alpha}{t^\alpha}, \,\,\,\,\text{for all}\,\,\,\, t\in(0,T].$$ $$(v)$$ For all $$t\in[0,T]$$, $$G(t)$$ commutes with the operator $$A^{\alpha}$$, that is, $$A^{\alpha}G(t)=G(t)A^{\alpha}$$ for each $$\alpha \in [0,1]$$. Proof. Assertion $$(i)$$ is obvious by Theorem 3.1 and the definition (9) of $$G(\cdot)$$. For Assertion $$(ii)$$, let $$\epsilon>0$$ very small, then the operator   $$\int_0^{t-\epsilon} S(t-s)L \left(G_s\right)ds=S(\epsilon)\int_0^{t-\epsilon} S(t-s-\epsilon)L \left(G_s\right)ds,$$ is clearly compact and, since $$\left(S(t)\right)_{t\geq 0}$$ is assumed to be compact, it converges to $$\int_0^{t} S(t-s)L \left(G_s\right)ds$$ uniformly as $$\epsilon\to 0^+$$, which shows that $$\int_0^{t} S(t-s)L \left(G_s\right)ds$$ is compact too. Consequently, from (10) we deduce that $$G(t)$$ is a compact operator on $$X$$ for any $$t> 0$$. $$(iii)$$ Let $$0<\epsilon< t_1< t_2\leq T$$, then from (10) we have   $$ \begin{aligned} \left\|G(t_2)-G(t_1)\right\| & \leq \left\| S(t_2)-S(t_1) \right\| + \int^{t_1-\epsilon}_0 \left\| S(t_2-s)-S(t_1-s) \right\| \left\|L(G_s)\right\|ds \\ &\quad{} + \int_{t_1-\epsilon}^{t_1} \left\|\left(S(t_2-s)-S(t_1-s)\right) \right\| \left\|L(G_s)\right\|ds +\int^{t_2}_{t_1}\left\|S(t_2-s)\right\|\left\|L(G_s)\right\|ds. \end{aligned} $$ By the analyticity, $$S(t)$$ is uniformly continuous for $$t>0$$, then the right hand side tends to zero for $$t_2 \to t_1$$ and $$\epsilon$$ small enough. Hence $$G(t)$$ is uniformly continuous for $$t\in (0,T]$$. Similarly, using the fact that, for all $$\alpha \in (0,1)$$, $$A^{\alpha}S(t)$$ is uniformly continuous for $$t>0$$, one can get the uniform continuity of $$A^{\alpha}G(t)$$ on $$(0,T]$$. Hence, assertion $$(iii)$$ holds. $$(iv)$$ First we note from (10) that $$\mathscr{R}(G)\subset D(A)$$ since the semigroup $$S(t)$$ is analytic. Then, from $$(i)$$, (10), $$(H_3)$$ and Lemma 2.1 $$(iii)$$ it follows that, for any $$\alpha \in (0,1)$$ and $$0<t\leq T$$,   \[ \begin{aligned} \left\|A^\alpha G(t)\right\|& \leq \left\|A^\alpha S(t)\right\| +\left\|\int_0^t A^\alpha S(t-s)L \left(G_s\right)ds\right\|\\ & \leq \left\|A^\alpha S(t)\right\| + l \overline{M} \int_0^t \frac{M_\alpha}{(t-s)^\alpha}ds\\ & \leq \frac{M_\alpha}{t^\alpha}+ l \overline{M} \frac{M_\alpha}{1-\alpha} \frac{T}{t^\alpha}\\ & = \left(M_\alpha + \frac{l \overline{M} M_\alpha T}{1-\alpha}\right) \frac{1}{t^\alpha}. \end{aligned} \] Let $$\overline{M}_\alpha= M_\alpha + \frac{l \overline{M} M_\alpha T}{1-\alpha}$$, then $$(iv)$$ is proved. $$(v)$$ Since the semigroup $$\left(S(t)\right)_{t\geq 0}$$ is analytic, we know that $$A^{\alpha} S(t)=S(t)A^{\alpha}, \text{for all} \alpha\in [0,1]$$ and $$t\in [0,T]$$. Now, from (10), we have that   \begin{align*} A^{\alpha} G(t) & = A^{\alpha}S(t)+\int_0^t S(t-s)L\left(A^{\alpha}G_s(\theta)\right)ds,\\ G(t)A^{\alpha} & = S(t)A^{\alpha}+\int_0^tS(t-s)L\left(G_s(\theta)A^{\alpha}\right) ds. \end{align*} Combined with $$(H_1)$$ and $$(H_3)$$ this yields   $$ \begin{aligned} \left\|A^{\alpha}G(t)-G(t)A^{\alpha} \right\| & \leq \int_0^t\left\|S(t-s)L \left(A^{\alpha}G_s(\theta)-G_s(\theta)A^{\alpha}\right)\right\|ds\\ & \leq M l \int_0^t \left\|A^{\alpha} G_s(\theta)-G_s(\theta)A^{\alpha}\right\|ds\\ & \leq M l \int_0^t\sup_{\nu\in[0,s]}\left\|A^{\alpha}G(\nu)-G(\nu)A^{\alpha}\right\|ds, \end{aligned} $$ hence   $$ \begin{aligned} \sup_{\tau\in[0,t]}\left\|A^{\alpha}G(\tau)-G(\tau)A^{\alpha}\right\|& \leq M l \int_0^t\sup_{\tau\in[0,s]}\left\|A^{\alpha}G(\tau)-G(\tau)A^{\alpha}\right\|ds. \end{aligned}$$ The Gronwall’s inequality implies that $$\sup\limits_{t\in[0,T]}\|A^{\alpha}G(t)-G(t)A^{\alpha}\|\equiv0$$, thus $$A^{\alpha} G(t)=G(t)A^{\alpha}$$ for all $$\alpha\in [0,1]$$ and $$t\in[0,T]$$. Then we get the assertion $$(v)$$. □ Now we consider the following linear deterministic inhomogeneous FDEs on $$X$$:   \begin{equation}\label{eq49}\left\{\begin{aligned} &\frac{d}{dt} y(t) =-Ay(t)+L \left(y_t\right)+f(t), \,\,\,\,t\geq 0,\\ &y(\theta)=\phi(\theta)\in C([-r,0];X), \,\,\,\,\theta\in [-r,0], \end{aligned}\right. \end{equation} (11) where the operators $$A$$ and $$L$$ are as described above and the function $$f(t)$$ belongs to $$L^1(\mathbb{R}^+;X)$$. The mild solutions of System (11) are represented through the $$C_0$$-semigroup $$(S(t))_{t\geq 0}$$ by   \begin{equation}\label{eq410} y(t,\phi)=\left\{ \begin{aligned} &S(t) \phi(0)+\int_0^tS(t-s)\Big(L \left(y_s(\cdot,\phi)\right)+f(s)\Big)ds, &t\geq 0,\\ &\phi(t), & t \in [-r,0]. \end{aligned} \right. \end{equation} (12) For the subsequent discussions, we need to represent the mild solutions (12) as an ‘explicit formula’ via the fundamental solution $$G(t)$$ defined above, that is Theorem 3.3 For $$\phi\in C([-r,0];X)$$, the mild solutions (12) of System (11) can be expressed equivalently by   \begin{equation*} y(t,\phi)=\left\{ \begin{aligned} &G(t)\phi(0)+\int_0^tG(t-s)\Big(L \left({\tilde{\phi}_s}\right)+f(s)\Big)ds, \,\,\,\,t\geq 0,\\ &\phi(t), \quad\quad t \in [-r,0], \end{aligned}\right. \end{equation*} where the function $$\tilde{\phi}(\cdot)$$ is defined as   \begin{equation}\label{eq411} \tilde{\phi}(t)=\left\{ \begin{aligned} & \phi(t), & t \in [-r,0],\\ &0, &t>0. \end{aligned} \right. \end{equation} (13) Proof. The proof is very similar to that of (Mokkedem & Fu (2017), Theorem 3.3), so we omit it here. □ Next, we define the mild solutions of the stochastic System (1) via the fundamental solution $$G(t)$$ as follows. Definition 3.1 A stochastic process $$y(\cdot)$$ defined on $$[-r,T], 0<r, T<\infty$$ is said to be a mild solution of System (1) if the following conditions are satisfied: (1) $$y(t,\omega)$$ is measurable as a function from $$[0,T]\times{\it{\Omega}}$$ to $$X$$ and $$y(t)$$ is $${\mathscr{F}_t}-$$adapted; (2) $$\mathbb{E}\|y(t)\|^p<\infty$$ for each $$t\in[-r,T]$$; (3) For each $$u\in L_p^{\mathscr{F}}([0,T];U)$$, the process $$y(\cdot)$$ satisfies the following integral equation:   \begin{equation*} y(t)=\left\{ \begin{aligned} &G(t)\phi(0)+\int_0^t G(t-s)\Big(L \left({\tilde{\phi}_s}\right)+f(s,y_s)+Bu(s)\Big)ds\\ & +\int_0^t G(t-s)g(s,y_s)dW(s), &t\in[0,T],\\ &\phi(t) \in ML(p,r,\alpha) ,& t \in [-r,0], \end{aligned} \right.