Controllability of a stochastic functional differential equation driven by a fractional Brownian motion
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Abstract
Let U, V and W be three Hilbert spaces and let \(B^{H}\) be a W-valued fractional Brownian motion with Hurst index \(H\in(\frac{1}{2},1)\). In this paper, we consider the approximate controllability of the Sobolev-type fractional stochastic differential equation
where \({}^{\mathrm{c}}D^{\alpha}\) is the Caputo fractional derivative of order \(\alpha\in(1-H,1)\), the time history \(x_{t}:(-\infty,0]\rightarrow x_{t}(\theta)=x(t+\theta)\) with \(t>0\) belonging to the phase space \(\mathscr{B}_{h}\), the control function \(u(\cdot)\) is given in \(L^{2}([0,T],V)\), B is a bounded linear operator from V into U. Under some suitable conditions, we show that the system is approximately controllable on \([0,T]\) and we give an example to illustrate the theory.
$$\textstyle\begin{cases} {}^{\mathrm{c}}D^{\alpha}_{t}[Lx(t)]=Ax(t)+f(t,x_{t})+ Bu(t)+\sigma(t)\frac {d}{dt}B^{H}(t), &t\in(0,T], \\ x(t)=\phi(t), &t\in(-\infty,0], \end{cases} $$
Keywords
Fractional stochastic functional differential equation Fractional Brownian motion Fractional calculus ControllabilityMSC
60G22 60H05 60H10 Download
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1 Introduction
The study of controllability is one of the important parts of mathematical control theory in both deterministic and stochastic control theory. There are lots of publications working on control problems of various systems [1, 2, 3, 4]. Complete controllability means the systems can be steered to arbitrary final state while the systems with approximate controllability just can be steered to a small neighborhood of the final state. Even though the concept of the approximate controllability is weaker than the complete controllability, it is prevalent to consider approximate controllable systems which can be adequate in application [5, 6, 7]. In fact, the approximate controllability of systems represented by nonlinear evolution equations has been studied by several authors. In [8], Sakthivel et al. studied a class of control systems governed by the semilinear fractional differential equations in Hilbert spaces by using the semigroup theory, the fractional power theory and fixed point theorem. Fu and Mei [9] investigated the approximate controllability of semilinear neutral functions differential systems with finite delay.
Due to the extensive applications in various fields such as science and engineering, fractional differential equations attract more and more attention of experts and scholars. Fractional differential equations may be derived from the particle sticking and trapping phenomena which would be more accurate to describe certain physical phenomena (see, for examples, [10, 11, 12]). In addition, Sobolev-type equation appears in all kinds of physical problems such as flow of fluid through fissured rocks, thermodynamics, propagation of long waves of small amplitude (see [13]). Therefore, it is necessary and significative to study fractional order differential equations of Sobolev-type (see [14, 15] and the references therein). The existence and uniqueness of mild solution to Sobolev-type fractional nonlocal dynamical equations in Banach spaces is shown in [16]. By using the fractional calculus, semigroup theory and stochastic analysis techniques, [17] considered a class of nonlinear fractional Sobolev-type stochastic differential equations in a Hilbert space.
On the other hand, the property of long memory is widely used in describing the phenomena in fields like hydrology and geophysics as well as economics and telecommunications. As an extension of Brownian motion, fractional Brownian motion (fBm) is a self-similar Gaussian processes which have the properties of long/short-range dependence. However, in comparison with Brownian motion, the process is neither a semi-martingale nor a Markov process. For this reason, there are a few publications leaning the systems which are driven by this type of perturbation. In [18], the authors first studied the fractional Brownian motion in Hilbert spaces and some related stochastic equations. We refer to [19, 20] and the references therein for the details of the theory of stochastic calculus for fractional Brownian motion. However, it should be emphasized that to the best of our knowledge the controllability of stochastic functional differential equation of Sobolev-type driven by fractional Brownian motion has not been studied yet and the aim of this paper is to do some further research on this problem.
Motivated by these results, in this paper we study the approximate controllability of the Sobolev-type fractional stochastic differential equations of the form
In the above system, we assume that
$$ \textstyle\begin{cases} {}^{\mathrm{c}}D^{\alpha}_{t}[Lx(t)]=Ax(t)+f(t,x_{t})+ Bu(t)+\sigma(t)\frac {d}{dt}B^{H}(t), & t\in(0,T], \\ x(t)=\phi(t), & t\in(-\infty,0]. \end{cases} $$
(1.1)
-
\({}^{\mathrm{c}}D^{\alpha}\) is the Caputo fractional derivative of order \(\alpha\in(1-H,1)\),
-
A, L are two linear bounded operators on a Hilbert space U,
-
B is a bounded linear operator from the Hilbert space V into Hilbert space U,
-
the time history \(x_{t}(\theta)=x(t+\theta)\), \(t>0\),
-
\(u(\cdot)\) is a control function on \(L^{2}([0,T],V)\),
-
\(B^{H}=\{B^{H}(t),t\in[0,T]\}\) is a cylindrical fractional Brownian motion with Hurst index \(H\in(\frac{1}{2},1)\),
-
the functions f and σ are two Borel functions with some suitable conditions.
The paper is organized as follows. In Sect. 2, we represent some preliminaries for stochastic integral of fractional Brownian motion in Hilbert space. In Sect. 3, we obtain the approximate controllability results of the Sobolev-type fractional stochastic system (1.1). In Sect. 4, we give an example as an application.
2 Preliminaries
In this section, we will introduce some definitions, lemmas and notions which will be used in the next section.
2.1 Fractional Brownian motion
Let \((\Omega,\mathscr{F},(\mathscr{F}_{t}),\mathrm{P})\) be a complete filtered probability space. A fractional Brownian motion (fBm) \(\beta ^{H}=\{\beta^{H}(t),t\in[0,T]\}\) with Hurst index \(H\in(0,1)\) is a mean zero Gaussian process such that \(\beta^{H}(0)=0\) and
for all \(t,s\geq0\). When \(H=1/2\), \(\beta^{H}\) coincides with the standard Brownian motion, and when \(H\neq\frac{1}{2}\) it is neither a semi-martingale nor a Markov process. The fBm \(\beta^{H}\) admits the following integral representation:
for all \(t\geq0\), where \(\{W(t),0\leq t\leq T\}\) is a standard Brownian motion and the kernel \(K_{H}(t,s)\) satisfies
with a normalizing constant \(\kappa_{H}>0\) such that \(E(\beta_{1}^{H})^{2}=1\). Throughout this paper we assume that \(\frac{1}{2}\leq H<1\) is arbitrary but fixed.
$$\mathrm{E} \bigl(\beta^{H}(t)\beta^{H}(s) \bigr)= \frac{1}{2} \bigl(s^{2H}+t^{2H}-|t-s|^{2H} \bigr) $$
$$\beta^{H}(t)= \int_{0}^{t}K_{H}(t,s)\,dW(t) $$
$$\frac{\partial K_{H}}{\partial t}(t,s)=\kappa_{H} \biggl(H-\frac{1}{2} \biggr) \biggl(\frac{s}{t} \biggr)^{\frac{1}{2}-H}(t-s)^{H-\frac{3}{2}} $$
Let \(\mathcal{H}\) be the completion of the linear space \(\mathcal{E}\) generated by the indicator functions \(1_{[0,t]}\), \(t\in[0,T]\) with respect to the inner product
The mapping
is an isometry from \({\mathcal {E}}\) to the Gaussian space generated by \(\beta^{H}\) and it can be extended to \({\mathcal {H}}\), which is called the Wiener integral with respect to \(\beta^{H}\). Consider the operator \(K^{*}_{H}\) from \({\mathcal {E}}\) to \(L^{2}([0,T])\) defined by
for \(\varphi\in{\mathcal {E}}\). Then the operator \(K_{H}^{*}\) is an isometry between \(\mathcal{E}\) and \(L^{2}([0,T])\) which can be also extended to the Hilbert space \(\mathcal{H}\).
