Boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fractional Brownian motion and Poisson jumps

Abstract

By using stochastic analysis, fractional analysis, compact semigroups and the Schauder fixed-point theorem, we discuss the approximate boundary controllability of a nonlocal Hilfer fractional stochastic differential system with fractional Brownian motion and a Poisson jump. In addition, we establish the sufficient conditions for exact null controllability for the same problem. Finally, an example is given to illustrate the results obtained.

1 Introduction

Fractional calculus has been applied to the description of problems that arise in a variety of fields, including finance, physics, geomagnetics, thermodynamics, and optimal control. Fractional Brownian motion (fBm) is for a family of Gaussian processes that is indexed by the Hurst parameter $H \in(0,1)$ (see [1]). When $H = 1/2$, the fBm is a standard Brownian motion. When $H \neq1/2$, it behaves in a way completely different from the standard Brownian motion, in particular it is neither a semi-martingale nor a Markov process. Especially, when $H >1/2$, fBm has a long range dependence. This property makes this process useful as driving noise in models appearing in finance markets, physics, telecommunication networks, hydrology and medicine etc. (see [2,3,4]). Stochastic differential equations driven by fractional Brownian motion have been considered extensively by research community in various aspects due to the salient features for real world problems (see [5,6,7,8,9,10,11,12]). In addition, controllability problems for different kinds of dynamical systems have been studied by several authors (see [13,14,15,16,17,18,19,20,21]), and the references therein. Few authors studied the controllability for linear and nonlinear systems when the control is on the boundary (see [22,23,24,25,26,27,28,29,30]). Also, few authors studied the stochastic fractional differential equations with Poisson jumps. Muthukumar and Thiagu [31] studied the existence of solutions and the approximate controllability of fractional nonlocal neutral impulsive stochastic differential equations of order $1 < q < 2$ with infinite delay and Poisson jumps. Rihan et al. [32] studied the fractional stochastic differential equations with Hilfer fractional derivative, with Poisson jumps and optimal control. Chadha and Bora [33] obtained the sufficient conditions for the approximate controllability of impulsive neutral stochastic differential equations driven by Poisson jumps. Ahmed and Wang [34] established the sufficient conditions for exact null controllability of Sobolev type Hilfer fractional stochastic differential equations with fractional Brownian motion and Poisson jumps. However, the approximate boundary controllability and the null boundary controllability results for Hilfer fractional stochastic differential system with fractional Browonian motion and Poisson jumps have not yet been considered in the literature. Motivated by these facts, we in this paper investigate the sufficient conditions for approximate boundary controllability and null boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fractional Brownian motion and Poisson jump in the following form:

$$ \textstyle\begin{cases} D^{\nu,\mu}_{0+} x(t) = \sigma x(t) + f_{1}(t,x(t))+ f_{2}(t,x(t))\frac {d\omega(t)}{dt}+G(t,x(t))\frac{dB^{H}(t)}{dt} \\ \hphantom{D^{\nu,\mu}_{0+} x(t) ={}}{}+ \int_{V} h(t,x(t),v)\tilde{N}(dt, dv),\quad t \in J = (0,b], \\ \tau x(t) = B_{1} u(t), \quad t \in[0,b], \\ I^{(1-\nu)(1-\mu)}_{0+}x(0)+g(x)=x_{0}, \end{cases} $$

(1.1)

where $D^{\nu,\mu}_{0+}$ is the Hilfer fractional derivative, $0 \leq\nu\leq1$, $\frac{1}{2} < \mu< 1$, the state $x(\cdot)$ takes values in the separable Hilbert space X with inner product $\langle \cdot,\cdot\rangle$ and norm $\|\cdot\|$ and the control function $u(\cdot)$ is given in $L_{2}(J, U)$, the Hilbert space of admissible control functions with U a Hilbert space. Let σ be a closed, densely defined linear operator with domain $D(\sigma) \subset C(J, L_{2}(\varOmega,X) )$ and range $R(\sigma) \subset X$ and let $\tau :D(\tau) \subset C(J, L_{2}(\varOmega,X) )\rightarrow R(\tau) \subset X$ is a linear operator. Here, $B_{1}: U \rightarrow X$ is a linear continuous operator. Let $A: X \rightarrow X$ be the linear operator defined by $D(A)= \{ x \in D(\sigma); \tau x = 0 \}$, $Ax = \sigma x$, for $x \in D(A)$.