\end{equation*} where the function $$\tilde{\phi}(\cdot)$$ is given by (13). Next, we turn to present the concept of approximate controllability. Definition 3.2 The System (1) is said to be approximately controllable on $$[0,T]$$ if $$\mathscr{R}(T,\phi)$$ is dense in $$L_p^{\mathscr{F}}\left({\it{\Omega}};X\right)$$, i.e.,   $$\overline{\mathscr{R}(T,\phi)}= L_p^{\mathscr{F}}\left({\it{\Omega}};X\right)\!,$$ where $$\mathscr{R}(T,\phi)=\left\{y(T,\phi,u): u(\cdot)\in L_p^{\mathscr{F}}([0,T];U)\right\}.$$ In the sequel, we shall study the approximate controllability for System (1) by assuming the approximate controllability of the linear deterministic system corresponding to (1). For this purpose, we introduce the following resolvent operator. Let   $${\it{\Gamma}}^t_s=\int_s^t G(t-\tau)BB^\ast G^\ast(t-\tau)d\tau,\,\,\,\, \text{for} \,\,\,\,0\leq s<t\leq T,$$ where $$B^*$$ and $$G^*$$ denote respectively the adjoint operators of $$B$$ and $$G$$, then the resolvent operator $$R(\lambda,-{\it{\Gamma}}^\top_s)\in {\mathscr{L}}(X)$$ for $$\lambda>0$$ is given by   $$R(\lambda,-{\it{\Gamma}}^\top_s)=(\lambda I+{\it{\Gamma}}^\top_s)^{-1}.$$ Since the operator $${\it{\Gamma}}^\top_s$$ is clearly positive, $$R(\lambda,-{\it{\Gamma}}^\top_s)$$ is well defined. We will always assume that $$(H_5)$$$$\lambda R(\lambda,-{\it{\Gamma}}^\top_0)\to 0$$ as $$\lambda\to 0^+$$ in the strong operator topology. The above condition $$(H_5)$$ is equivalent to the approximate controllability of the following deterministic linear system corresponding to (1):   \begin{equation}\label{eq413} \left\{\begin{aligned} &\frac{d}{dt}y(t) =-Ay(t)+L \left(y_t\right)+ Bu(t),\,\,\,\, t\in [0,T],\\ &y(\theta)=0, \,\,\,\,\theta\in[-r,0]. \end{aligned}\right. \end{equation} (14) More precisely, we have that Theorem 3.4 The following statements are equivalent: $$(i)$$ The control system (14) is approximately controllable on $$[0,T]$$. $$(ii)$$ If $$B^*G^*(t)y=0$$ for all $$t\in [0,T]$$, then $$y=0$$. $$(iii)$$ The condition ($$H_5$$) holds. Proof. The proof of this theorem is similar to that of (Curtain & Zwart, 1995, Theorem 4.4.17) and (Bashirov & Mahmudov, 1999, Theorem 2), so we omit it here. □ By this theorem the condition $$(H_5)$$ implies that the deterministic linear system (14) is approximately controllable on interval $$[0,T]$$ and hence on $$[s,T]$$ for all $$0\leq s <T$$, we thus infer that   $$\lambda R(\lambda,-{\it{\Gamma}}^\top_s)\to 0,\,\,\,\, \text{strongly as} \,\,\,\,\lambda \to 0^+, \text{ for any } s\in[0,T),$$ from which we can assume that   $$\left\|R(\lambda,-{\it{\Gamma}}^\top_s)\right\|\leq \frac{1}{\lambda}, \qquad\text{ for any } s\in[0,T), \,\,\,\,\lambda\in(0,1).$$ Finally, to study the existence of mild solutions of System (1), we need the following lemmas: Lemma 3.1 (DA Prato & Zabbczyk, 1992, Proposition 4.15) If $$\phi\in L_2^{\mathscr{F}}(0,T;{{\mathscr L}_2^0}(K;X))$$, $$A^\alpha \phi\in L_2^{\mathscr{F}}(0,T;{{\mathscr L}_2^0}(K;X))$$ and $$\phi(t)k\in X_\alpha, t\geq 0,$$ for arbitrary $$k\in K$$, then   $$A^\alpha \int_0^t \phi(s)dW(s)=\int_0^t A^\alpha \phi(s)dW(s).$$ Lemma 3.2 (DA Prato & Zabbczyk, 1992, Lemma 7.2) For any $$p>2$$ and $$\phi\in L_p^{\mathscr{F}}({\it{\Omega}};L_2(0,T;{{\mathscr L}_2^0}(K;X))),$$ we have   \begin{eqnarray*} \mathbb{E}\left( \sup_{s\in[0,t]} \left\| \int_0^s \phi(r)dW(r) \right\|^p \right) &\leq& c_p \sup_{s\in[0,t]} \mathbb{E}\left\| \int_0^s \phi(r)dW(r) \right\|^p\\ &\leq& C_p \mathbb{E}\left(\int_0^t \left\| \phi(r) \right\|^2_{{\mathscr L}_2^0} dr \right)^{\frac{p}{2}}, t\in [0,T], \end{eqnarray*} where $$c_p=\left(\frac{p}{p-1}\right)^p, C_p=\left(\frac{p}{2}(p-1)\right)^{\frac{p}{2}}\left(\frac{p}{p-1}\right)^{\frac{p^2}{2}}.$$ Lemma 3.3 (Dauer & Mahmudov, 2004, Lemma 6) Let $$p\geq 2$$ and $$h\in L_p\left({\it{\Omega}};X\right)$$ be fixed, then there exists a function $$\varphi$$ in space $$L_p^{\mathscr{F}}({\it{\Omega}};L_2(0,T;{{\mathscr L}_2^0}(K;X)))$$ such that   $$h=\mathbb{E}h+\int_0^\top \varphi(s) dW(s).$$ Moreover, we can prove that Lemma 3.4 Let $$p>2$$, $$\frac{1}{p}+\alpha<\frac12$$ and $$g\in L_p^{\mathscr{F}}(0,T;{{\mathscr L}_2^0}(K;X)).$$ Then there exists a constant $$N_2>0$$ such that   $$ \mathbb{E}\sup_{\theta\in[-r,0]} \left\| \int_0^{t+\theta} A^\alpha G(t+\theta-\tau) g(\tau)dW(\tau) \right\|^p \leq N_2 \int_0^t \mathbb{E}\left\|g(\tau)\right\|^p_{{\mathscr L}_2^0} d\tau, $$ where   \begin{eqnarray*} N_2 &=&2^{p-1}M^p_\alpha \Bigg({\it{\Gamma}}(1+q(\beta-1-\alpha))(aq)^{q(1+\alpha-\beta)}\Bigg)^{\frac{p}{q}} C_{p} \frac{T^{\frac{p(1-2\beta)}{2}}}{(1-2\beta)^{\frac{p}{2}}}\left(1+T^{\frac{p}{q}+1}\left(l \overline{M}\right)^p\right)\!, \end{eqnarray*} with $$\frac{1}{p}+\alpha<\beta<\frac12, \frac{1}{p}+\frac{1}{q}=1$$ and $${\it{\Gamma}}(\cdot)$$ is the well known Gamma function. Proof. From (10) we have   \begin{align*} & \sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-\tau)g(\tau)dW(\tau)\right\|^p\\ &\quad{} = \sup_{\theta\in[-r,0]} \Bigg\|\int_0^{t+\theta} A^\alpha S(t+\theta-\tau)g(\tau)dW(\tau)\\ &\qquad{} +\int_0^{t+\theta}\int_0^{t+\theta-\tau}A^\alpha S(t+\theta-\tau-u)L\left(G_u\right)dug(\tau)dW(\tau)\Bigg\|^p\\ & \quad{} = \sup_{\theta\in[-r,0]}\Bigg\| \int_0^{t+\theta} A^\alpha S(t+\theta-\tau)g(\tau)dW(\tau)\\ &\qquad{} +\int_0^{t+\theta}\int_{\tau}^{t+\theta} A^\alpha S(\nu-\tau) L\left(G_{t+\theta-\nu}\right)d\nu g(\tau)dW(\tau)\Bigg\|^p\\ &\quad{} = \sup_{\theta\in[-r,0]}\Bigg\| \int_0^{t+\theta} A^\alpha S(t+\theta-\tau)g(\tau)dW(\tau)\\ &\qquad +\int_0^{t+\theta}\int_{0}^{\nu} A^\alpha S(\nu-\tau)L\left(G_{t+\theta-\nu}\right)g(\tau)dW(\tau)d\nu\Bigg\|^p, \end{align*} then   \begin{align*} & \mathbb{E}\sup_{\theta\in[-r,0]} \left\| \int_0^{t+\theta} A^\alpha G(t+\theta-\tau) g(\tau)dW(\tau) \right\|^p \\ &\quad{} \leq 2^{p-1} \mathbb{E} \sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha S(t+\theta-\tau)g(\tau)dW(\tau) \right\|^p \\ &\qquad{} +2^{p-1} \mathbb{E}\sup_{\theta\in[-r,0]} \left\|\int_0^{t+\theta}\int_{0}^{\nu} A^\alpha S(\nu-\tau)L\left(G_{t+\theta-\nu}\right)g(\tau)dW(\tau)d\nu\right\|^p\\ & = 2^{p-1}(I_1+I_2), \end{align*} where from (Dauer & Mahmudov, 2004, Lemma 7)   \begin{align}\label{eq414} I_1 \leq N \int_0^t \mathbb{E}\left\|g(\tau)\right\|^{p}_Q d\tau, \end{align} (15) with   $$N = M^p_\alpha \Bigg({\it{\Gamma}}(1+q(\beta-1-\alpha))(aq)^{q(1+\alpha-\beta)}\Bigg)^{\frac{p}{q}} C_{p} \frac{T^{\frac{p(1-2\beta)}{2}}}{(1-2\beta)^{\frac{p}{2}}}, \,\,\,\,\frac{1}{p}+\alpha<\beta<\frac12,$$ where $$\frac{1}{p}+\frac{1}{q}=1.