$$\langle1_{[0,s]},1_{[0,t]}\rangle_{\mathcal{H}}=\frac{1}{2} \bigl(t^{2H}+s^{2H}-|t-s|^{2H} \bigr). $$
$${\mathcal {E}}\ni\varphi\to\beta^{H}(\varphi):= \int_{0}^{T}\varphi(s)\,d\beta^{H}(s) $$
$$\bigl(K_{H}^{*}\varphi\bigr) (s)= \int^{T}_{s}\varphi(t)\frac{\partial K_{H}}{\partial t}(t,s)\,dt $$
Lemma 2.1
For every\(\varphi\in\mathcal{H}\), we have
$$\int^{T}_{0}\varphi(s)\,d\beta^{H}(s)= \int^{T}_{0}\bigl(K_{H}^{*}\varphi\bigr) (s) \,dW(s). $$
We now recall that the definition of stochastic integral of fBm in the Hilbert space V. Let \(\{B^{H}(t),0\leq t\leq T\}\) be a W-valued \(\mathscr{F}_{t}\)-adapted fBm defined on \((\Omega,\mathscr{F},(\mathscr {F}_{t}),\mathrm{P})\) with the representation of the form
where \(\{e_{n}\}_{n\in \mathbb {N}}\) is a complete orthogonal basis in W, and Let \(\varphi:[0,T]\rightarrow L^{0}_{2}(W,U)\) such that
where \(L^{0}_{2}(W,U)\) is the space of all Hilbert–Schmidt operators from \(Q^{\frac{1}{2}}W\) to U with norm \(\|\cdot\|_{L^{0}_{2}(W,U)}\) defined by
$$B^{H}(t)=\sum_{n=1}^{\infty}\sqrt{ \lambda_{n}}\beta^{H}_{n}(t)e_{n}, \quad t \geq0, $$
-
\(\{\beta^{H}_{n},n=1,2,\ldots\}\) is a sequence of independent fBms with the same Hurst index \(H\in(\frac{1}{2},1)\),
-
\(\{\lambda_{n}; n\in\mathbb{N}\}\) is a bounded sequence of non-negative real numbers such that \(Qe_{n}=\lambda_{n}e_{n}\),
- Q is a non-negative self-adjoint trace class operator with finite trace$$\operatorname{Tr}Q=\sum^{\infty}_{n=1} \lambda_{n}< +\infty. $$
$$ \sum^{\infty}_{n=1} \bigl\Vert K^{*}_{H}\bigl(\varphi Q^{\frac{1}{2}}e_{n}\bigr) \bigr\Vert _{L^{0}_{2}([0,T];U)}< \infty, $$
(2.1)
$$\Vert \xi \Vert _{L^{0}_{2}(W,U)}= \bigl\Vert \xi Q^{\frac{1}{2}} \bigr\Vert _{H-S}^{2}= \operatorname{Tr}\bigl(\xi Q\xi^{*}\bigr)= \sum^{\infty}_{n=1} \Vert \sqrt{ \lambda_{n}}\xi e_{n} \Vert ^{2}. $$
Definition 2.1
Let \(\varphi:[0,T]\rightarrow L^{0}_{2}(W,U)\) satisfy (2.1). We define the stochastic integral \(\int^{t}_{0} \varphi(s)\,dB^{H}(s)\) by
$$\begin{aligned} \int^{t}_{0} \varphi(s)\,dB^{H}(s)&:=\sum ^{\infty}_{n=1} \int^{t}_{0}\varphi (s)Q^{\frac{1}{2}}e_{n} \,d\beta_{n}^{H} \\ &=\sum^{\infty}_{n=1} \int^{t}_{0}\bigl(K_{H}^{*}\bigl(\varphi Q^{\frac{1}{2}}e_{n}\bigr)\bigr) (s)\,dB(s). \end{aligned}$$
Lemma 2.2
Let\(\varphi:[0,T]\to L^{0}_{2}(W,U)\)satisfy (2.1). Then, for any\(a,b\in[0,T]\)with\(a< b\)we haveIn addition, \(\sum^{\infty}_{n=1}\|\varphi(s)Q^{\frac{1}{2}}e_{n}\|\)is uniformly convergent in\(t\in[0,T]\), then we have
$$\mathrm{E} \biggl\Vert \int^{b}_{a}\varphi(s)\,dB^{H}(s) \biggr\Vert ^{2}\leq cH(2H-1) (b-a)^{2H-1}\sum ^{\infty}_{n=1} \int^{b}_{a} \bigl\Vert \varphi(s)Q^{\frac{1}{2}}e_{n} \bigr\Vert ^{2}\,ds. $$
$$\mathrm{E} \biggl\Vert \int^{b}_{a}\varphi(s)\,dB^{H}(s) \biggr\Vert ^{2}\leq cH(2H-1) (b-a)^{2H-1}\sum ^{\infty}_{n=1} \int^{b}_{a} \bigl\Vert \varphi(s) \bigr\Vert _{L^{0}_{2}(W,U)}^{2}\,ds. $$
2.2 Some assumptions
In this subsection, we recall that some notions of fractional calculus and give some assumptions for the stochastic system (1.1). Recall that the fractional integral \(I^{\alpha}\) of order α for a function \(f:[0,\infty)\rightarrow \mathbb {R}\) is defined as
provided the right side is point-wise defined on \([0,\infty)\), where \(\Gamma(\cdot)\) is the gamma function, which is defined by \(\Gamma(x):=\int^{\infty}_{0}t^{x-1}e^{-t}\,dt\). Moreover, the Caputo derivative \({}^{\mathrm{c}}D^{\alpha}\) of order α for a function \(f\in \mathbf{C}^{n}([0,\infty))\) is defined as
If f is an abstract function with values in U, then the integrals appearing in the above definitions are taken in Bochner’s sense.
$$I^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)} \int^{t}_{0}\frac{f(s)}{(t-s)^{1-\alpha }}\,ds,\quad t>0, \alpha>0, $$
$${}^{\mathrm{c}}D^{\alpha}_{t}f(t)=\frac{1}{\Gamma(n-\alpha)} \int^{t}_{0}\frac{f^{(n)}(s)}{ (t-s)^{1+\alpha-n}}\,ds=I^{n-\alpha}f^{(n)}(t), \quad t>0,n-1< \alpha< n. $$
To study the stochastic system (1.1), we need some assumptions. Throughout this paper we assume that U, V, W is three real separable Hilbert spaces with inner products \(\langle\cdot, \cdot\rangle_{U}\), \(\langle\cdot,\cdot\rangle_{V}\) and \(\langle\cdot,\cdot\rangle_{W}\), respectively. We first give some conditions about the three operators L, A, B as follows: Based on the above assumptions (\(\mathbb{A}1\)), (\(\mathbb{A}2\)) and the closed graph theorem, the linear operator \(AL^{-1}:U\to U\) the bounded, and \(AL^{-1}\) generates a semigroup \(\{S(t),t>0\}\) in U. Denote \(M=\max_{t>0}\|S(t)\|\), \(\|L\|=M_{1}\) and \(\|L^{-1}\|=\widetilde{M}_{1}\).
- (\(\mathbb{A}1\))
-
A and L are two linear unbounded operators on U such that \(D(A)\subset U\), \(D(L)\subset U\), and A is closed,
- (\(\mathbb{A}2\))
-
\(D(L)\subset D(A)\),
- (\(\mathbb{A}3\))
-
\(L^{-1}:U\to D(U)\) is compact,
- (\(\mathbb{A}4\))
-
B is a bounded linear operator from V into U.
For \(x\in U\), we define two families \(\{\mathscr{T}_{L}(t),t\geq0\}\) and \(\{\mathscr{S}_{L}(t),t\geq0\}\) of operators by
and
where
is a probability density function defined on \((0,\infty)\).
$$\mathscr{T}_{L}(t)x:= \int^{\infty}_{0}L^{-1}\xi_{\alpha}( \theta)S\bigl(t^{\alpha}\theta \bigr)x\, d\theta $$
$$\mathscr{S}_{L}(t)x:=\alpha \int^{\infty}_{0}L^{-1}\theta \xi_{\alpha}(\theta) S\bigl(t^{\alpha}\theta\bigr)x\, d\theta, $$
$$\theta\mapsto\xi_{\alpha}(\theta)=\frac{1}{\pi\alpha}\sum _{n=1}^{\infty }(-\theta)^{n-1} \frac{1}{n!} \Gamma(n\alpha+1)\sin(n\pi\alpha) $$
Lemma 2.3
([21])
The operators\(\mathscr{T}_{L}(t)\)and\(\mathscr{S}_{L}(t)\)have the following properties:
- (i)For every\(t\geq0\), \(\mathscr{T}_{L}(t)\)and\(\mathscr {S}_{L}(t)\)are linear and bounded, and, moreover, for every\(x\in U\)$$ \begin{aligned} & \bigl\Vert \mathscr{T}_{L}(t)x \bigr\Vert \leq M\widetilde{M}_{1} \Vert x \Vert , \\ & \bigl\Vert \mathscr{S}_{L}(t)x \bigr\Vert \leq \frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \Vert x \Vert . \end{aligned} $$(2.2)
- (ii)
\(\mathscr{T}_{L}(t)\)and\(\mathscr{S}_{L}(t)\)are strong continuous and compact.
We now introduce the abstract phase space. For a continuous function \(h:(-\infty,0]\to(0,\infty)\) satisfying
we define a phase space \(\mathscr{B}_{h}\) associated with h as follows (see, Cui and Yan [11]):
Clearly, \((\mathscr{B}_{h},\|\cdot\|_{\mathscr{B}_{h}})\) is a Banach space if \(\mathscr{B}_{h}\) is endowed with the norm
for \(\phi\in\mathscr{B}_{h}\). According to the definition of the fractional derivative and Caputo derivative, we rewrite (1.1) as the equivalent integral equation
$$l:= \int^{0}_{-\infty}h(t)\,dt< \infty, $$
$$\begin{aligned} \mathscr{B}_{h} =& \biggl\{ \phi:(-\infty,0]\to U, \text{for any } a>0, \bigl(\mathrm{E} \bigl\Vert \phi(\theta) \bigr\Vert ^{2} \bigr)^{1/2} \text{ is bounded} \\ & \text{and measurable functions on } [-a,0] \text{ with } \phi(0)=0 \\ & \text{and } \int^{0}_{-\infty}h(s)\sup_{s\leq\theta\leq0}\bigl( \mathrm{E} \bigl\Vert \phi(\theta) \bigr\Vert ^{2} \bigr)^{1/2}\,ds< \infty \biggr\} . \end{aligned}$$
$$\Vert \phi \Vert _{\mathscr{B}_{h}}= \int^{0}_{-\infty}h(s)\sup_{s\leq\theta\leq 0}\bigl( \mathrm{E} \bigl\Vert \phi(\theta) \bigr\Vert ^{2} \bigr)^{1/2}\,ds $$
$$\begin{aligned} Lx(t) =&\phi(0)+\frac{1}{\Gamma(\alpha)} \int^{t}_{0}(t-s)^{\alpha -1}\bigl[Ax(s)+f(s,x_{s})+Bu(s) \bigr]\,ds \\ &{}+\frac{1}{\Gamma(\alpha)} \int^{t}_{0}(t-s)^{\alpha-1}\sigma(s) \,dB^{H}(s). \end{aligned}$$
(2.3)
We present the definition of mild solutions of (1.1).