Suppose $\{ \omega(t) \}_{t \geq0}$ is a Wiener process defined on $(\varOmega, F, \{F_{t} \}_{t \geq0}, P)$ with values in the Hilbert space K and $\{B^{H}(t)\}_{t \geq0}$ is a fractional Brownian motion (fBm) with Hurst parameter $H \in(\frac {1}{2}, 1)$ defined on $(\varOmega, F, \{F_{t} \}_{t \geq0}, P)$ with values in Hilbert space Y. The nonlinear operators $f_{1} : J \times X \rightarrow X$, $f_{2} : J \times X \rightarrow L(K,X)$, $h: J \times X \times V \rightarrow X$, $G : J \times X \rightarrow L^{0}_{2}(Y,X)$ and $g:C(J,X)\rightarrow X$ are given.

2 Preliminaries

In order to study the approximate boundary controllability and the null boundary controllability of nonlocal Hilfer fractional stochastic differential equations with fractional Brownian motion and Poisson jumps, we need the following basic definitions and lemmas.

Definition 2.1

(see [35])

The fractional integral operator of order $\mu> 0$ for a function f can be defined as

$$I^{\mu}f(t)=\frac{1}{\varGamma(\mu)} \int^{t}_{0} \frac{f(s)}{(t-s)^{1-\mu }} \,ds, \quad t > 0, $$

where $\varGamma(\cdot)$ is the Gamma function.

Definition 2.2

(see [36])

The Hilfer fractional derivative of order $0\leq\nu\leq1$ and $0 < \mu< 1$ is defined as

$$D^{\nu,\mu}_{0+} f(t)= I^{\nu(1-\mu)}_{0+} \frac{d}{dt}I^{(1-\nu )(1-\mu)}_{0+} f(t). $$

Fix a time interval $[0,b]$ and let $(\varOmega,\eta,P)$ be a complete probability space furnished with a complete family of right continuous increasing sub σ-algebras $\{\eta_{t}:t\in[0,b]\}$ satisfying $\eta_{t}\subset\eta$. Let $(V,\varPhi,\rho(dv))$ be a σ-finite measurable space. Consider the stationary Poisson point process $(p_{t})_{t \geq0}$, which is defined on $(\varOmega,\eta,P)$ with values in V and with characteristic measure ρ. We will denote by $N(t,dv)$ the counting measure of $p_{t}$ such that $\tilde{N}(t,\varTheta):= E(N(t,\varTheta))=t\rho(\varTheta)$ for $\varTheta \in\varPsi$. Define $\tilde{N}(t,dv):= N(t,dv)-t\lambda(dv)$, the Poisson martingale measure generated by $p_{t}$.

Let X, K and Y be real, separable Hilbert spaces. For the sake of convenience, we shall use the same notation $\|\cdot\|$ to denote the norms in X, K, Y, $L(K, X)$ and $L(Y,X)$ where $L(K,X)$ and $L(Y,X)$ denote, respectively, the space of all bounded linear operators from K into X and Y into X. Let $Q \in L(Y,Y)$ be an operator defined by $Qe_{n}=\lambda_{n} e_{n}$ with finite trace $\operatorname{Tr}(Q)= \sum_{n=1}^{\infty}\lambda_{n} < \infty$ where $\lambda_{n} \geq0$ ($n=1,2,\ldots$) are non-negative real numbers and $\{e_{n}\}$ ($n=1,2,\ldots$) is a complete orthonormal basis in Y.