$$ On the other hand, using $$(H_3)$$, Theorem 3.2 $$(i)$$ and H$$\ddot{o}$$lder inequality, we have   \begin{align}\label{eq415} I_2 & \leq T^{\frac{p}{q}}\mathbb{E}\sup_{\theta\in[-r,0]} \left(\int_0^{t+\theta}\left\|\int_{0}^{\nu} A^\alpha S(\nu-\tau)L\left(G_{t+\theta-\nu}\right)g(\tau)dW(\tau)\right\|^p d\nu\right) \nonumber\\ & \leq T^{\frac{p}{q}}\left(l \overline{M}\right)^p \mathbb{E} \sup_{\theta\in[-r,0]} \left(\int_0^{t+\theta}\left\|\int_{0}^{\nu} A^\alpha S(\nu-\tau)g(\tau)dW(\tau)\right\|^p d\nu\right)\nonumber\\ & \leq T^{\frac{p}{q}}\left(l \overline{M}\right)^p \mathbb{E}\left(\int_0^{t}\left\|\int_{0}^{\nu} A^\alpha S(\nu-\tau)g(\tau)dW(\tau)\right\|^p d\nu\right)\!, \end{align} (16) or from (Dauer & Mahmudov, 2004, Lemma 7) it yields that   \begin{align}\label{eq41*} \mathbb{E} \left\|\int_0^t A^\alpha S(t-\tau)g(\tau)dW(\tau) \right\|^p & \leq N \int_0^t \mathbb{E} \left\|g(\tau)\right\|^{p}_Q d\tau, \end{align} (17) substituting (17) into (16), we get   \begin{align}\label{eq416} I_2 & \leq T^{\frac{p}{q}}\left(l \overline{M}\right)^p N \int_0^t\int_{0}^{\nu} \mathbb{E}\left\|g(\tau)\right\|^{p}_{Q} d\tau d\nu\nonumber\\ & \leq T^{\frac{p}{q}}\left(l \overline{M}\right)^p N \int_0^t (t-\tau) \mathbb{E}\left\|g(\tau)\right\|^{p}_{Q} d\tau \nonumber\\ & \leq T^{\frac{p}{q}+1}\left(l \overline{M}\right)^p N \int_0^t \mathbb{E}\left\|g(\tau)\right\|^{p}_{Q} d\tau. \end{align} (18) Combining (15) and (18) gives that   $$\mathbb{E} \left\| \int_0^t A^\alpha G(t-\tau) g(\tau)dW(\tau) \right\|^p \leq N_2 \int_0^t\mathbb{E}\left\|g(\tau)\right\|^{p}_{Q} d\tau,$$ with   $$N_2=2^{p-1}N \left(1+T^{\frac{p}{q}+1}\left(l \overline{M}\right)^p\right)\!.$$ Hence Lemma 3.4 is proved. □ 4. Approximate controllability In this section, we present the main result of this article on the approximate controllability of System (1). That is, for any $$h\in L_p^{\mathscr{F}}\left({\it{\Omega}};X\right)$$, by selecting proper control $$u^\lambda$$ in $$L_p^{\mathscr{F}}([0,T];U)$$ (for any given $$\lambda \in (0,1)$$), there exists a mild solution $$y^\lambda(\cdot,\phi,u^\lambda)\in C([-r,T];L_p({\it{\Omega}};X_\alpha))$$ for System (1), such that $$y^\lambda(T,\phi,u^\lambda)\to h$$ in $$L_p({\it{\Omega}};X)$$ as $$\lambda\to 0^+$$. For this purpose, besides the previous assumptions $$(H_1)-(H_5)$$, we will assume that the analytic semigroup $$\left(S(t)\right)_{t\ge 0}$$ is compact for all $$t>0$$. First, we define $$D_p$$ to be the closed subspace of $\(C([-r,T];L_p({\it{\Omega}};X))\)$ consisting of measurable and $${\mathscr{F}_t}-$$adapted processes $$Z$$ with $$\|Z\|_{D_p}<\infty$$, where   $$\|Z\|_{D_p}:= \left(\sup_{t\in[0,T]}\mathbb{E}\left\|Z_t\right\|^p_{C}\right)^{\frac{1}{p}} =\left(\sup_{t\in[0,T]}\mathbb{E} \sup_{\theta\in[-r,0]}\left\|Z_t(\theta)\right\|^p\right)^{\frac{1}{p}},$$ and put   $$D_{p,\alpha}:= \left\{z\in D(A^{\alpha}): A^{\alpha}z\in D_p\right\}\!.$$ Then, let $$\lambda\in(0,1)$$, $$\phi\in ML(p,r,\alpha)$$ and $$h\in L_p\left({\it{\Omega}};X\right)$$ be fixed. For any $$t\in [0,T]$$ and $$y\in D_{p,\alpha}$$, we define the control function $$u^{\lambda}(t,y)$$ as   \begin{align}\label{eq417} u^{\lambda}(t,y):=& B^*G^*(T-t)R(\lambda,-{\it{\Gamma}}^\top_0)\Big(\mathbb{E}h-G(T)\phi(0)\Big)\nonumber\\ &- B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^\top_s)G(T-s)\left(L\left({\tilde{\phi}_s}\right)+f(s,y_s)\right)ds\nonumber\\ &- B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^\top_s)\Big(G(T-s)g(s,y_s)-\varphi(s)\Big)dW(s), \end{align} (19) where $$\tilde{\phi}$$ is given by (13), $${\it{\Gamma}}^\top_s=\int_s^\top G(T-\tau)BB^\ast G^\ast(T-\tau)d\tau$$ is the controllability Grammian and $$h=\mathbb{E} h+\int_0^\top \varphi(s)dW(s)$$ by Lemma 3.3. Then, we can obtain the following estimates. Lemma 4.1 There exists a positive real constant $$N_3>0$$ such that, for all $$(x, y) \in D_{p,\alpha}^2$$,   \begin{align*} \mathbb{E}\left\|u^{\lambda}(t,x)-u^{\lambda}(t,y)\right\|^p & \leq \frac{1}{\lambda^p} N_3 \mathbb{E}\int_0^t\|x_s-y_s\|^p_{C_{\alpha}} ds,\\ \mathbb{E}\left\|u^{\lambda}(t,x)\right\|^p & \leq \frac{1}{\lambda^p} N_3\left(1+\mathbb{E} \int_0^t\|x_s\|^p_{C_{\alpha}} ds\right)\!. \end{align*} Proof. We will only prove the first inequality since the proof of the second is similar. Let $$x$$ and $$y$$ be two fixed functions in $$D_{p,\alpha}$$. Then, from (19), we have   \begin{eqnarray*} \mathbb{E}\left\|u^{\lambda}(t,x)-u^{\lambda}(t,y)\right\|^p &\leq& 2^{p-1}\Bigg(\mathbb{E}\left\|B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^\top_s)G(T-s)\Big(f(s,x_s)-f(s,y_s)\Big)ds\right\|^p\\ &&+\mathbb{E}\left\|B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^\top_s)G(T-s)\Big(g(s,x_s)-g(s,y_s)\Big)dW(s)\right\|^p\Bigg). \end{eqnarray*} According to $$(H_2)-(H_5)$$, Theorem 3.2 $$(i)$$, Lemma 3.2 and H$$\ddot{o}$$lder inequality we have   \begin{eqnarray*} \mathbb{E}\left\|u^{\lambda}(t,x)-u^{\lambda}(t,y)\right\|^p &\leq& 2^{p-1}\|B\|^p \Bigg[\frac{1}{\lambda^p}\overline{M}^{2p} T^{\frac{p}{q}} N_1 \mathbb{E}\int_0^t\|x_s-y_s\|^p_{C_{\alpha}} ds \\ &&+\overline{M}^{p}C_p\mathbb{E}\left(\int_0^t \left\|R(\lambda,-{\it{\Gamma}}^\top_s)G(T-s)\right\|^2\left\|g(s,x_s)-g(s,y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}}\Bigg]\\ &\leq&\frac{1}{\lambda^p} 2^{p-1}\|B\|^p {\overline{M}}^{2p} T^{\frac{p}{q}} N_1 \mathbb{E}\int_0^t\|x_s-y_s\|^p_{C_{\alpha}} ds \\ &&+\frac{1}{\lambda^p}2^{p-1}\|B\|^p \overline{M}^{2p} C_p N_1\mathbb{E}\left(\int_0^t\|x_s-y_s\|^2_{C_{\alpha}} ds\right)^{\frac{p}{2}} \\ &\leq& \frac{1}{\lambda^p} N_3 \mathbb{E}\int_0^t\|x_s-y_s\|^p_{C_{\alpha}} ds, \end{eqnarray*} for some constant $$N_3>0$$. Hence the lemma is proved. □ Next, for any $$\phi\in ML(p,r,\alpha)$$, we define the following mapping $$P^{\lambda}$$ on $$D_p$$ as   \begin{equation}\label{eq418} (P^{\lambda} y)(t) =\left\{ \begin{aligned} &G(t)A^\alpha\phi(0)+\int_0^t A^\alpha G(t-s)\Big(L \left({\tilde{\phi}_s}\right)+f(s,A^{-\alpha}y_s)+Bu^{\lambda}(s,A^{-\alpha}y)\Big)ds \\ & +\int_0^t A^\alpha G(t-s)g(s,A^{-\alpha}y_s)dW(s), \,\,\,\,t\in[0,T],\\ &A^\alpha \phi(t), \,\,\,\,\,\,\,\,t\in [-r,0], \end{aligned} \right. \end{equation} (20) with   \begin{align}\label{eq419} u^{\lambda}(t,A^{-\alpha}y):=& B^*G^*(T-t)R(\lambda,-{\it{\Gamma}}^\top_0)\Big(\mathbb{E}h-G(T)\phi(0)\Big)\nonumber\\ &- B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^\top_s)G(T-s)\Big(L\left({\tilde{\phi}_s}\right)+f(s,A^{-\alpha}y_s)\Big)ds \nonumber\\ &- B^*G^*(T-t)\int_0^tR(\lambda,-{\it{\Gamma}}^T_s)\Big(G(T-s)g(s,A^{-\alpha}y_s)-\varphi(s)\Big)dW(s). \end{align} (21) In the sequel, we need to prove the existence of mild solutions of System (1). Clearly, if the operator $$P^{\lambda}$$ has a fixed point $$y(\cdot,\phi)$$ on $$D_{p}$$, then $$x(\cdot,\phi)=A^{-\alpha}y(\cdot,\phi)$$ is a mild solution of System (1) belonging to $\(C([-r,T];L_p({\it{\Omega}};X_\alpha))\)$. To make our arguments clear we first prove several auxiliary lemmas before presenting the existence theorem. Lemma 4.2 Assume that $$(H_1)-(H_4)$$ hold. Let $$2<p<\infty$$ and $$0<\alpha<\frac{p-2}{2p}$$. Then, for any $$\phi\in ML(p,r,\alpha)$$ and $$y\in D_p, (P^{\lambda} y)(\cdot)$$ is continuous on the interval $$[0,T]$$ in the $$L_p$$ sense. Proof. Let $$0 \leq t_1 < t_2 < T$$. Then for any fixed $$y\in D_p$$, define   $$J_1 := \mathbb{E}\left\|\left(P^{\lambda}y\right)(t_1)-\left(P^{\lambda}y\right)(t_2)\right\|^p.$$ We will show that $$J_1$$ tends to zero when $$t_2\to t_1$$ which implies the right continuity of $$(P^{\lambda} y)(\cdot)$$ on $$[0,T)$$ in the $$L_p$$ sense. By definition (20) of $$P^{\lambda} y$$ we have   \begin{eqnarray*} J_1 &\leq&5^{p-1}\mathbb{E} \left\|\left(G(t_1)-G(t_2)\right)A^\alpha\phi(0)\right\|^p \\ &&+ 5^{p-1}\mathbb{E} \left\|\int_0^{t_2} A^\alpha G(t_2-s)L \left({\tilde{\phi}_s}\right) ds-\int_0^{t_1} A^\alpha G(t_1-s)L \left({\tilde{\phi}_s}\right)ds\right\|^p \\ &&+ 5^{p-1}\mathbb{E} \left\|\int_0^{t_2} A^\alpha G(t_2-s)f(s,A^{-\alpha}y_s)ds-\int_0^{t_1} A^\alpha G(t_1-s)f(s,A^{-\alpha}y_s)ds\right\|^p \\ &&+ 5^{p-1}\mathbb{E} \left\|\int_0^{t_2} A^\alpha G(t_2-s)Bu^{\lambda}(s,A^{-\alpha}y)ds-\int_0^{t_1} A^\alpha G(t_1-s)Bu^{\lambda}(s,A^{-\alpha}y)ds \right\|^p \\ &&+ 5^{p-1}\mathbb{E} \left\|\int_0^{t_2} A^\alpha G(t_2-s)g(s,A^{-\alpha}y_s)dW(s)-\int_0^{t_1} A^\alpha G(t_1-s)g(s,A^{-\alpha}y_s)dW(s)\right\|^p, \end{eqnarray*} hence   \begin{eqnarray*} J_1 &\leq&5^{p-1}\mathbb{E} \left\|\left(G(t_1)-G(t_2)\right)A^\alpha\phi(0)\right\|^p +10^{p-1}\Bigg(\mathbb{E}\left\|\int_{t_1}^{t_2} A^\alpha G(t_2-s)L \left({\tilde{\phi}_s}\right)ds\right\|^p \\ &&+\mathbb{E}\left\|\int_{t_1}^{t_2} A^\alpha G(t_2-s)f(s,A^{-\alpha}y_s)ds\right\|^p +\mathbb{E}\left\|\int_{t_1}^{t_2} A^\alpha G(t_2-s)Bu^{\lambda}(s,A^{-\alpha}y)ds\right\|^p\\ &&+\mathbb{E}\left\|\int_{t_1}^{t_2} A^\alpha G(t_2-s)g(s,A^{-\alpha}y_s)dW(s)\right\|^p +\mathbb{E}\left\|\int_0^{t_1} A^\alpha \left(G(t_2-s)-G(t_1-s)\right)L \left({\tilde{\phi}_s}\right) ds\right\|^p\\ &&+\mathbb{E}\left\|\int_0^{t_1} A^\alpha \left(G(t_2-s)-G(t_1-s)\right)f(s,A^{-\alpha}y_s)ds\right\|^p \\ &&+\mathbb{E}\left\|\int_0^{t_1} A^\alpha \left(G(t_2-s)-G(t_1-s)\right)Bu^{\lambda}(s,A^{-\alpha}y)ds\right\|^p \\ &&+\mathbb{E} \left\|\int_0^{t_1} A^\alpha \left(G(t_2-s)-G(t_1-s)\right)g(s,A^{-\alpha}y_s)dW(s)\right\|^p \Bigg)\\ &=&\sum_{\imath=1}^9 I_{\imath}. \end{eqnarray*} According to Theorem 3.2 $$(iv)$$ and $$(H_3)$$ together with H$$\ddot{o}$$lder inequality we can find a positive constant $$l_{2}>0$$ such that   \begin{align*} I_{2} &\leq 10^{p-1}\overline{M}^p_\alpha\mathbb{E}\left(\int_{t_1}^{t_2}(t_2-s)^{-\alpha} \left\|L \left({\tilde{\phi}_s}\right)\right\|ds\right)^p \\ &\leq 10^{p-1} \overline{M}^p_\alpha \mathbb{E}\left[\left(\int_{t_1}^{t_2}(t_2-s)^{-\alpha q}ds\right)^{\frac{p}{q}}\left(\int_{t_1}^{t_2}\left\|L \left({\tilde{\phi}_s}\right)\right\|^p ds\right)\right]\\ &\leq 10^{p-1} \overline{M}^p_\alpha \mathbb{E}\left[\left(\frac{(t_2-t_1)^{1-\alpha q}}{1-\alpha q}\right)^{\frac{p}{q}}l^p \left\|A^{-\alpha}\right\|^p \int_{t_1}^{t_2}\|{\tilde{\phi}_s}\|^p_{{\mathcal{B}}_{\alpha}}ds\right]\\ &\leq l_{2} \left(\frac{1}{1-\alpha q}\right)^{\frac{p}{q}}(t_2-t_1)^{p(1-\alpha)}\mathbb{E}\|\phi\|^p_{C_{\alpha}}, \end{align*} where $$\frac{1}{p}+\frac{1}{q}=1$$. Similarly, using Theorem 3.2 $$(iv)$$, Lemma 4.1, $$(H_2)$$, $$(H_4)$$ and H$$\ddot{o}$$lder inequality, one has that   \begin{eqnarray*} I_{3} &\leq& l_{3}\left(\frac{1}{1-\alpha q}\right)^{\frac{p}{q}} (t_2-t_1)^{p(1-\alpha)}\left(1+\|y\|^p_{D_p}\right)\!, \end{eqnarray*} and   \[ I_{4} \leq l_{4} \left(\frac{1}{1-\alpha q}\right)^{\frac{p}{q}}(t_2-t_1)^{p(1-\alpha)}\left(1+\|y\|^p_{D_p}\right)\!, \] for some constants $$l_{3}, l_4 >0$$. On the other hand, by using $$(H_4)$$, Theorem 3.2 $$(iv)$$ and Lemma 3.2, we obtain that for some $$C'_p$$,   \begin{eqnarray*} I_5 &\leq & 10^{p-1}C'_p\mathbb{E}\left( \int_{t_1}^{t_2}\left\| A^\alpha G(t_2-s)g(s,A^{-\alpha}y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}}\\ &\leq& 10^{p-1}C'_p \overline{M}^p_\alpha \mathbb{E}\left(\int_{t_1}^{t_2}(t_2-s)^{-2\alpha} \left\|g(s,A^{-\alpha}y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}} \\ &\leq& 10^{p-1}C'_p \overline{M}^p_\alpha \mathbb{E}\left[\left(\int_{t_1}^{t_2}(t_2-s)^{-2\alpha {\frac{p}{p-2}}}ds\right)^{\frac{p-2}{2}} \left(\int_{t_1}^{t_2}\left\|g(s,A^{-\alpha}y_s)\right\|^p_{{\mathscr L}_2^0} ds\right)\right] \\ &\leq& 10^{p-1}C'_p N_1 \overline{M}^p_\alpha \left(\frac{p-2}{p-2-2\alpha p}\right)^{\frac{p-2}{2}}(t_2-t_1)^{\frac{p-2-2\alpha p}{2}} \mathbb{E}\int_{t_1}^{t_2}\left(1+\left\|y_s\right\|^p_{C}\right)ds \\ &\leq& l_{5}\left(\frac{p-2}{p-2-2\alpha p}\right)^{\frac{p-2}{2}}(t_2-t_1)^{\frac{p-2\alpha p}{2}}\left(1+\|y\|^p_{D_p}\right)\!, \end{eqnarray*} for $$l_{5}>0$$ and $$p-2-2\alpha p>0.