Definition 2.2
An U-valued stochastic process \(\{x(t),t\in[0,T]\}\) is a mild solution of (1.1) if the next conditions hold:
- (i)
\(x(t)\) is measurable and \(\mathscr{F}_{t}\)-adapted, and \(x_{t}\) is \(\mathscr{B}_{h}\)-valued,
- (ii)for each \(t\in[0,T]\), \(x(t)\) satisfies the equation$$\begin{aligned} x(t) =&\mathscr{T}_{L}(t) \bigl(L\phi(0)\bigr)+ \int^{t}_{0}(t-s)^{\alpha-1} \mathscr {S}_{L}(t-s)\bigl[f(s,x_{s})+Bu(s)\bigr]\,ds \\ &{}+ \int^{t}_{0}(t-s)^{\alpha-1}\mathscr{S}_{L}(t-s) \sigma(s)\,dB^{H}(s), \end{aligned}$$(2.4)
- (iii)
\(x(t)=\phi(t)\) on \((-\infty,0]\) such that \(\|\phi\| ^{2}_{\mathscr{B}_{h}}<\infty\).
Finally, in order to prove our main statement, we need some conditions on the functions f and σ as follows.
- (\(\mathbb{B}1\))
- Let the function \(f:[0,T]\times\mathscr {B}_{h}\rightarrow U\) is continuous and there exist some constants \(N_{f}>0\), \(k_{f}>0\) such that, for \(t\in[0,T]\) and \(\xi,\eta\in\mathscr{B}_{h}\)for all \(t\in[0,T]\) and \(k_{f}=\sup_{t\in[0,T]}\|f(t,0)\|^{2}\).$$\mathrm{E} \bigl\Vert f(t,\xi)-f(t,\eta) \bigr\Vert ^{2}\leq N_{f} \Vert \xi-\eta \Vert _{\mathscr{B}_{h}} $$
- (\(\mathbb{B}2\))
- For the complete orthogonal basis \(\{e_{n}\}_{n\in \mathbb {N}}\) in W, the function \(\sigma:[0,T]\to L^{0}_{2}(W,U)\) satisfyand \(\sum_{n=1}^{\infty}\|\sigma(t) Q^{\frac{1}{2}}e_{n}\|\) is uniformly convergent in \(t\in[0,T]\). In addition, there exist some \(t_{0}\) and \(\delta>0\) such that$$\sum^{\infty}_{n=1} \bigl\Vert \sigma Q^{\frac{1}{2}}e_{n} \bigr\Vert _{L^{2}([0,T],U)}< \infty $$$$\int_{0}^{t_{0}} \int_{0}^{t_{0}}r^{-\delta}s^{-\delta} \bigl\Vert \sigma(r) \bigr\Vert _{L_{2}^{0}(W,U)} \bigl\Vert \sigma(s) \bigr\Vert _{L_{2}^{0}(W,U)}\,dr\,ds< \infty. $$
3 Main results
In this section, we will show the approximate controllability of the stochastic system (1.1). We need to establish the existence of the solution for the stochastic control system and to show that the corresponding linear part is approximate controllability.
Definition 3.1
The system (1.1) is called approximately controllable on \([0,T]\) if
with \(\mathfrak{R}(t)=\{x(t)=x(t,u):u\in L^{2}([0,T],V)\}\).
$$\overline{\mathfrak{R}(T)}=U $$
Consider the corresponding linear fractional deterministic control system to (1.1)
and define the relevant operators
and
where \(B^{*}\) and \(\mathscr{S}^{*}_{L}(T-s)\) denote the adjoint operators of B and \(\mathscr{S}_{L}(T-s)\), respectively. It is clear that the operator \(\Gamma_{0}^{T}\) is a linear bounded operator. The fact that the linear Sobolev-type fractional control system (3.1) is approximately controllable on \([0,T]\) is equivalent to the next hypothesis (see, for example, Mahmudov and Denker [22]):
$$ \textstyle\begin{cases} {}^{\mathrm{c}}D^{\alpha}_{t}[Lx(t)]=Ax(t)+ Bu(t),& t\in[0,T], \\ x(0)=\phi(0), \end{cases} $$
(3.1)
$$\Gamma_{0}^{T}= \int^{T}_{0}(T-s)^{\alpha-1}\mathscr{S}_{L}(T-s) BB^{*}\mathscr{S}^{*}_{L}(T-s)\,ds $$
$$R\bigl(\alpha,\Gamma_{0}^{T}\bigr)=\bigl(\alpha I+ \Gamma_{0}^{T}\bigr)^{-1}, $$
- (\(\mathbb{H}_{0}\))
-
\(\alpha R(\alpha,\Gamma_{0}^{T})\to0\) in the strong operator topology, as \(\alpha\to0^{+}\).
Lemma 3.1
(Guendouzi and Idrissi [23])
For any\(z_{T}\in L^{2}(\Omega;U)\), there exists\(\widehat{\varphi}(t)\in L^{2}(\Omega,L^{2}(0,T;L_{2}^{0}))\)such that
$$z_{T}=Ez_{T}+ \int_{0}^{T}\widehat{\varphi}(s)\,dB^{H}(s). $$
For any \(\lambda>0\) and \(z_{T}\in L^{2}(\Omega;U)\), we now define the control function \(u^{\lambda}\) as follows:
$$\begin{aligned} u^{\lambda}(t) =&B^{*}\mathscr{S}^{*}_{L}(T-t) \bigl(\lambda I+ \Gamma_{0}^{T}\bigr)^{-1} \biggl[- \mathscr{T}_{L}(T) \bigl(L\phi(0) \bigr)+\mathrm{E}z_{T}+ \int^{T}_{0}\widehat {\varphi}(s) \,dB^{H}(s) \biggr] \\ &{}-B^{*}\mathscr{S}^{*}_{L}(T-t) \int^{t}_{0}\bigl(\lambda I+\Gamma _{s}^{T}\bigr)^{-1}(T-s)^{\alpha-1} \mathscr{S}_{L}(T-s)f(s,x_{s})\,ds \\ &{}-B^{*}\mathscr{S}^{*}_{L}(T-t) \int^{t}_{0}\bigl(\lambda I+\Gamma _{s}^{T}\bigr)^{-1}(T-s)^{\alpha-1} \mathscr{S}_{L}(T-s)\sigma(s)\,dB^{H}(s). \end{aligned}$$
(3.2)
Theorem 3.1
(Pachpatte [24])
LetNbe a convex subset of a normed linear spaceXand let\(0\in N\). If\(\Phi:N\rightarrow N\)is a completely continuous operator andthen either Φ has a fixed point or\(\Pi(\Phi)\)is bounded.
$$\Pi(\Phi):=\bigl\{ x\in N:x=\gamma\Phi x\textit{ for some } \gamma\in(0,1)\bigr\} , $$
Define the space
and let \(\|\cdot\|_{a}\) be a seminorm defined by
where \(C((-\infty,T],U)\) denotes the space of all continuous U-valued stochastic process \(\{\xi(t),t\in(-\infty,T]\}\).
$$\mathscr{B}_{a}= \bigl\{ x:x\in C\bigl((-\infty,T],U\bigr) \text{ with } x_{0}=\phi\in \mathscr{B}_{h} \bigr\} $$
$$\Vert x \Vert _{a}= \Vert \phi \Vert _{\mathscr{B}_{h}}+\sup _{s\in[0,T]} \bigl(\mathrm{E} \bigl\Vert x(s) \bigr\Vert ^{2} \bigr)^{\frac{1}{2}},\quad x\in\mathscr{B}_{a}, $$
Lemma 3.2
(Li and Liu [25])
Assume that\(x\in\mathscr{B}_{a}\), then, for all\(t\in[0,T]\), \(x_{t}\in \mathscr{B}_{h}\). Moreover,
where\(l=\int^{0}_{-\infty} h(s)\,ds\)is given in Sect. 2.
$$l\bigl(\mathrm{E} \bigl\Vert x(t) \bigr\Vert ^{2} \bigr)^{1/2}\leq l\sup_{s\in[0,t]}\bigl(\mathrm{E} \bigl\Vert x(s) \bigr\Vert ^{2}\bigr)^{1/2}+ \Vert x_{0} \Vert _{\mathscr{B}_{h}}, $$
Theorem 3.2
Assume the conditions (\(\mathbb{B}_{1}\)), (\(\mathbb{B}_{2}\)) hold, then, for each\(\lambda>0\)there exists a mild solution of (1.1) on\((-\infty ,T]\), provided that there is a constant\(\mathscr{K}>0\)such that\(\frac {\mathcal{K}_{1}\mathscr{K}}{\mathcal{K}_{2}}>1\), whereand\(M_{B}=\|B\|\).