We define the infinite dimensional fBm on Y with covariance Q as

$$B^{H}(t)= B_{Q}^{H}(t)= \sum _{n=1}^{\infty}\sqrt{\lambda_{n} } e_{n} \beta_{n}^{H}(t), $$

where $\beta_{n}^{H}$ are real, independent fBm’s. The Y-valued process is Gaussian, starts from 0, and has mean zero and covariance:

$$E \bigl\langle B^{H}(t),x\bigr\rangle \bigl\langle B^{H}(s),y\bigr\rangle = R(s,t)\bigl\langle Q(x),y\bigr\rangle \quad \mbox{for all } x, y \in Y \mbox{ and } t, s \in[0,b]. $$

In order to define Wiener integrals with respect to the Q-fBm, we introduce the space $L^{0}_{2}:= L^{0}_{2}(Y,X)$ of all Q-Hilbert Schmidt operators $\psi: Y \rightarrow X$. We recall that $\psi\in L(Y,X)$ is called a Q-Hilbert-Schmidt operator, if

$$\| \psi\|^{2}_{L^{0}_{2}}:= \sum_{n=1}^{\infty}\| \sqrt{\lambda}_{n} \psi e_{n} \|^{2} < \infty $$

and that the space $L^{0}_{2}$ equipped with the inner product $\langle \vartheta, \psi\rangle_{L_{2}^{0}} = \sum_{n=1}^{\infty}\langle \vartheta e_{n}, \psi e_{n} \rangle$ is a separable Hilbert space.

Let $\phi(s)$; $s \in[0,b]$ be a function with values in $L_{2}^{0}(Y,X)$, the Wiener integral of ϕ with respect to $B^{H}$ is defined by

$$ \int_{0}^{t} \phi(s)\, d B^{H}(s)= \sum _{n=1}^{\infty}\int_{0}^{t} \sqrt{\lambda}\phi(s) e_{n} \, d \beta^{H}_{n} = \sum_{n=1}^{\infty}\int_{0}^{t} \sqrt{\lambda}K^{*}(\phi e_{n}) (s)\, d \beta_{n}(s), $$

(2.1)

where $\beta_{n}$ is the standard Brownian motion.

Lemma 2.1

(see [1])

If $\psi: [0,b]\rightarrow L^{0}_{2}(Y,X)$ satisfies $\int_{0}^{b} \| \psi(s) \|^{2}_{L^{0}_{2} }< \infty$ then the above sum in (2.1) is well defined as an X-valued random variable and we have

$$E \biggl\Vert \int_{0}^{t} \psi(s) \,dB^{H} (s) \biggr\Vert ^{2} \leq2H t^{2H-1} \int_{0}^{t} \bigl\Vert \psi(s) \bigr\Vert ^{2}_{L^{0}_{2} } \,ds. $$

The collection of all strongly measurable, square-integrable, X-valued random variables, denoted by $L_{2}(\varOmega,X)$, is a Banach space equipped with norm

$$\bigl\Vert x(\cdot) \bigr\Vert _{L_{2}(\varOmega,X)} = \bigl(E \bigl\Vert x( \cdot,\omega) \bigr\Vert ^{2}\bigr)^{\frac {1}{2}}, $$

where the expectation, E, is defined by $E(x)= \int_{\varOmega}x(\omega ) \,dP$.

Let $C(J, L_{2}(\varOmega, X))$ be the Banach space of all continuous maps from J into $L_{2}(\varOmega,X)$ satisfying the condition $\sup_{t\in J}E\|x(t)\|^{2} <\infty$. Define $\bar{C} =\{x:\cdot^{(1- \nu)(1-\mu)}x(\cdot)\in C(J,L_{2}(\varOmega,X))\}$, with norm $\|\cdot\|_{\bar{C}}$ defined by

$$\Vert \cdot \Vert _{\bar{C}}=\Bigl(\sup_{t \in J} E \bigl\Vert t^{(1-\nu)(1-\mu)}x(t) \bigr\Vert ^{2}\Bigr)^{\frac{1}{2}}. $$

Obviously, C̄ is a Banach space.

Like [35], we denote $\bar {C}^{\nu+\mu-\nu\mu} =\{x:x\in\bar{C}, D^{\nu+\mu-\nu\mu }_{0+}x\in\bar{C}\}$, $\bar{C}^{\nu,\mu} =\{x:x\in\bar{C}, D^{\nu,\mu}_{0+}x\in\bar {C}\}$. Obviously, $\bar{C}^{\nu+\mu-\nu\mu} \subseteq\bar {C}^{\nu,\mu}$.

Also, let us introduce the set $B_{r} = \{ \nu\in\bar{C} : \| \nu\| ^{2}_{\bar{C}} \leq r \}$, where $r> 0$.