$$ Also, there holds   \begin{align*} I_{6} &\leq 20^{p-1} \mathbb{E}\left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right) L \left({\tilde{\phi}_s}\right)\right\| ds\right)^p\\ &\quad{} +20^{p-1}\mathbb{E}\left(\int_{t_1-\epsilon}^{t_1}\left\|A^\alpha \left(G(t_2-s)-G(t_1-s)\right)L \left({\tilde{\phi}_s}\right)\right\|ds\right)^p\\ &\leq 20^{p-1}\mathbb{E}\left[\left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}} \left(\int_0^{t_1-\epsilon}\left\| L \left({\tilde{\phi}_s}\right)\right\|^p ds\right)\right]\\ &\quad{} +20^{p-1}\mathbb{E}\left[\left(\int_{t_1-\epsilon}^{t_1}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}} \left(\int_{t_1-\epsilon}^{t_1}\left\| L \left({\tilde{\phi}_s}\right)\right\|^p ds\right)\right]\\ &\leq 20^{p-1}\left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}}l^p \left\|A^{-\alpha}\right\|^p \mathbb{E}\int_{0}^{t_1-\epsilon}\|\tilde{\phi}_s\|^p_{{\mathcal{B}}_{\alpha}}ds\\ &\quad{} +20^{p-1}\left[ \int_{t_1-\epsilon}^{t_1}\left(\frac{\overline{M}_\alpha}{(t_2-s)^\alpha}+\frac{\overline{M}_\alpha}{(t_1-s)^\alpha}\right)^q ds\right]^{\frac{p}{q}}l^p \left\|A^{-\alpha}\right\|^p \mathbb{E}\int_{t_1-\epsilon}^{t_1}\|\tilde{\phi}_s\|^p_{{\mathcal{B}}_{\alpha}}ds\\ &\leq l_{61}\left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}}\\ &\quad{} +l_{62}\frac{\overline{M}_\alpha^p}{(1-\alpha q)^{\frac{p}{q}}}\left[\left(t_2-t_1\right)^{1-\alpha q}-\left(t_2-t_1+\epsilon\right)^{1-\alpha q}-\epsilon^{1-\alpha q}\right]^{\frac{p}{q}}\!, \end{align*} where $$l_{61},l_{62}>0$$. Similarly, there exist $$l_{71}, l_{72}, l_{81}, l_{82}>0$$ such that   \begin{eqnarray*} I_7 &\leq & l_{71} \left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}}\\ &&+l_{72}\frac{\overline{M}_\alpha^p}{(1-\alpha q)^{\frac{p}{q}}}\left[\left(t_2-t_1\right)^{1-\alpha q}-\left(t_2-t_1+\epsilon\right)^{1-\alpha q}-\epsilon^{1-\alpha q}\right]^{\frac{p}{q}}\!, \end{eqnarray*} and   \begin{eqnarray*} I_8 &\leq & l_{81} \left( \int_0^{t_1-\epsilon}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^q ds\right)^{\frac{p}{q}}\\ &&+l_{82}\frac{\overline{M}_\alpha^p}{(1-\alpha q)^{\frac{p}{q}}}\left[\left(t_2-t_1\right)^{1-\alpha q}-\left(t_2-t_1+\epsilon\right)^{1-\alpha q}-\epsilon^{1-\alpha q}\right]^{\frac{p}{q}}\!. \end{eqnarray*} Finally, from Lemma 3.2, it follows immediately that, for some constant $$C''_p\geq 0$$,   \begin{eqnarray*} I_9 &\leq & 10^{p-1}C''_p\mathbb{E} \left(\int_0^{t_1}\left\| A^\alpha \left(G(t_2-s)-G(t_1-s)\right)g(s,A^{-\alpha}y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}} \\ &\leq& 10^{p-1}C''_p \mathbb{E}\left(\int_{0}^{t_1-\epsilon} \left\|A^\alpha \left(G(t_2-s)-G(t_1-s)\right)g(s,A^{-\alpha}y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}} \\ &&+ 10^{p-1}C''_p \mathbb{E}\left(\int_{t_1-\epsilon}^{t_1} \left\|A^\alpha G(t_2-s)-G(t_1-s)g(s,A^{-\alpha}y_s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}} \\ &=& I_{91}+I_{92}. \end{eqnarray*} By virtue of (3), we see easily that there exists a positive constant $$l_{91}>0$$ such that   \begin{eqnarray*} I_{91} &\leq& 10^{p-1} C''_p \left(\int_{0}^{t_1-\epsilon}\left\|A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^{2\frac{p}{p-2}} ds\right)^{\frac{p-2}{p}}\mathbb{E}\int_{0}^{t_1-\epsilon} \left\|g(s,A^{-\alpha}y_s)\right\|^p_{{\mathscr L}_2^0} ds\\ &\leq& l_{91} \left(\int_{0}^{t_1-\epsilon}\left\|A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^{2\frac{p}{p-2}} ds\right)^{\frac{p-2}{p}} \left(1+\|y\|^p_{D_p}\right)\!. \end{eqnarray*} Similarly, we have, for some constant $$l_{92}>0$$,   \begin{eqnarray*} I_{92} &\leq & l_{91} \left(\int_{t_1-\epsilon}^{t_1}\left\|A^\alpha \left(G(t_2-s)-G(t_1-s)\right)\right\|^{2\frac{p}{p-2}} ds\right)^{\frac{p-2}{p}} \left(1+\|y\|^p_{D_p}\right)\\ &\leq & l_{91} \left[\int_{t_1-\epsilon}^{t_1}\left(\frac{\overline{M}_\alpha}{(t_2-s)^\alpha}+\frac{\overline{M}_\alpha}{(t_1-s)^\alpha}\right)^{\frac{2p}{p-2}} ds\right]^{\frac{p-2}{p}} \left(1+\|y\|^p_{D_p}\right)\\ &\leq & l_{92} \overline{M}_\alpha^2 \left(\frac{p-2}{p-2-2\alpha p}\right)^{\frac{p-2}{p}}\left[\left(t_2-t_1\right)^{\frac{p-2-2\alpha p}{p-2}}-\left(t_2-t_1+\epsilon\right)^{\frac{p-2-2\alpha p}{p-2}}-\epsilon^{\frac{p-2-2\alpha p}{p-2}}\right]^{\frac{p-2}{p}}\!. \end{eqnarray*} By Theorem 3.2 $$(i)$$, $$G(t)$$ is strongly continuous for all $$t\geq 0$$. Hence $$I_1$$ tends to zero when $$t_2\to t_1$$. On the other hand, since from Theorem 3.2 $$(iii)$$, $$A^\alpha G(t)$$ is uniformly continuous for $$t\in (0,T]$$, it follows that $$I_\imath, \imath=2,\cdots,9,$$ tend to zero as $$t_2\to t_1$$ and $$\epsilon$$ small enough. Hence, for $$2<p<\infty$$ and $$0<\alpha<\frac{p-2}{2p} (<\frac{p-1}{p}<1)$$, $$(P^\lambda y)(\cdot)$$ is continuous from the right in $$[0,T)$$. A similar reasoning shows that it is also continuous from the left in $$(0,T]$$. Therefore, the proof of the lemma is complete. □ Lemma 4.3 Assume that all the hypotheses of Lemma 4.2 hold. Then the operator $$P^\lambda$$ sends $$D_p$$ into itself, i.e., $$P^\lambda(D_p) \subset D_p$$. Proof. Let $$y\in D_p$$ and let $$t\in[0,T]$$. Then   \begin{align*} \mathbb{E} \left\|\left(P^\lambda y\right)_t \right\|^p_C &\leq 5^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|G(t+\theta)A^\alpha\phi(0)\right\|^p\\ &\quad{} +5^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s) L \left({\tilde{\phi}_s}\right)ds\right\|^p\\ &\quad{} +5^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s) f(s,A^{-\alpha}y_s)ds\right\|^p\\ &\quad{} +5^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s) Bu^{\lambda}(s,A^{-\alpha}y)ds\right\|^p\\ &\quad{} +5^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s) g(s,A^{-\alpha}y_s)dW(s)\right\|^p\\ &= \sum_{\imath=10}^{14} I_{\imath}, \end{align*} where, from Theorem 3.2 $$(i)$$,   $$ I_{10}\leq 5^{p-1}\overline{M}^p \mathbb{E} \|\phi\|^p_{C_\alpha}. $$ Moreover, using $$(H_3)$$, Theorem 3.2 $$(iv)$$ and H$$\ddot{o}$$lder inequality we get   \begin{eqnarray*} I_{11} &\leq& 5^{p-1}\overline{M}^p_\alpha\mathbb{E}\sup_{\theta\in[-r,0]}\left(\int_0^{t+\theta} (t+\theta-s)^{-\alpha}\left\|L \left({\tilde{\phi}_s}\right)\right\|ds\right)^p\\ &\leq& 5^{p-1}\overline{M}^p_\alpha \mathbb{E}\sup_{\theta\in[-r,0]}\left[\left(\int_0^{t+\theta}(t+\theta-s)^{-\alpha q}ds \right)^{\frac{p}{q}}\left(\int_0^{t+\theta}\left\|L \left({\tilde{\phi}_s}\right)\right\|^p ds\right)\right]\\ &\leq& l_{11}\mathbb{E}\left\|\phi\right\|^p_{C_{\alpha}}, \end{eqnarray*} for a positive constant $$l_{11}$$. Similarly, due to $$(H_4)$$, Lemma 4.1, Theorem 3.2 $$(iv)$$ and H$$\ddot{o}$$lder inequality, there exist $$l_{12}, l_{13}>0$$ such that   \begin{eqnarray*} I_{12} &\leq& l_{12}\left(1+\|y\|^p_{D_p}\right)\!,\\ I_{13} &\leq& l_{13}\left(1+\|y\|^p_{D_p}\right)\!, \end{eqnarray*} and from $$(H_4)$$ and Lemma 3.4 it follows that   \begin{eqnarray*} I_{14} &\leq& 5^{p-1}N_2 \int_0^t \mathbb{E}\left\|g(s,A^{-\alpha}y_s)\right\|^p_{{\mathscr L}_2^0} ds\\ &\leq&l_{14}\left(1+\|y\|^p_{D_p}\right)\!, \end{eqnarray*} for a positive constant $$l_{14}$$. The above computations show well that   $$\left\|P^\lambda y\right\|^p_{D_p}=\sup_{t\in[0,T]}\mathbb{E} \left\|\left(P^\lambda y\right)_t \right\|^p_C <+\infty.