$$\begin{aligned}& \mathcal{K}_{1}=1-24l^{2} \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{\alpha\Gamma (\alpha)} \biggr)^{2}N_{f} \biggl(1+4 \biggl(\frac{M^{2}\widetilde{M}_{1}^{2}M_{B}T^{\alpha}}{ \alpha\lambda\Gamma(\alpha)^{2}} \biggr)^{2} \biggr)>0, \\& \mathcal{K}_{2}=4l^{2}(MM_{1} \widetilde{M}_{1})^{2}\mathrm{E}\|\phi_{0} \|^{2} \biggl[1+12 \biggl(\frac{M^{2}\widetilde{M}_{1}^{2}M_{B}^{2}T^{\alpha}}{\alpha\lambda\Gamma (\alpha)^{2}} \biggr)^{2} \biggr]+4\|\phi\|^{2}_{\mathscr{B}_{h}} \\& \hphantom{\mathcal{K}_{2}={}}{}+12l^{2} \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} \biggl[1+4 \biggl(\frac{M^{2}\widetilde{M}_{1}^{2}M_{B}^{2}T^{\alpha}}{\alpha\lambda\Gamma (\alpha)^{2}} \biggr)^{2} \biggr]\times \biggl(\frac{2k_{f}T^{2\alpha}}{\alpha ^{2}}+cH(2H-1)N_{\sigma}\biggr) \\& \hphantom{\mathcal{K}_{2}={}}{}+96l^{2} \biggl(\frac{M^{2}\widetilde{M}_{1}^{2}M_{B}^{2}T^{\alpha}}{\alpha\lambda \Gamma(\alpha)^{2}} \biggr)^{2} \biggl(\mathrm{E}\|z_{T}\|^{2}+cH(2H-1)T^{2H-1} \int ^{T}_{0}\bigl\| \widehat{\varphi}(s) \bigr\| ^{2}_{L_{2}^{0}(W,U)} \,ds \biggr) \end{aligned}$$
Proof
Define the operator \(\Phi:\mathscr{B}_{a}\to\mathscr{B}_{a}\) by
for \(x\in{\mathscr {B}}_{a}\).
$$ \Phi x(t)= \textstyle\begin{cases} \phi(t), &t\in(-\infty,0], \\ \mathscr{T}_{L}(t)(L\phi(0))+\int^{t}_{0}(t-s)^{\alpha-1}\mathscr{S}_{L}(t-s) [f(s,x_{s})+Bu^{\lambda}(s)]\,ds \\ \quad {}+\int^{t}_{0}(t-s)^{\alpha-1}\mathscr{S}_{L}(t-s)\sigma(s)\,dB^{H}(s), &t\in(0,T], \end{cases} $$
We will show that Φ has a fixed point which is a mild solution for system (1.1). For \(\phi\in\mathscr{B}_{h}\), define
Then \(\widetilde{\phi}(t)\in\mathscr{B}_{a}\). Let \(x(t)=\widetilde{\phi }(t)+y(t)\), \(t\in(-\infty,T]\). It is easy to check that \(x(t)\) satisfies (1.1) if and only if \(y_{0}=0\) and
Denote \(\mathscr{B}_{b}=\{y\in\mathscr{B}_{a},y_{0}=0\in\mathscr{B}_{h}\}\) and let \(\|\cdot\|_{b}\) be the seminorm in \(\mathscr{B}_{b}\), defined by
For \(r\geq0\), we set \(B_{r}=\{y\in\mathscr{B}_{b},\|y\|_{b}^{2}\leq r\}\). Then \(B_{r}\) is a bounded closed convex set in \(\mathscr{B}_{b}\) for each r. According to Lemma 3.2, we get
for \(y\in B_{r}\). Define the mapping \(\Psi:\mathscr{B}_{b}\to\mathscr {B}_{b}\) by
for \(t\in[0,T]\). It is evident that the operator Φ has a fixed point if and only if the operator Ψ has a fixed point. Now, we turn to prove Ψ has a fixed point.
$$ \widetilde{\phi}(t)= \textstyle\begin{cases} \phi(t), &t\in(-\infty,0], \\ \mathscr{T}_{L}(t)(L\phi(0)), &t\in(0,T]. \end{cases} $$
$$\begin{aligned} y(t) =& \int^{t}_{0}(t-s)^{\alpha-1}\mathscr{S}_{L}(t-s) \bigl[f(s,y_{s}+\widetilde{\phi }_{s}) +Bu^{\lambda}(s) \bigr]\,ds \\ &{} + \int^{t}_{0}(t-s)^{\alpha-1}\mathscr{S}_{L}(t-s) \sigma(s)\,dB^{H}(s). \end{aligned}$$
$$\Vert y \Vert _{b}= \Vert y_{0} \Vert _{\mathscr{B}_{h}}+\sup_{s\in[0,T]}\bigl(\mathrm{E} \bigl\Vert y(s) \bigr\Vert ^{2}\bigr)^{1/2}=\sup_{s\in[0,T]} \bigl(\mathrm{E} \bigl\Vert y(s) \bigr\Vert ^{2} \bigr)^{1/2}. $$
$$\begin{aligned} \Vert y_{t}+\widetilde{\phi}_{t} \Vert _{\mathscr{B}_{h}}^{2} \leq&2 \bigl(\| y_{t} \|_{\mathscr{B}_{h}}^{2}+\|\widetilde{\phi}_{t} \|_{\mathscr{B}_{h}}^{2} \bigr) \\ \leq&4 \Bigl(l^{2}\sup_{s\in[0,t]}\mathrm{E}\bigl\| y(s) \bigr\| ^{2}+\|y_{0}\|_{\mathscr {B}_{h}}^{2}+l^{2} \sup_{s\in[0,t]}\mathrm{E}\bigl\| \widetilde{\phi}(s)\bigr\| ^{2}+\| \widetilde{\phi}_{0}\|_{\mathscr{B}_{h}}^{2} \Bigr) \\ \leq&4 \Bigl(l^{2}\sup_{s\in[0,t]}\mathrm{E}\bigl\| y(s) \bigr\| ^{2}+l^{2}\sup_{s\in [0,t]}\mathrm{E}\bigl\| \mathscr{T}_{L}(t) \bigl(L\phi(0)\bigr)\bigr\| ^{2}+\|\phi \|_{\mathscr {B}_{h}}^{2} \Bigr) \\ \leq&4 \bigl(l^{2}r+l^{2}M^{2}M_{1}^{2} \widetilde{M}_{1}^{2}\mathrm{E}\bigl\| \phi(0)\bigr\| ^{2}+\| \phi\|^{2}_{\mathscr{B}_{h}} \bigr)\equiv r' \end{aligned}$$
(3.3)
$$\begin{aligned} \Psi y(t) =& \int^{t}_{0}(t-s)^{\alpha-1}\mathscr{S}_{L}(t-s) \bigl[f(s,\widetilde {\phi}_{s}+y_{s}) +Bu^{\lambda}(s) \bigr]\,ds \\ &{} + \int^{t}_{0}(t-s)^{\alpha-1}\mathscr{S}_{L}(t-s) \sigma(s)\,dB^{H}(s) \end{aligned}$$
Step I. We claim that Ψ maps bounded sets into bounded sets of \(\mathscr{B}_{b}\). This is equivalent to show that there exists a positive constant ρ such that, for each \(y\in B_{r}\), one has
By (3.2), we obtain
By Lemma 2.2, it is easy to show that
By using the Hölder inequality, the assumption (\(\mathbb{B}1\)), (3.3) and Lemma 2.3, we have
For the last part \(I_{4}\), when \(\alpha>1-H\), we have
and
where \(B(\cdot,\cdot)\) denotes the Beta function, which imply that
Consequently, we get
It follows that
This shows that the first statement holds, i.e., for each \(y\in B_{r}\), \(\|\Psi y(t)\|_{a}^{2}\leq\rho\).