To establish the results, we need the following hypotheses.

• (H1) $D(\sigma) \subset D(\tau)$ and the restriction of τ to $D(\sigma)$ is continuous relative to graph norm of $D(\sigma)$.

• (H2) The operator A is the infinitesimal generator of a compact semigroup $T(t)$ on X and there exists a constant $M > 0$ such that $\| T(t)\| \leq M$.

• (H3) There exists a linear operator $B: U \rightarrow X$ such that for all $u \in U$ we have $B u \in D(\sigma)$, $\tau(Bu)= B_{1} u$ and $\| B u \| \leq C \| B_{1} u \|$, where C is a constant.

• (H4) For all $t \in(0,b]$ and $u \in U$, $T(t)Bu \in D(A)$. Moreover, there exists a positive constant $M_{1} >0$ such that $\| A T(t)\| \leq M_{1}$.

• (H5) The function $f_{1} : J \times X \rightarrow X$ satisfies the following two conditions:

  • (i) The function $f_{1} : J \times X \rightarrow X$ is continuous;

  • (ii) for each positive number $r \in N$, there is a positive function $\rho_{r}(\cdot):J \rightarrow R^{+}$ such that

    $$\sup_{\|x\|^{2} \leq r} E \bigl\Vert f_{1}\bigl(t,x(t)\bigr) \bigr\Vert ^{2} \leq\rho _{r}(t), $$

    the function $s\rightarrow\rho_{r}(s)\in L^{1}([0,t], R^{+})$, and there exists a $\delta_{1} > 0$ such that

    $$\lim_{r\rightarrow\infty}\inf\frac{\int_{0}^{t} (t-s)^{\mu-1}\rho _{r}(s) \,ds}{r}=\delta_{1} < \infty, \quad t\in J. $$

• (H6) The function $f_{2} : J \times X \rightarrow L(K,X)$ satisfies the following two conditions:

  • (i) The function $f_{2} : J \times X \rightarrow L(K,X)$ is continuous;

  • (ii) for each positive number $r \in N$, there is a positive function $h_{r}(\cdot):J \rightarrow R^{+}$ such that

    $$\sup_{\|x\|^{2} \leq r} E \bigl\Vert f_{2}\bigl(t, x(t)\bigr) \bigr\Vert _{Q}^{2} \leq h_{r}(t), $$

    the function $s\rightarrow h_{r}(s)\in L^{1}([0,t], R^{+})$, and there exists a $\delta_{2} > 0$ such that

    $$\lim_{r\rightarrow\infty}\inf\frac{\int_{0}^{t} (t-s)^{\mu-1} h_{r}(s) \,ds}{r}=\delta_{2} < \infty, \quad t\in J. $$

• (H7) The function $G : J \times X \rightarrow L^{0}_{2}(Y,X)$ satisfies the following two conditions:

  • (i) The function $G : J \times X \rightarrow L^{0}_{2}(Y,X)$ is continuous;

  • (ii) for each positive number $r \in N$, there is a positive function $k_{r}(\cdot):J \rightarrow R^{+}$ such that

    $$\sup_{\|x\|^{2} \leq r} E \bigl\Vert G\bigl(t,x(t)\bigr) \bigr\Vert _{L_{2}^{0}}^{2} \leq k_{r}(t), $$

    the function $s\rightarrow k_{r}(s)\in L^{1}([0,t], R^{+})$, and there exists a $\delta_{3} > 0$ such that

    $$\lim_{r\rightarrow\infty}\inf\frac{\int_{0}^{t} (t-s)^{\mu-1} k_{r}(s) \,ds}{r}=\delta_{3} < \infty,\quad t\in J. $$

• (H8) The function $h:J\times X\times V\rightarrow X$ satisfies the following two conditions:

  • (i) The function $h:J\times X\times V\rightarrow X$ is continuous;

  • (ii) for each positive number $r\in N$, there is a positive function $\chi_{r}(\cdot):J \rightarrow R^{+}$ such that

    $$\sup_{\|x\|^{2} \leq r} \int_{V} E \bigl\Vert h(t, x,v) \bigr\Vert ^{2} \lambda(dv)\leq\chi_{r}(t), $$

    the function $s\rightarrow(t -s)^{\mu-1} \chi_{r}(s) \in L^{1}([0,t], R^{+})$, and there exists a $\delta_{4}>0$ such that

    $$\lim_{r\rightarrow\infty} \inf\frac{\int_{0}^{t} (t-s)^{\mu-1}\chi _{r}(s) \,ds}{r} = \delta_{4} < \infty,\quad t\in J. $$