$$ Therefore, we obtain that $$P^\lambda(D_p)\subset D_p$$ and this completes the proof. □ Now, we are on the position to prove the existence and uniqueness of mild solution of (1). Theorem 4.1 Let $$2<p<\infty$$ and $$0<\alpha<\frac{p-2}{2p}$$. Suppose that the assumptions $$(H_1)-(H_5)$$ hold. Then, for any $$\phi\in ML(p,r,\alpha)$$, the operator $$P^\lambda$$ has a unique fixed point in $$D_p$$. Proof. We use the classical Banach fixed point theorem to prove this theorem. By Lemmas 4.2 and 4.3, $$P^\lambda$$ is a continuous operator on $$[0,T]$$ that maps $$D_p$$ into itself. It remains to show that there exists an $$n\in \mathbb{N}$$ such that $${P^\lambda}^n$$ is a contraction. Let $$(x, y) \in D_p^2$$, then, for any fixed $$t\in[0,T]$$, we have   \begin{eqnarray*} J_2&:=& \mathbb{E} \left\| \left(P^\lambda x\right)_t -\left(P^\lambda y\right)_t \right\|^p_C \\[1ex] &\leq& 3^{p-1} \mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s)\left(f(s,A^{-\alpha}x_s)-f(s,A^{-\alpha}y_s)\right)ds\right\|^p \\[1ex] &&+3^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s)B\left(u^{\lambda}(s,A^{-\alpha}x)-u^{\lambda}(s,A^{-\alpha}y)\right)ds\right\|^p \\[1ex] &&+3^{p-1}\mathbb{E}\sup_{\theta\in[-r,0]}\left\|\int_0^{t+\theta} A^\alpha G(t+\theta-s)\left(g(s,A^{-\alpha}x_s)-g(s,A^{-\alpha}y_s)\right)dW(s)\right\|^p \\[1ex] &=&\sum_{\imath=15}^{17} I_{\imath}, \end{eqnarray*} which by using $$(H_4)$$, Theorem 3.2 $$(iv)$$ and H$$\ddot{o}$$lder inequality yields that   \begin{eqnarray*} I_{15} &\leq & 3^{p-1}\overline{M}_\alpha^p \mathbb{E}\sup_{\theta\in[-r,0]}\left(\int_0^{t+\theta}(t+\theta-s)^{-\alpha} \left\|f(s,A^{-\alpha}x_s)-f(s,A^{-\alpha}y_s)\right\|ds \right)^p\\[1ex] &\leq & 3^{p-1}\overline{M}_\alpha^p\mathbb{E}\sup_{\theta\in[-r,0]} \left[\left(\int_0^{t+\theta} (t+\theta-s)^{-\alpha q} ds\right)^{\frac{p}{q}}\left(\int_0^{t+\theta}\left\|f(s,A^{-\alpha}x_s)-f(s,A^{-\alpha}y_s)\right\|^p ds\right)\right] \\[1ex] &\leq & 3^{p-1}\overline{M}_\alpha^p N_1 \sup_{\theta\in[-r,0]}\left(\frac{(t+\theta)^{1-\alpha q}}{1-\alpha q}\right)^{\frac{p}{q}} \mathbb{E} \int_0^t \left\|x_s-y_s\right\|^p_{C}ds \\[1ex] &\leq & 3^{p-1}\overline{M}_\alpha^p N_1 \left(\frac{ T^{1-\alpha q}}{1-\alpha q}\right)^{\frac{p}{q}} \mathbb{E} \int_0^t \left\|x_s-y_s\right\|^p_{C}ds . \end{eqnarray*} Similarly, from $$(H_2)$$, Lemma 4.1, Theorem 3.2 $$(iv)$$ and H$$\ddot{o}$$lder inequality, we obtain   \begin{eqnarray*} I_{16} &\leq & \frac{1}{\lambda^p} 3^{p-1}\overline{M}_\alpha^p \|B\|^p N_3 \left(\frac{T^{1-\alpha q}}{1-\alpha q}\right)^{\frac{p}{q}}T \mathbb{E} \int_0^t \left\|x_s-y_s\right\|^p_{C}ds. \end{eqnarray*} Finally, $$(H_4)$$ and Lemma 3.4 yields to   \begin{eqnarray*} I_{17} &\leq & 3^{p-1}N_2 \int_0^t \mathbb{E} \left\|g(s,A^{-\alpha}x_s)-g(s,A^{-\alpha}y_s)\right\|^p_{{\mathscr L}_2^0} ds \\[1ex] &\leq & 3^{p-1}N_1 N_2\mathbb{E} \int_0^t \left\|x_s-y_s\right\|^p_{C}ds. \end{eqnarray*} Hence, there exists a $$B(\lambda)>0$$ such that   \begin{eqnarray*} \mathbb{E} \left\| \left(P^\lambda x\right)_t -\left(P^\lambda y\right)_t \right\|^p_C &\leq& B(\lambda)\mathbb{E}\int_0^t \left\|x_s-y_s\right\|^p_{C}ds\\ &\leq& T B(\lambda) \left\|x-y\right\|^p_{D_p}, \end{eqnarray*} for any $$t\in[0,T]$$. Then   $$\left\| P^\lambda x-P^\lambda y \right\|^p_{D_p} \leq T B(\lambda)\left\|x-y\right\|^p_{D_p},\,\,\,\,\,\,\,\, \mbox{for any}\,\,\,\, (x,y)\in D_p.$$ For any integer $$n\geq 1$$, by iteration, it follows that, for any $$x,y\in D_p$$,   $$\left\|{P^\lambda}^n x-{P^\lambda}^n y \right\|^p_{D_p} \leq \frac{(T B(\lambda))^n}{n!}\left\|x-y\right\|^p_{D_p}.$$ Since for sufficiently large $$n$$, $$\frac{(T B(\lambda))^n}{n!}< 1$$, $${P^\lambda}^n$$ is a contraction map on $$D_p$$ and therefore $${P^\lambda}$$ itself has a unique fixed point $$y(\cdot,\phi)$$ in $$D_p$$. The theorem is proved. □ Thus, by Theorem 4.1, for any $$\lambda\in(0,1)$$, the operator $${P^\lambda}$$ defined by (20) has a unique fixed point $$y^{\lambda}\in D_p$$, from which by setting $$x^{\lambda}(t,\phi)=A^{-\alpha}y^{\lambda}(t,\phi), t\in [-r,T],$$ we get a mild solution of System (1) which belongs to $\(C([-r,T];L_p({\it{\Omega}};X_\alpha))\)$. Using the Gronwall’s inequality, one can easily derive the uniqueness of the mild solution of System (1). Next we prove the main result of this work on the approximate controllability of System (1). That is Theorem 4.2 Let $$2<p<\infty$$ and $$0<\alpha<\frac{p-2}{2p}$$. Assume that the hypotheses $$(H_1)-(H_3), (H'_4)$$ and $$(H_5)$$ are all satisfied and suppose in addition that the semigroup $$\left(S(t)\right)_{t>0}$$ is compact. Then, for any $$\phi\in ML(p,r,\alpha)$$, System (1) is approximately controllable on $\([0,T]\)$. Proof. Let $$x^{\lambda}(\cdot,\phi)=A^{-\alpha}y^{\lambda}(\cdot,\phi)$$ be the mild solution of System (1) obtained in Theorem 4.1 under the control function $$u^{\lambda}(\cdot,A^{-\alpha}y)$$ given by (21). Then substituting (21) into (20) immediately yields that $$x^{\lambda}(\cdot,\phi)=\phi(\cdot)$$ on $$[-r,0]$$ and, for $$t\in[0,T]$$,   \begin{equation*} \begin{aligned} x^{\lambda}(t,\phi)&= G(t)\phi(0)+\int_0^t G(t-s)B B^* G^*(T-s)R(\lambda,-{\it{\Gamma}}_0^\top)\Big(\mathbb{E}h-G(T)\phi(0)\Big)\\ &\quad{} +\!\int_0^t\!\left(\!G(t-s)-R(\lambda,-{\it{\Gamma}}_s^\top)G(T-s)\int_s^t G(t-s)BB^*G^*(T-s)ds \!