$$\bigl\Vert \Psi y(t) \bigr\Vert _{a}^{2}\leq\rho. $$
$$\begin{aligned} \mathrm{E} \bigl\Vert u^{\lambda}(s) \bigr\Vert ^{2} \leq& \frac{4(M_{B}M\widetilde{M}_{1})^{2}}{\lambda^{2}\Gamma(\alpha)^{2}}\biggl(\mathrm{E} \bigl\Vert \mathscr {T}_{L}(T) \bigl(L\phi(0)\bigr) \bigr\Vert ^{2}+\mathrm{E} \biggl\Vert \mathrm{E}z_{T}+ \int^{T}_{0}\widehat {\varphi}(s) \,dB^{H}(s) \biggr\Vert ^{2} \\ &{} +\mathrm{E} \biggl\Vert \int^{T}_{0}(T-s)^{\alpha-1}\mathscr{S}_{L}(T-s)f (s,y_{s}+\widetilde{\phi}_{s} )\,ds \biggr\Vert ^{2} \\ &{} +\mathrm{E} \biggl\Vert \int^{T}_{0}(T-s)^{\alpha-1}\mathscr {S}_{L}(T-s)\sigma(s)\,dB^{H}(s) \biggr\Vert ^{2}\biggr) \\ \equiv&\frac{4(M_{B}M\widetilde{M}_{1})^{2}}{\lambda^{2}\Gamma(\alpha)^{2}}\sum_{i=1}^{4} I_{i}. \end{aligned}$$
$$\begin{aligned} I_{1}+I_{2} \leq&(M\widetilde{M}_{1}M_{1})^{2} \mathrm{E} \bigl\Vert \phi(0) \bigr\Vert ^{2}+2\mathrm{E} \Vert z_{T} \Vert ^{2} \\ &{} +2cH(2H-1)T^{2H-1} \int^{T}_{0} \bigl\Vert \widehat{\varphi}(s) \bigr\Vert ^{2}_{L_{2}^{0}(W,U)}\,ds. \end{aligned}$$
$$\begin{aligned} I_{3}&\leq \int^{T}_{0}(T-s)^{\alpha-1} \bigl\Vert \mathscr{S}_{L}(T-s) \bigr\Vert \,ds \int ^{T}_{0}(T-s)^{\alpha-1} \bigl\Vert \mathscr{S}_{L}(T-s) \bigr\Vert \mathrm{E} \bigl\Vert f(s,y_{s}+\widetilde{\phi}_{s}) \bigr\Vert ^{2} \,ds \\ &\leq2 \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} \frac {T^{\alpha}}{\alpha} \int^{T}_{0}(T-s)^{\alpha-1}\bigl(N_{f} \Vert y_{s} +\widetilde{\phi }_{s} \Vert _{\mathscr{B}_{h}}^{2}+k_{f}\bigr)\,ds \\ &\leq2 \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{\alpha\Gamma(\alpha)} \biggr)^{2}\bigl(N_{f}r'+k_{f} \bigr)\equiv\mu_{f}. \end{aligned}$$
$$\int^{T}_{0} \int^{T}_{0}(T-s)^{\alpha-1}(T-t)^{\alpha-1}|s-t|^{2H-2} \,ds\,dt =\frac {B(\alpha,2H-1)}{2(\alpha+H-1)}T^{2(\alpha+H-1)} $$
$$N_{\sigma}:=\sup_{t\in[0,T]} \int_{0}^{t} \int_{0}^{t}(t-s)^{\alpha-1} (t-r)^{\alpha -1}|s-r|^{2H-2} \bigl\Vert \sigma(s) \bigr\Vert _{L_{2}^{0}(W,U)} \bigl\Vert \sigma(r) \bigr\Vert _{L_{2}^{0}(W,U)}\,ds\,dr< \infty, $$
$$\begin{aligned} I_{4} =&\mathrm{E} \Biggl\Vert \sum^{\infty}_{n=1} \int^{T}_{0}(T-s)^{\alpha -1}\mathscr{S}_{L}(T-s) \sigma(s)Q^{\frac{1}{2}}e_{n}d\beta_{n}^{H}(s) \Biggr\Vert ^{2} \\ =&\sum^{\infty}_{n=1}\mathrm{E} \biggl\Vert \int^{T}_{0}(T-s)^{\alpha-1}\mathscr {S}_{L}(T-s)\sigma(s)Q^{\frac{1}{2}}e_{n}d \beta_{n}^{H}(s) \biggr\Vert ^{2} \\ =&\sum^{\infty}_{n=1}H(2H-1) \int^{T}_{0} \int^{T}_{0} \bigl\Vert (T-s)^{\alpha-1} \mathscr {S}_{L}(T-s)\sigma(s)Q^{\frac{1}{2}}e_{n} \bigr\Vert \\ &{} \times \bigl\Vert (T-t)^{\alpha-1}\mathscr{S}_{L}(T-t) \sigma(t)Q^{\frac{1}{2}}e_{n} \bigr\Vert |s-t|^{2H-2}\,ds\,dt \\ \leq&cH(2H-1) \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} \int ^{T}_{0} \int^{T}_{0}(T-s)^{\alpha-1}(T-t)^{\alpha-1}|s-t|^{2H-2} \\ &{} \times \bigl\Vert \sigma(s) \bigr\Vert _{L_{2}^{0}(W,U)} \bigl\Vert \sigma(t) \bigr\Vert _{L_{2}^{0}(W,U)}\,ds\,dt. \end{aligned}$$
$$\begin{aligned} \mathrm{E} \bigl\Vert u^{\lambda}(s) \bigr\Vert ^{2} \leq& \frac{4(M_{B}M\widetilde{M}_{1})^{2}}{ \lambda^{2}\Gamma(\alpha)^{2}}\biggl\{ (M\widetilde{M}_{1}M_{1})^{2} \mathrm{E} \bigl\Vert \phi (0) \bigr\Vert ^{2}+2\mathrm{E} \Vert z_{T} \Vert ^{2} \\ &{} +2cH(2H-1)T^{2H-1} \int^{T}_{0} \bigl\Vert \widehat{\varphi}(s) \bigr\Vert ^{2}_{L_{2}^{0}(W,U)}\,ds+\mu_{f} \\ &{} +cH(2H-1) \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} N_{\sigma}\biggr\} \\ \equiv&\frac{4(M_{B}M\widetilde{M}_{1})^{2}}{\lambda^{2}\Gamma(\alpha)^{2}}N_{u}. \end{aligned}$$
$$\begin{aligned} \mathrm{E} \bigl\Vert \Psi y(t) \bigr\Vert ^{2} \leq&3\mathrm{E} \biggl\Vert \int^{t}_{0}(t-s)^{\alpha -1}\mathscr{S}_{L}(t-s) f(s,y_{s}+\widetilde{\phi}_{s})\,ds \biggr\Vert ^{2} \\ &{} +3\mathrm{E} \biggl\Vert \int^{t}_{0}(t-s)^{\alpha-1}\mathscr {S}_{L}(t-s)Bu^{\lambda}(s)\,ds \biggr\Vert ^{2} \\ &{} +3\mathrm{E} \biggl\Vert \int^{t}_{0}(t-s)^{\alpha-1}\mathscr{S}_{L}(t-s) \sigma (s)\,dB^{H}(s) \biggr\Vert ^{2} \\ \leq&3 \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} \biggl[ \frac {2T^{2\alpha}}{\alpha^{2}}\bigl(N_{f}r'+k_{f}\bigr) \\ &{} + \biggl(\frac{M_{B}T^{\alpha}}{\alpha} \biggr)^{2} \frac {4(M_{B}M\widetilde{M}_{1})^{2}}{\lambda^{2}\Gamma(\alpha)^{2}}N_{u} +cH(2H-1)N_{\sigma}\biggr]\equiv\rho< \infty. \end{aligned}$$
Step II. We claim that Ψ maps bounded sets into equicontinuous sets of \(\mathscr{B}_{b}\). Similar to the calculation in Step I, for \(0\leq t_{1}< t_{2}\leq T\), we have
We estimate \(J_{1}\), \(J_{2}\), \(J_{3}\). Let \(0<\varepsilon<t<T\) and \(\delta>0\) such that
for every \(s_{1},s_{2}\in[0,T]\) with \(|s_{1}-s_{2}|<\delta\). Then, for \(J_{1}\) we have
Analogously, for \(J_{2}\) we also have
As \(\alpha>1-H\), for \(J_{3}\) Thus, we see that \(J_{j}\) (\(j=1,2,3\)) tends to zero, as \(t_{1}\to t_{2}\). It follows from the Arzela–Ascoli theorem that Ψ is completely continuous.