• (H9) The function $g:C(J,X)\rightarrow X$ satisfies the following two conditions:

  • (i) There exist positive constants $M_{2}$ and $M_{3}$ such that $\| g(x )\|\leq M_{2} \| x\| + M_{3}$ for all $x \in X$;

  • (ii) g is completely continuous map.

Let $x(t)$ be the solution of the system (1.1). Then we can define a function $z(t)= x(t)- B u(t)$ and from our assumption we see that $z(t) \in D(A)$. Hence (1.1) can be written in terms of A and B as

$$ \textstyle\begin{cases} D^{\nu,\mu}_{0+} z(t)= A z(t) +\sigma B u(t)- B D^{\nu,\mu}_{0+} u(t)+ f_{1}(t,x(t))+ f_{2}(t,x(t))\frac{d\omega(t)}{dt} \\ \hphantom{D^{\nu,\mu}_{0+} z(t)={}}{}+G(t,x(t))\frac{dB^{H}(t)}{dt}+ \int_{V} h(t,x(t),v)\tilde{N}(dt, dv),\quad t \in J = (0,b], \\ I^{(1-\nu)(1-\mu)}_{0+}[z(0)+B u(0)] =I^{(1-\nu)(1-\mu )}_{0+}x(0)=x_{0}-g(x) . \end{cases} $$

(2.2)

Lemma 2.2

From (2.2), problem (1.1) is equivalent to the integral equation

$$\begin{aligned} x(t) =&\frac{x_{0}-g(x)}{\varGamma(\nu+ \mu-\nu\mu)} t^{(\nu-1)(1-\mu )}+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1}A x(s)\,ds \\ &{}+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1}[\sigma-A] B u(s) \,ds+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1}f_{1} \bigl(s,x(s)\bigr) \,ds \\ &{}+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu -1}f_{2} \bigl(s,x(s)\bigr)\,d\omega(s)+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1}G\bigl(s,x(s)\bigr) \,dB^{H}(s) \\ &{}+\frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1} \int_{V} h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv). \end{aligned}$$

(2.3)

Proof

Applying $I^{\mu}$ to both sides of (2.2), we obtain

$$\begin{aligned} I^{\mu}D^{\nu,\mu}_{0+}\bigl[ z(t)+B u(t)\bigr] =& I^{\mu}\biggl[A z(t) +\sigma B u(t)+ f_{1}\bigl(t,x(t)\bigr)+ f_{2}\bigl(t,x(t)\bigr)\frac{d\omega(t)}{dt} \\ &{}+G\bigl(t,x(t)\bigr)\frac{dB^{H}(t)}{dt}+ \int_{V} h\bigl(t,x(t),v\bigr)\tilde{N}(dt, dv)\biggr]. \end{aligned}$$

Since $I^{\mu}D^{\nu,\mu}_{0+}[ z(t)+B u(t)]=I^{\mu+\nu-\mu\nu} D^{\mu +\nu-\mu\nu}_{0+}[ z(t)+B u(t)]$,

$$\begin{aligned} I^{\mu+\nu-\mu\nu} D^{\mu+\nu-\mu\nu}_{0+}\bigl[ z(t)+B u(t)\bigr] =& I^{\mu}\biggl[A z(t) +\sigma B u(t)+ f_{1}\bigl(t,x(t)\bigr)+ f_{2}\bigl(t,x(t)\bigr)\frac{d\omega (t)}{dt} \\ &{}+G\bigl(t,x(t)\bigr)\frac{dB^{H}(t)}{dt}+ \int_{V} h\bigl(t,x(t),v\bigr)\tilde{N}(dt, dv)\biggr]. \end{aligned}$$