\right)\left(L \left({\tilde{\phi}_s}\right)+f(s,x^{\lambda}_s)\right)\!ds\\ &\quad{} +\int_0^t\left(G(t-s)-R(\lambda,-{\it{\Gamma}}_s^\top)G(T-s)\int_s^t G(t-s)BB^*G^*(T-s)ds \right)g(s,x^{\lambda}_s)dW(s)\\ &\quad{} +\int_0^t\int_s^tG(t-\nu)BB^*G^*(T-\nu) d\nu R(\lambda,-{\it{\Gamma}}_s^\top)\varphi(s)dW(s), \end{aligned} \end{equation*} which at $$t=T$$ gives   \begin{align*} x^{\lambda}(T,\phi)& = G(T)\phi(0)+{\it{\Gamma}}_0^\top R(\lambda,-{\it{\Gamma}}_0^\top)\Big(\mathbb{E} h-G(T)\phi(0)\Big)\\ &\quad{} +\int_0^\top\left(I-{\it{\Gamma}}_s^\top R(\lambda,-{\it{\Gamma}}_s^\top) \right)G(T-s)\left(L \left({\tilde{\phi}_s}\right)+f(s,x^{\lambda}_s)\right)ds\\ &\quad{} +\int_0^\top\left(I-{\it{\Gamma}}_s^\top R(\lambda,-{\it{\Gamma}}_s^\top) \right)G(T-s)g(s,x^{\lambda}_s)dW(s)\\ &\quad{} +\int_0^\top {\it{\Gamma}}_s^\top R(\lambda,-{\it{\Gamma}}_s^\top)\varphi(s)dW(s), \end{align*} hence, using the fact that $$I-{\it{\Gamma}}_s^\top R(\lambda,-{\it{\Gamma}}_s^\top)=\lambda R(\lambda,-{\it{\Gamma}}_s^\top)$$, we get   $$ \begin{aligned} x^{\lambda}(T,\phi) & = h-\lambda R(\lambda,-{\it{\Gamma}}_0^\top)\Big(\mathbb{E} h-G(T)\phi(0)\Big)\\ &\quad{} +\int_0^\top\lambda R(\lambda,-{\it{\Gamma}}_s^\top)G(T-s)\left(L \left({\tilde{\phi}_s}\right)+f(s,x^{\lambda}_s)\right)ds\\ &\quad{} +\int_0^\top\lambda R(\lambda,-{\it{\Gamma}}_s^\top)\left(G(T-s)g(s,x^{\lambda}_s)-\varphi(s)\right)dW(s). \end{aligned} $$ By assumption $$(H'_4)$$, we know that $$\left\{f(s,x^{\lambda}_s):\lambda\in(0,1)\right\}$$ and $$\left\{g(s,x^{\lambda}_s):\lambda\in(0,1)\right\}$$ are uniformly bounded in $$\lambda\in(0,1)$$ in $$X$$ and $${{\mathscr L}_2^0}(K;X)$$ respectively, from which it follows that there exist subsequences, still denoted by $$f(s,x^{\lambda}_s)$$ and $$g(s,x^{\lambda}_s)$$, that converge weakly to, say, $$f(s)$$ and $$g(s)$$ in $$X$$ and $${{\mathscr L}_2^0}(K;X)$$, respectively, for each $$s\in[0,T]$$. Then the compactness of $$G(t), t > 0,$$ which is guaranteed by Theorem 3.2 $$(ii)$$, implies that   \begin{align*} G(T-s)f(s,x^{\lambda}_s)&\to G(T-s)f(s),\\ G(T-s)g(s,x^{\lambda}_s)&\to G(T-s)g(s), \mbox{in} (0,T]\times {\it{\Omega}}. \end{align*} Hence   \begin{align*} \mathbb{E}\left\|x^{\lambda}(T,\phi)-h\right\|^p &\leq 7^{p-1}\left\|\lambda R(\lambda,-{\it{\Gamma}}_0^\top)\left(\mathbb{E} h-G(T)\phi(0)\right)\right\|^p\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top\left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top)G(T-s) L \left({\tilde{\phi}_s}\right)\right\|ds\right)^p\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top\left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top)\right\|\left\|G(T-s)\left(f(s,x^{\lambda}_s)-f(s)\right)\right\|ds\right)^p\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top\left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top)G(T-s)f(s)\right\|ds\right)^p\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top\left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top)\right\|^2 \left\|G(T-s)\left(g(s,x^{\lambda}_s)-g(s)\right)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}}\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top \left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top) G(T-s)g(s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}}\\ &\quad{} +7^{p-1} \mathbb{E}\left(\int_0^\top\left\|\lambda R(\lambda,-{\it{\Gamma}}_s^\top)\varphi(s)\right\|^2_{{\mathscr L}_2^0} ds\right)^{\frac{p}{2}}\\ &\to 0 \,\,\,\,\mbox{as} \,\,\,\, \lambda\to 0^+. \end{align*} Consequently, System (1) is approximately controllable on $$[0,T]$$. □ 5. Example To illustrate the obtained results, we present in this section an example of controllable SPDE. Consider the following stochastic boundary value problem.   \begin{equation}\label{eq420}\left\{ \begin{aligned} &d z(t,x)= \left[-\frac{\partial^2}{\partial x^2}z(t,x)+z(t-1,x)+F_1(t,z(t-r_1(t),x))+Bu(t,x)\right]dt\\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +F_2(t,z(t-r_2(t),x)) d\beta(t), \,\,\,\,0< t\leq 2, \,\,\,\,0\leq x\leq\pi,\\ &z(t,0)=z(t,\pi)=0, \,\,\,\,0\leq t\leq 2,\\ &z(\theta,x)=\phi_0(\theta,x), \,\,\,\,-1\leq \theta\leq 0, \,\,\,\,0\leq x\leq\pi, \end{aligned}\right. \end{equation} (22) where $$r_1, r_2$$ are continuous functions with $$0<r_1(\cdot), r_2(\cdot)\le r$$, and $$\phi_0$$ is $$\mathscr{F}_0-$$measurable. The functions $$F_1$$ and $$F_2$$ will be described below. $$\beta(t)$$ denotes a one-dimensional standard Brownian motion. System (22) arises in the study of stochastic heat flow in materials of the so-called retarded type. Here, $$z(t,x)$$ represents the temperature of the point $$x$$ at time $$t$$. For this stochastic equation without the second term $$z(t-1,x)$$ on the right hand side, reference Dauer & Mahmudov (2004) has already established the sufficient conditions for its approximate controllability. Clearly, the results in Dauer & Mahmudov (2004) can not be applied to the system (22) as $$z(t-1,x)$$ is not uniformly bounded. However, we may explore here the approximate controllability of the stochastic system (22) by using the results achieved in the last section. Let $$X=L_2([0,\pi]), K=\mathbb{R}$$ and $$U=L_2([0,2])$$. Let $$A: D(A)\to X$$ be the operator given by   $$A\xi=-\xi'',$$ with the domain   \begin{align*} D(A)= \Bigg\{&\xi\in X \,\,\,\,\mbox{such that}\,\,\,\, \;\xi(0)=\xi(\pi)=0 \,\,\,\,\mbox{with}\,\,\,\, \frac{\partial\xi}{\partial x},\;\frac{\partial^2\xi }{\partial x^2} \,\,\,\,\mbox{all in}\,\,\,\, X, \\ & \xi\,\,\,\, \mbox{and}\,\,\,\, \frac{\partial\xi}{\partial x}\,\,\,\, \mbox{are both absolutely continuous on} \,\,\,\,[0,\pi]\;\Bigg\}. \end{align*} Then $$-A$$ generates a strongly continuous semigroup $$(S(t))_{t\ge 0}$$ which is analytic, compact and self-adjoint. Furthermore, $$-A$$ has a discrete spectrum, the eigenvalues are $$-n^2,\;n\in \mathbb{N^+}$$, with the corresponding normalized eigenvectors $$e_n(x)= \sqrt{\frac{2}{\pi}}\sin(nx)$$, $$n=1,2,\cdots$$. Then the following properties hold: (i) If $$\xi\in D(A)$$, then   $$A\xi= \sum_{n=1}^{\infty} n^2 \langle \xi,e_n \rangle e_n.