$$\begin{aligned}& \mathrm{E} \bigl\Vert \Psi y(t_{2})-\Psi y(t_{1}) \bigr\Vert ^{2} \\& \quad \leq3\mathrm{E} \biggl\Vert \int^{t_{2}}_{0}(t_{2}-s)^{\alpha-1} \mathscr {S}_{L}(t_{2}-s) f (s,y_{s}+\widetilde{ \phi}_{s} )\,ds \\& \qquad {} - \int^{t_{1}}_{0}(t_{1}-s)^{\alpha-1} \mathscr{S}_{L}(t_{1}-s)f (s,y_{s}+\widetilde{ \phi}_{s} )\,ds \biggr\Vert ^{2} \\& \qquad {} +3\mathrm{E} \biggl\Vert \int^{t_{2}}_{0}(t_{2}-s)^{\alpha-1} \mathscr{S}_{L}(t_{2}-s) Bu^{\lambda}(s)\,ds- \int^{t_{1}}_{0}(t_{1}-s)^{\alpha-1} \mathscr{S}_{L}(t_{1}-s) Bu^{\lambda}(s)\,ds \biggr\Vert ^{2} \\& \qquad {} +3\mathrm{E} \biggl\Vert \int^{t_{2}}_{0}(t_{2}-s)^{\alpha-1} \mathscr{S}_{L}(t_{2}-s) \sigma(s)\,dB^{H}(s) \\& \qquad {} - \int^{t_{1}}_{0}(t_{1}-s)^{\alpha-1} \mathscr{S}_{L}(t_{1}-s)\sigma (s)\,dB^{H}(s) \biggr\Vert ^{2} \\& \quad \equiv\sum_{j=1}^{3} J_{j}. \end{aligned}$$
$$\bigl\Vert \mathscr{S}_{L}(s_{2})-\mathscr{S}_{L}(s_{1}) \bigr\Vert < \varepsilon $$
$$\begin{aligned} J_{1} \leq&3\mathrm{E} \biggl\Vert \int^{t_{1}}_{0}\bigl[(t_{2}-s)^{\alpha -1}-(t_{1}-s)^{\alpha-1} \bigr]\mathscr{S}_{L}(t_{2}-s)f (s,y_{s}+ \widetilde{\phi }_{s} )\,ds \biggr\Vert ^{2} \\ &{} +3\mathrm{E} \biggl\Vert \int^{t_{1}}_{0}(t_{1}-s)^{\alpha-1} \bigl[\mathscr {S}_{L}(t_{2}-s)-\mathscr{S}_{L}(t_{1}-s) \bigr]f (s,y_{s}+\widetilde{\phi }_{s} )\,ds \biggr\Vert ^{2} \\ &{} +3\mathrm{E} \biggl\Vert \int^{t_{2}}_{t_{1}}(t_{2}-s)^{\alpha-1} \mathscr {S}_{L}(t_{2}-s)f (s,y_{s}+\widetilde{ \phi}_{s} )\,ds \biggr\Vert ^{2} \\ \leq&3 \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} \int ^{t_{1}}_{0}\bigl[(t_{1}-s)^{\alpha-1}-(t_{2}-s)^{\alpha-1} \bigr]\,ds \\ &{} \times \int^{t_{1}}_{0}\bigl[(t_{1}-s)^{\alpha -1}-(t_{2}-s)^{\alpha-1} \bigr]\mathrm{E} \bigl\Vert f (s,y_{s}+\widetilde{\phi }_{s} ) \bigr\Vert ^{2}\,ds \\ &{} +\frac{3 t_{1}^{\alpha}\varepsilon^{2}}{\alpha} \int^{t_{1}}_{0}(t_{1}-s)^{\alpha -1} \mathrm{E} \bigl\Vert f (s,y_{s}+\widetilde{\phi}_{s} ) \bigr\Vert ^{2}\,ds \\ &{} +\frac{3(t_{2}-t_{1})^{\alpha}}{\alpha} \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} \int^{t_{2}}_{t_{1}}(t_{2}-s)^{\alpha-1} \mathrm {E} \bigl\Vert f (s,y_{s}+\widetilde{\phi}_{s} ) \bigr\Vert ^{2}\,ds \\ \leq&3 \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2}\bigl(N_{f}r'+k_{f} \bigr) \biggl( \int^{t_{1}}_{0}\bigl[(t_{1}-s)^{\alpha-1}-(t_{2}-s)^{\alpha -1} \bigr]\,ds \biggr)^{2} \\ &{}+\frac{3 t_{1}^{2\alpha}\varepsilon^{2}}{\alpha^{2}}\bigl(N_{f}r'+k_{f} \bigr)+\frac {3(t_{2}-t_{1})^{2\alpha}}{\alpha^{2}} \biggl(\frac{M\widetilde{M}_{1}}{\Gamma (\alpha)} \biggr)^{2} \bigl(N_{f}r'+k_{f}\bigr). \end{aligned}$$
$$\begin{aligned} J_{2} \leq&3\mathrm{E} \biggl\Vert \int^{t_{1}}_{0}\bigl[(t_{2}-s)^{\alpha -1}-(t_{1}-s)^{\alpha-1} \bigr]\mathscr{S}_{L}(t_{2}-s)Bu^{\lambda}(s)\,ds \biggr\Vert ^{2} \\ &{} +3\mathrm{E} \biggl\Vert \int^{t_{1}}_{0}(t_{1}-s)^{\alpha-1} \bigl[\mathscr {S}_{L}(t_{2}-s)-\mathscr{S}_{L}(t_{1}-s) \bigr]Bu^{\lambda}(s)\,ds \biggr\Vert ^{2} \\ &{} +3\mathrm{E} \biggl\Vert \int^{t_{2}}_{t_{1}}(t_{2}-s)^{\alpha-1} \mathscr {S}_{L}(t_{2}-s)Bu^{\lambda}(s)\,ds \biggr\Vert ^{2} \\ \leq&3 \biggl(\frac{M\widetilde{M}_{1}M_{B}}{\Gamma(\alpha)} \biggr)^{2} \int ^{t_{1}}_{0}\bigl[(t_{1}-s)^{\alpha-1}-(t_{2}-s)^{\alpha-1} \bigr]\,ds \\ &{} \times \int^{t_{1}}_{0}\bigl[(t_{1}-s)^{\alpha-1}-(t_{2}-s)^{\alpha-1} \bigr]\mathrm{E}\bigl\| u^{\lambda}(s)\bigr\| ^{2}\,ds \\ &{} +\frac{3 M_{B}^{2}t_{1}^{\alpha}\varepsilon^{2}}{\alpha} \int ^{t_{1}}_{0}(t_{1}-s)^{\alpha-1} \mathrm{E} \bigl\Vert u^{\lambda}(s) \bigr\Vert ^{2}\,ds \\ &{} +\frac{3(t_{2}-t_{1})^{\alpha}}{\alpha} \biggl(\frac{M\widetilde{M}_{1}M_{B}}{\Gamma(\alpha)} \biggr)^{2} \int^{t_{2}}_{t_{1}}(t_{2}-s)^{\alpha -1} \mathrm{E} \bigl\Vert u^{\lambda}(s) \bigr\Vert ^{2}\,ds \\ \leq&3 \biggl(\frac{M\widetilde{M}_{1}M_{B}}{\Gamma(\alpha)} \biggr)^{2}\frac {4(M_{B}M\widetilde{M}_{1})^{2}}{\lambda^{2}\Gamma(\alpha)^{2}}N_{u} \biggl( \int ^{t_{1}}_{0}\bigl[(t_{1}-s)^{\alpha-1}-(t_{2}-s)^{\alpha-1} \bigr]\,ds \biggr)^{2} \\ &{} +\frac{3 t_{1}^{2\alpha}M_{B}^{2}\varepsilon^{2}}{\alpha^{2}}\frac {4(M_{B}M\widetilde{M}_{1})^{2}}{\lambda^{2}\Gamma(\alpha)^{2}}N_{u} +\frac{3(t_{2}-t_{1})^{2\alpha}}{\alpha^{2}} \biggl(\frac{M\widetilde{M}_{1}M_{B}}{\Gamma(\alpha)} \biggr)^{2}\frac{4(M_{B}M\widetilde{M}_{1})^{2}}{\lambda^{2}\Gamma(\alpha)^{2}}N_{u}. \end{aligned}$$
$$\begin{aligned} J_{3} \leq&3\mathrm{E} \biggl\Vert \int^{t_{1}}_{0}\bigl[(t_{2}-s)^{\alpha -1}-(t_{1}-s)^{\alpha-1} \bigr]\mathscr{S}_{L}(t_{2}-s)\sigma(s)\,dB^{H}(s) \biggr\Vert ^{2} \\ &{} +3\mathrm{E} \biggl\Vert \int^{t_{1}}_{0}(t_{1}-s)^{\alpha-1} \bigl[\mathscr {S}_{L}(t_{2}-s)-\mathscr{S}_{L}(t_{1}-s) \bigr]\sigma(s)\,dB^{H}(s) \biggr\Vert ^{2} \\ &{} +3\mathrm{E} \biggl\Vert \int^{t_{2}}_{t_{1}}(t_{2}-s)^{\alpha-1} \mathscr {S}_{L}(t_{2}-s)\sigma(s)\,dB^{H}(s) \biggr\Vert ^{2} \\ \leq&3cH(2H-1) \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} \int^{t_{1}}_{0} \int^{t_{1}}_{0}\bigl[(t_{1}-s)^{\alpha-1}-(t_{2}-s)^{\alpha -1} \bigr] \\ &{} \times\bigl[(t_{1}-t)^{\alpha-1}-(t_{2}-t)^{\alpha-1} \bigr] \\ &{} \times \bigl\Vert \sigma(s) \bigr\Vert _{L_{2}^{0}(W,U)} \bigl\Vert \sigma(t) \bigr\Vert _{L_{2}^{0}(W,U)}|s-t|^{2H-2}\,ds\,dt \\ &{} +3cH(2H-1)\varepsilon^{2}N_{\sigma}\\ &{} +3cH(2H-1) \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} \int ^{t_{2}}_{t_{1}} \int^{t_{2}}_{t_{1}}(t_{2}-s)^{\alpha-1}(t_{2}-t)^{\alpha -1}|s-t|^{2H-2} \\ &{} \times \bigl\Vert \sigma(s) \bigr\Vert _{L_{2}^{0}(W,U)} \bigl\Vert \sigma(t) \bigr\Vert _{L_{2}^{0}(W,U)}\,ds\,dt. \end{aligned}$$
Step III. We show that there exists an open set \(\Pi\subseteq \mathscr{B}_{b}\) with \(y\neq\gamma\Psi y\) for \(\gamma\in(0,1)\) and \(y\in \partial\Pi\).
If \(y=\gamma\Psi y\in\mathscr{B}_{b}\) for some \(\gamma\in(0,1)\). We then have
for every \(t\in[0,T]\). It follows that
which implies that
Denote the right side of the above inequality by \(\psi(t)\). There exists \(t^{*}\in[0,t]\) such that
Then, for \(t\in[0,T]\), we have
Taking the norm on both sides above, it follows that
which implies that \(\frac{\mathcal{K}_{1}\|\psi(t)\|^{2}_{a}}{\mathcal {K}_{2}}\leq1\), where
and
$$\begin{aligned} y(t) =&\gamma\biggl\{ \int^{t}_{0}(t-s)^{\alpha-1}\mathscr{S}_{L}(t-s) \bigl[f(s,y_{s}+\widetilde{\phi}_{s})+Bu^{\lambda}(s) \bigr]\,ds \\ &{} + \int^{t}_{0}(t-s)^{\alpha-1}\mathscr{S}_{L}(t-s) \sigma (s)\,dB^{H}(s)\biggr\} \end{aligned}$$
$$\begin{aligned} \mathrm{E} \bigl\Vert y(t) \bigr\Vert ^{2} \leq&3\mathrm{E} \biggl\Vert \int^{t}_{0}(t-s)^{\alpha -1}\mathscr{S}_{L}(t-s)f(s,y_{s}+ \widetilde{\phi}_{s})\,ds \biggr\Vert ^{2} \\ &{} +3\mathrm{E} \biggl\Vert \int^{t}_{0}(t-s)^{\alpha-1}\mathscr {S}_{L}(t-s)Bu^{\lambda}(s)\,ds \biggr\Vert ^{2} \\ &{} +3\mathrm{E} \biggl\Vert \int^{t}_{0}(t-s)^{\alpha-1}\mathscr{S}_{L}(t-s) \sigma (s)\,dB^{H}(s) \biggr\Vert ^{2} \\ \leq&6 \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{\alpha\Gamma(\alpha)} \biggr)^{2} N_{f} \Vert y_{t}+\widetilde{\phi}_{t} \Vert _{\mathscr{B}_{h}}^{2}+6 \biggl(\frac {M\widetilde{M}_{1}T^{\alpha}}{\alpha\Gamma(\alpha)} \biggr)^{2}k_{f} \\ &{} +3 \biggl(\frac{M\widetilde{M}_{1}M_{B}T^{\alpha}}{\alpha\Gamma(\alpha)} \biggr)^{2} \mathrm{E} \bigl\Vert u^{\lambda}(t) \bigr\Vert ^{2} +3 \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2}cH(2H-1)N_{\sigma}, \end{aligned}$$
$$\begin{aligned} \Vert y_{t}+\widetilde{\phi}_{t} \Vert _{\mathscr{B}_{h}}^{2} \leq&2 \bigl( \Vert y_{t} \Vert _{\mathscr{B}_{h}}^{2}+ \Vert \widetilde{\phi}_{t} \Vert _{\mathscr{B}_{h}}^{2} \bigr) \\ \leq&4 \Bigl(l^{2}\sup_{s\in[0,t]}\mathrm{E} \bigl\Vert y(s) \bigr\Vert ^{2}+l^{2}M^{2}M_{1}^{2} \widetilde{M}_{1}^{2}\mathrm{E} \bigl\Vert \phi(0) \bigr\Vert ^{2}+ \Vert \phi \Vert ^{2}_{\mathscr{B}_{h}} \Bigr). \end{aligned}$$
$$\psi(t)=4 \bigl(l^{2}\mathrm{E} \bigl\Vert y\bigl(t^{*}\bigr) \bigr\Vert ^{2}+l^{2}M^{2}M_{1}^{2} \widetilde{M}_{1}^{2}\mathrm{E} \bigl\Vert \phi(0) \bigr\Vert ^{2}+ \Vert \phi \Vert ^{2}_{\mathscr{B}_{h}} \bigr). $$
$$\begin{aligned} \psi(t) \leq&4 \biggl\{ l^{2} \biggl[6 \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{ \alpha\Gamma(\alpha)} \biggr)^{2} N_{f}\psi(t)+6 \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{\alpha\Gamma(\alpha)} \biggr)^{2} k_{f} \\ &{}+12 \biggl(\frac{M^{2}\widetilde{M}_{1}^{2}M_{B}^{2}T^{\alpha}}{\lambda\alpha\Gamma (\alpha)^{2}} \biggr)^{2}\times \biggl((M \widetilde{M}_{1}M_{1})^{2}\mathrm{E} \bigl\Vert \phi (0) \bigr\Vert ^{2}+2\mathrm{E} \Vert z_{T} \Vert ^{2} \\ &{}+2cH(2H-1)T^{2H-1} \int^{T}_{0} \bigl\Vert \widehat{\varphi}(s) \bigr\Vert ^{2}_{L_{2}^{0}(W,U)}\,ds+2 \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{\alpha\Gamma (\alpha)} \biggr)^{2}N_{f}\psi(t) \\ &{}+2 \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{\alpha\Gamma(\alpha)} \biggr)^{2}k_{f}+cH(2H-1) \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} N_{\sigma}\biggr) \\ &{}+3 \biggl(\frac{M\widetilde{M}_{1}}{ \Gamma(\alpha)} \biggr)^{2}c H(2H-1)N_{\sigma}\biggr]+l^{2}M^{2}M_{1}^{2} \widetilde{M}_{1}^{2}\mathrm{E} \bigl\Vert \phi(0) \bigr\Vert ^{2}+ \Vert \phi \Vert ^{2}_{\mathscr{B}_{h}} \biggr\} . \end{aligned}$$
$$\begin{aligned} \begin{aligned} \bigl\Vert \psi(t) \bigr\Vert ^{2}_{a}\leq{}&4 \biggl\{ l^{2} \biggl[6 \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{ \alpha\Gamma(\alpha)} \biggr)^{2} N_{f} \bigl\Vert \psi(t) \bigr\Vert ^{2}_{a}+6 \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{\alpha\Gamma(\alpha)} \biggr)^{2} k_{f} \\ &{}+12 \biggl(\frac{M^{2}\widetilde{M}_{1}^{2}M_{B}^{2}T^{\alpha}}{\lambda\alpha\Gamma (\alpha)^{2}} \biggr)^{2}\times \biggl((M \widetilde{M}_{1}M_{1})^{2}\mathrm{E} \bigl\Vert \phi (0) \bigr\Vert ^{2}+2\mathrm{E} \Vert z_{T} \Vert ^{2} \\ &{}+2cH(2H-1)T^{2H-1} \int^{T}_{0} \bigl\Vert \widehat{\varphi}(s) \bigr\Vert ^{2}_{L_{2}^{0}(W,U)}\,ds+2 \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{\alpha\Gamma (\alpha)} \biggr)^{2}N_{f} \bigl\Vert \psi(t) \bigr\Vert ^{2}_{a} \\ &{}+2 \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{\alpha\Gamma(\alpha)} \biggr)^{2}k_{f}+cH(2H-1) \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} N_{\sigma}\biggr) \\ &{}+3 \biggl(\frac{M\widetilde{M}_{1}}{ \Gamma(\alpha)} \biggr)^{2}c H(2H-1)N_{\sigma}\biggr]+l^{2}M^{2}M_{1}^{2} \widetilde{M}_{1}^{2}\mathrm{E} \bigl\Vert \phi(0) \bigr\Vert ^{2}+ \Vert \phi \Vert ^{2}_{\mathscr{B}_{h}} \biggr\} , \end{aligned} \end{aligned}$$
$$\mathcal{K}_{1}=1-24l^{2} \biggl(\frac{M\widetilde{M}_{1}T^{\alpha}}{ \alpha\Gamma (\alpha)} \biggr)^{2}N_{f} \biggl(1+4 \biggl(\frac{M^{2}\widetilde{M}_{1}^{2} M_{B}T^{\alpha}}{\alpha\lambda\Gamma(\alpha)^{2}} \biggr)^{2} \biggr) $$
$$\begin{aligned} \mathcal{K}_{2} =&4l^{2}(MM_{1} \widetilde{M}_{1})^{2}\mathrm{E} \Vert \phi_{0} \Vert ^{2} \biggl[1+12 \biggl(\frac{M^{2}\widetilde{M}_{1}^{2}M_{B}^{2}T^{\alpha}}{\alpha\lambda\Gamma (\alpha)^{2}} \biggr)^{2} \biggr]+4 \Vert \phi \Vert ^{2}_{\mathscr{B}_{h}} \\ &{}+12l^{2} \biggl(\frac{M\widetilde{M}_{1}}{\Gamma(\alpha)} \biggr)^{2} \biggl[1+4 \biggl(\frac{M^{2}\widetilde{M}_{1}^{2}M_{B}^{2}T^{\alpha}}{\alpha\lambda\Gamma (\alpha)^{2}} \biggr)^{2} \biggr]\times \biggl( \frac{2k_{f}T^{2\alpha}}{\alpha ^{2}}+cH(2H-1)N_{\sigma}\biggr) \\ &{}+96l^{2} \biggl(\frac{M^{2}\widetilde{M}_{1}^{2}M_{B}^{2}T^{\alpha}}{\alpha\lambda \Gamma(\alpha)^{2}} \biggr)^{2} \biggl( \mathrm{E} \Vert z_{T} \Vert ^{2}+cH(2H-1)T^{2H-1} \int ^{T}_{0} \bigl\Vert \widehat{\varphi}(s) \bigr\Vert ^{2}_{L_{2}^{0}(W,U)} \,ds \biggr). \end{aligned}$$
From the assumption, \(\|\psi\|_{a}^{2}\neq\mathscr{K}\). Set \(\Pi=\{y\in \mathscr{B}_{b},\|y\|^{2}_{a}<\mathscr{K}+1\}\). Then there is no \(y\in\partial \Pi\) such that \(y=\gamma\Psi y\) for some \(\gamma\in(0,1)\). By Theorem 3.1, we find that Ψ has a fixed point. Hence Φ has a fixed point which is a solution to the system (1.1). □
Theorem 3.3
Assume that the conditions of Theorem 3.2and (\(\mathbb{H}_{0}\)) hold. In addition, the functionsfis uniformly bounded on its domain. Then the fractional control system (1.1) is approximately controllable on\([0,T]\).