$$ (ii) For every $$\xi\in X$$,   \begin{equation*} S(t)\xi=\sum_{n=1}^{\infty} e^{-n^2t}\langle \xi,e_n \rangle e_n. \end{equation*} Define $$A^\alpha$$ for self-adjoint operator $$A$$ by the classical spectral theorem and it is easy to deduce that   $$|A^{\alpha}|S(t)\xi=\sum\limits_{n=1}^{\infty}\left(n^2\right)^{\alpha} e^{-n^2 t}\langle \xi,e_n \rangle e_n ,$$ which immediately implies   \begin{align*} \left\|A^{\alpha}S(t)\xi\right\|^2&=\sum\limits_{n=1}^{\infty} n^{4\alpha} e^{-2 n^2 t} \left|\langle \xi,e_n \rangle \right|^2\\ &=e^{-2 a t} t^{-2\alpha}\sum\limits_{n=1}^{\infty} (n^2 t)^{2\alpha} e^{-\left(2 n^2-2a\right)t} \left|\langle \xi,e_n \rangle \right|^2\!. \end{align*} Then $$(H_1)$$ holds. Let $$2<p<\infty$$ and $$0<\alpha<\frac{p-2}{2p}$$ such that there exists some number $$\beta$$ satisfies $$\frac{1}{p}+\alpha<\beta<\frac12$$. Suppose that the following conditions hold for System (22): (i) The functions $$F_i:[0,2]\times \mathbb{R}\to \mathbb{R}$$ are Lipschitz continuous in the second variable and uniformly bounded. i.e., there exist positive constants $$c_1$$ and $$c_2$$ such that   $$|F_{i}(t,x_1)-F_{i}(t,x_2)| \leq c_i |x_1-x_2|, \,\,\,\,i=1,2,$$ for any $$t\in [0,2], x_i\in \mathbb{R}$$. Moreover, there exists a constant $$c_0> 0$$ such that, for any $$t\in [0,2], x\in \mathbb{R}$$,   $$|F_{i}(t,x,y)|\leq c_0.$$ (ii) The function $$\phi_0(\theta,x)$$ belongs to $$ML(p,r,\alpha)$$. To represent problem (22) as the abstract form of System (1), we define $$Z(t)(\cdot):= z(t,\cdot)$$ and $$\phi(t)(\cdot):= \phi(t,\cdot)$$. Now define the operator $$L: \mathcal{B}([-1,0];X) \to X$$, the maps $$f(\cdot,\cdot):[0,2]\times C_\alpha\to X$$ and $$g(\cdot,\cdot):[0,2]\times C_\alpha \to {{\mathscr L}_2^0}(\mathbb{R};X)$$, respectively, as   \begin{align*} L(\phi)(x) & =L(\phi(\cdot,x))=\phi(-1,x),\\ f(t,\psi)(x) & = f(t,\psi(\cdot,x))=F_1\left(t,\psi(-r_1(t),x)\right)\!,\\ g(t,\psi)(x) & =g(t,\psi(\cdot,x))=F_2\left(t,\psi(-r_2(t),x)\right)\!, \end{align*} for any $$t\in [0,2]$$, $$\phi\in \mathcal{B}([-1,0];X)$$ and $$\psi\in C_\alpha$$. Then under these notations System (22) can be rewritten into the form of System (1). Clearly, $$L$$ is a linear bounded operator satisfying $$(H_3)$$. On the other hand, from the computations in Taniguchi et al. (2002) we see that the functions $$f(\cdot,\cdot)$$ and $$g(\cdot,\cdot)$$ are Lipschitz continuous with respect to the second variable. Moreover, their uniform boundedness follows directly from the uniform boundedness of $$F_{\imath}(\cdot,\cdot), \imath=1,2$$. Therefore, hypotheses $$(H_4')$$ is satisfied. Here, as usual, we take   $$ U=\left\{u=\sum\limits_{n=2}^{\infty}u_ne_n: \sum\limits_{n=2}^{\infty}u_n^2< +\infty\right\}\!, $$ with the norm   $$ \|u\|= \bigg(\sum\limits_{n=2}^{\infty}u_n^2\bigg)^{\frac{1}{2}}. $$ Then $$U$$ is a Hilbert space. Now define the linear continuous operator $$B$$ from $$U$$ into $$X$$ as   $$ Bu=2u_2e_1(x)+\sum\limits_{n=2}^{\infty}u_ne_n(x), \,\,\,\,\text{for}\,\,\,\, u=\sum\limits_{n=2}^{\infty}u_ne_n\in U. $$ It is easy to compute that   \begin{equation}\label{eq421} B^*v=(2v_1+v_2)e_2(x)+\sum\limits_{n=3}^{\infty}v_ne_n(x), \end{equation} (23) with $$v=\sum\limits_{n=1}^{\infty}v_ne_n(x)\in X.$$ So $$(H_2)$$ is verified too. According to Theorem 4.2, to obtain the approximate controllability for System (22), it remains to us to verify the resolvent condition ($$H_5$$). As one can see, it is difficult for us to obtain the explicit expression of the fundamental solution $$G(t)$$ associated to the linear system. Fortunately, however, we are able to calculate the expression of $$G(t)$$ on the interval $$[0,1]$$, and this is enough to guarantee that the condition ($$H_5$$) holds in this situation. Indeed, the solution on the interval $$[0,1]$$ of the corresponding linear deterministic equation   $$\left\{ \begin{aligned} &\frac{d}{dt}Z(t)=-AZ(t)+L(Z_t)+ f(t),\,\,\,\, t\in [0,2],\\ &Z_0=0, \end{aligned} \right.$$ is given by the $$C_0-$$semigroup $$S(t)$$ as follows:   $$Z(t)=\int_0^t S(t-s)f(s)ds,\,\,\,\,\,\,\,\, t\in [0,1].$$ This indicates that $$G(t)=S(t)$$ for $$t\in [0,1]$$. Thus we have   $$G^*(t)=S^*(t)=S(t)=G(t),\,\,\,\,\text{for}\,\,\,\, t\in [0,1].$$ Hence combining (23) we calculate directly that   $$B^*G^*(t)\xi =(2\xi_1e^{-t}+\xi_2e^{-4t})e_2(x)+\sum\limits_{n=3}^{+\infty}\xi_ne^{-n^2t}e_n(x),$$ for $$\xi=\sum\limits_{n=1}^{+\infty}\xi_ne_n(x)\in X$$ and $$t\in [0,1]$$. Now let $$\|B^*G^*(t)\xi\|=0,\text{for all} t\in [0,2]$$, then   $$ \|B^*G^*(t)\xi\|=0,\,\,\,\, t\in [0,1], $$ it follows that   $$\|2\xi_1e^{-t}+\xi_2e^{-4t}\|^2+\sum\limits_{n=3}^{+\infty}\|\xi_ne^{-n^2t}\|^2=0, \,\,\,\,t\in [0,1],$$ which implies $$\xi_n=0, n=1,2,\cdots.$$ and hence $$\xi =0$$. So, by virtue of Theorem 3.4, $$(H_5)$$ holds. Therefore by Theorem 4.2 we infer that System (22) is approximately controllable on the interval [0,2]. 6. Conclusion In this article, we studied the approximate controllability of the stochastic system (1) with finite delay in $$L_p$$ spaces ($$2<p<\infty$$). The resolvent condition was used to get the desired results by equivalently assuming that the linear deterministic part of the system (1) is approximately controllable. Since this approach requires the nonlinear terms of the system be uniformly bounded and since system (1) involves an additional linear term which is not uniformly bounded, we established the fundamental solution theory corresponding to the associated deterministic linear system and we have used it to describe in an explicit way the mild solutions of the considered stochastic system. Thus we could discuss the approximate controllability for System (1) with partially non-uniformly bounded term under the resolvent condition and obtained sufficient conditions for it. Therefore, this work extends somewhat the existing related results in the literature. It is worth pointing out that, as the fundamental solution $$G(t)$$ has the same regular properties as the semigroup and the expression of mild solutions becomes much simpler due to the use of $$G(t)$$, it is convenient for us to apply the fundamental solution theory to study qualitative issues and other topics on control theory such as stabilization, optimal control for deterministic and stochastic partial functional differential equations (PFDEs) with finite delay, which are our subsequent works in the recent future. There are some direct problems for our further discussion. First we may extend the results of this article to neutral stochastic PFDEs with finite delay by establishing the fundamental solution theory for the associated neutral linear system. We will also explore the optimal control problems for stochastic PFDEs with finite delay by utilizing the fundamental solution theory founded here. 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Published: Nov 6, 2017

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