Proof
Let \(x^{\lambda}\) be a fixed point of the operator \(\Phi^{\lambda}\). Using the stochastic Fubini theorem, we can get
It follows from the property of f that there exists \(C>0\) such that \(\|f(s,x_{s}^{\lambda})\|^{2}\leq C\). Then there is a subsequence denoted by \(\{f(s,x_{s}^{\lambda})\}\) weakly converging to \(f(s)\). Thus, from the above equation, we obtain
On the other hand, by assumption (\(\mathbb{H}_{0}\)) for all \(0\leq s\leq T\), the operator \(\lambda(\lambda I+\Gamma_{s}^{T})^{-1}\to0\) strongly as \(\lambda\to0+\), and, moreover, \(\|\lambda(\lambda I+\Gamma _{s}^{T})^{-1}\|<1\). Thus, by the Lebesgue dominated convergence theorem and the compactness of \(\mathscr{S}_{L}(t)\), we can get \(\mathrm{E}\| x^{\lambda}(T)-z_{T}\|^{2}\to0\) as \(\lambda\to0+\). This gives the approximate controllability of (1.1). □
$$\begin{aligned} x^{\lambda}(T) =&z_{T}-\lambda\bigl(\lambda I+ \Gamma^{T}_{0}\bigr)^{-1} \biggl[-\mathscr {T}_{L}(T) \bigl(L\phi(0) \bigr)+\mathrm{E}z_{T}+ \int^{T}_{0}\widehat{\varphi }(s) \,dB^{H}(s) \biggr] \\ &{}+\lambda \int^{T}_{0}\bigl(\lambda I+\Gamma_{s}^{T} \bigr)^{-1}(T-s)^{\alpha-1}\mathscr {S}_{L}(T-s)f \bigl(s,x_{s}^{\lambda}\bigr)\,ds \\ &{}+\lambda \int^{T}_{0}\bigl(\lambda I+\Gamma_{s}^{T} \bigr)^{-1}(T-s)^{\alpha-1}\mathscr {S}_{L}(T-s)\sigma(s) \,dB^{H}(s). \end{aligned}$$
$$\begin{aligned}& \mathrm{E} \bigl\Vert x^{\lambda}(T)-z_{T} \bigr\Vert ^{2} \\& \quad \leq 5 \bigl\Vert \lambda\bigl(\lambda I+\Gamma^{T}_{0} \bigr)^{-1} \bigl[\mathrm{E}z_{T}-\mathscr {T}_{L}(T) \bigl(L\phi(0) \bigr) \bigr] \bigr\Vert ^{2} \\& \qquad {} +5cH(2H-1)T^{2H-1}\mathrm{E} \int^{T}_{0} \bigl\Vert \lambda\bigl(\lambda I+ \Gamma ^{T}_{0}\bigr)^{-1}\widehat{\varphi}(s) \bigr\Vert ^{2}_{L^{0}_{2}(W,U)}\,ds \\& \qquad {} +5\mathrm{E} \biggl( \int^{T}_{0}(T-s)^{\alpha-1} \bigl\Vert \lambda \bigl(\lambda I+\Gamma _{s}^{T}\bigr)^{-1} \mathscr{S}_{L}(T-s)\bigl[f\bigl(s,x_{s}^{\lambda}\bigr)-f(s)\bigr] \bigr\Vert \,ds \biggr)^{2} \\& \qquad {} +5\mathrm{E} \biggl( \int^{T}_{0}(T-s)^{\alpha-1} \bigl\Vert \lambda \bigl(\lambda I+\Gamma _{s}^{T}\bigr)^{-1} \mathscr{S}_{L}(T-s)f(s) \bigr\Vert \,ds \biggr)^{2} \\& \qquad {} +5cH(2H-1) \int^{T}_{0} \int^{T}_{0}(T-s)^{\alpha-1}(T-t)^{\alpha -1}|s-t|^{2H-2} \\& \qquad {} \times \bigl\Vert \lambda\bigl(\lambda I+\Gamma_{s}^{T} \bigr)^{-1}\mathscr{S}_{L}(T-s)\sigma (s) \bigr\Vert ^{2}_{L^{0}_{2}(W,U)} \\& \qquad {} \times\bigl\Vert \lambda\bigl(\lambda I+ \Gamma_{t}^{T}\bigr)^{-1}\mathscr {S}_{L}(T-t)\sigma(t) \bigr\Vert ^{2}_{L^{0}_{2}(W,U)}\,ds \,dt. \end{aligned}$$
4 Example
In this section, we will show an example to apply our results above. Consider the Sobolev-type fractional stochastic functional equation with infinite delay of the following form:
\(B^{H}(t)\) is a cylindrical fractional Brownian motion with Hurst index \(H\in(\frac{1}{2},1)\).
$$ \textstyle\begin{cases} {}^{\mathrm{c}}D^{\frac{3}{4}}_{t}[x(t,y)-x_{yy}(t,y)]=\frac{\partial^{2}}{\partial y^{2}}x(t,y)+f(t,x(t-h,y)) \\ \hphantom{{}^{\mathrm{c}}D^{\frac{3}{4}}_{t}[x(t,y)-x_{yy}(t,y)]={}}{}+ \nu(t,y)+\sigma(t)\frac{dB^{H}(t)}{dt}, &0\leq t\leq1,h>0,0\leq y\leq 1, \\ x(t,0)=x(t,1), &0\leq t\leq1, \\ x(t,y)=\phi(t,y), &0\leq y\leq1,-\infty< t\leq0, \end{cases} $$
(4.1)
Take \(U=L^{2}([0,1])\). Define the operators \(A:D(A)\subset U\rightarrow U\) and \(L:D(L)\subset U\rightarrow U\) by \(Ax=x''\) and \(Lx=x-x''\), where \(D(A)\) and \(D(L)\) are given by
Furthermore, A and L can be written as
where \(x_{n}(y)=\sqrt{\frac{2}{\pi}}\sin(ny)\), \(n=1,2,\ldots\) , is the orthogonal set of eigenvectors of A. Also, for \(x\in U\),
Clearly, \(L^{-1}\) is compact, bounded with \(\|L^{-1}\|\leq1\) and \(AL^{-1}\) generates a strongly continuous semigroup \(S(t)\) on U with \(\|S(t)\|\leq1\).
$$\bigl\{ x\in U, x \text{ and } x' \text{ are absolutely continuous}, x''\in U,x(0)=x(1)=0\bigr\} . $$
$$\begin{aligned}& Ax=\sum^{\infty}_{n=1}n^{2}\langle x, x_{n}\rangle x_{n},\quad x\in D(A), \\& Lx=\sum^{\infty}_{n=1}\bigl(1+n^{2} \bigr)\langle x, x_{n}\rangle x_{n},\quad x\in D(L), \end{aligned}$$
$$\begin{aligned}& L^{-1}x=\sum^{\infty}_{n=1} \frac{1}{1+n^{2}}\langle x,x_{n}\rangle x_{n},\qquad AL^{-1}x=\sum^{\infty}_{n=1} \frac{n^{2}}{1+n^{2}}\langle x,x_{n}\rangle x_{n}, \\& S(t)x=\sum^{\infty}_{n=1}e^{\frac{-n^{2}}{1+n^{2}}t} \langle x,x_{n}\rangle x_{n}, \\& \mathscr{S}_{L}(t) (x)=\frac{3}{4}\sum ^{\infty}_{n=1}\frac{1}{1+n^{2}} \int^{\infty }_{0} \theta\xi_{\frac{3}{4}}( \theta)e^{\frac{-n^{2}}{1+n^{2}}t^{\frac{3}{4}}\theta} \, d\theta\langle x,x_{n}\rangle x_{n}. \end{aligned}$$
Let \(h(s)=e^{2s}\), \(s<0\), then \(l=\int^{0}_{-\infty}h(s)\,ds=\frac{1}{2}\). Let \(\mathscr{B}_{h}\) be a phase space endowed with the norm
Then \((\mathscr{B}_{h},\|\cdot\|_{\mathscr{B}_{h}})\) is a Banach space. Define an infinite-dimensional space V by \(V=\{u|u=\sum^{\infty }_{n=2}u_{n}x_{n}, \text{with } \sum^{\infty}_{n=2}u_{n}^{2}<\infty\}\). The norm in V is defined by \(\|u\|_{V}= (\sum^{\infty}_{n=2}u_{n}^{2} )^{\frac{1}{2}}\). Then define a continuous linear mapping B from V into U as \(Bu=2u_{2}x_{1}+\sum^{\infty}_{n=2}u_{n}x_{n}\) for \(u=\sum^{\infty }_{n=2}u_{n}x_{n}\in V\). We assume the operator \(B:V\rightarrow U\) is a bounded linear operator by \(Bu(t)(y)=\nu(t,y)\). In addition, the linear part corresponding to (4.1) is approximately controllable. Then the system (4.1) can be written in the abstract form of (1.1). Imposing all the conditions of corresponding coefficients of Theorem 3.3, we can conclude that the fractional control system (4.1) is approximately controllable on \([0,1]\).
$$\Vert \phi \Vert _{\mathscr{B}_{h}}= \int^{0}_{-\infty}h(s)\sup_{s\leq\theta\leq 0}\bigl( \mathrm{E} \bigl\Vert \phi(\theta) \bigr\Vert ^{2} \bigr)^{\frac{1}{2}}\,ds. $$
Notes
Acknowledgements
The Project-sponsored by National Natural Science Foundation of China (11571071).
Authors’ contributions
JH and LY carried out the mathematical studies, participated in the sequence alignment, drafted the manuscript and participated in the design of the study and performed proof of results. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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