From properties of fractional integral, we obtain

$$\begin{aligned} z(t)+B u(t) =&\frac{I^{(1-\nu)(1-\mu)}_{0+}[z(0)+B u(0)] }{\varGamma (\nu+ \mu-\nu\mu)} t^{(\nu-1)(1-\mu)}+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1}A x(s)\,ds \\ &{}+ \frac{1}{\varGamma(\mu )} \int_{0}^{t} (t-s)^{\mu-1}[\sigma-A] B u(s) \,ds+ \frac{1}{\varGamma(\mu )} \int_{0}^{t} (t-s)^{\mu-1}f_{1} \bigl(s,x(s)\bigr) \,ds \\ &{}+ \frac {1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1}f_{2} \bigl(s,x(s)\bigr)\,d\omega(s) \\ &{}+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1}G\bigl(s,x(s)\bigr) \,dB^{H}(s) \\ &{}+\frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1} \int_{V} h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv). \end{aligned}$$

Since $z(t)+B u(t)=x(t)$,

$$\begin{aligned} x(t) =&\frac{x_{0}-g(x)}{\varGamma(\nu+ \mu-\nu\mu)} t^{(\nu-1)(1-\mu )}+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1}A x(s)\,ds \\ &{}+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1}[\sigma-A] B u(s) \,ds+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1}f_{1} \bigl(s,x(s)\bigr) \,ds \\ &{}+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu -1}f_{2} \bigl(s,x(s)\bigr)\,d\omega(s)+ \frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1}G\bigl(s,x(s)\bigr) \,dB^{H}(s) \\ &{}+\frac{1}{\varGamma(\mu)} \int_{0}^{t} (t-s)^{\mu-1} \int_{V} h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv). \end{aligned}$$

For more details see [37]. □

Lemma 2.3

If the integral equation (2.3) holds, then we have

$$\begin{aligned} x(t) =& S_{\nu,\mu}(t)\bigl[x_{0}-g(x)\bigr]+ \int_{0}^{t} \bigl[P_{\mu}(t-s)\sigma- AP_{\mu}(t-s)\bigr] B u(s)\,ds \\ &{}+ \int_{0}^{t} P_{\mu}(t-s)f_{1} \bigl(s,x(s)\bigr) \,ds+ \int_{0}^{t} P_{\mu}(t-s)f_{2} \bigl(s,x(s)\bigr)\,d\omega(s) \\ &{}+ \int_{0}^{t} P_{\mu}(t-s) G\bigl(s,x(s) \bigr) \,dB^{H}(s) \\ &{}+ \int_{0}^{t} P_{\mu}(t-s) \int_{V} h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv), \end{aligned}$$

(2.4)

where

$$ S_{\nu, \mu}(t)=I^{\nu(1-\mu)}_{0+}P_{\mu}(t), \qquad P_{\mu}(t)= t^{\mu -1} T_{\mu}(t), \qquad T_{\mu}(t)= \int_{0}^{\infty}\mu\theta\varPsi_{\mu}(\theta )T \bigl(t^{\mu}\theta\bigr)\,d\theta, $$

with

$$ \varPsi_{\mu}(\theta)= \sum_{n=1}^{\infty} \frac{(- \theta )^{n-1}}{(n-1)! \varGamma(1-n \mu)}, \quad 0< \mu< 1, \theta\in(0, \infty), $$

is a function of Wright-type which satisfies

$$ \int_{0}^{\infty}\theta^{\delta}\varPsi_{\mu}(\theta)\,d\theta=\frac {\varGamma(1+ \delta)}{\varGamma(1+ \mu\delta)}\quad \textit{for } \theta \geq 0. $$

Proof

The proof of this lemma similar to the proof of Lemma 2.12 in [38]. □

Lemma 2.4

(see [38])

The operator $S_{\nu,\mu}$ and $P_{\mu}$ have the following properties:

• (i) $\{ P_{\mu}(t): t > 0\}$ is continuous in the uniform operator topology.

• (ii) For any fixed $t >0$, $S_{\nu, \mu}(t)$ and $P_{\mu}(t)$ are linear and bounded operators, and

  $$ \bigl\Vert P_{\mu}(t)x \bigr\Vert \leq\frac{M t^{\mu-1}}{\varGamma(\mu)} \Vert x \Vert , \qquad \bigl\Vert S_{\nu ,\mu}(t)x \bigr\Vert \leq \frac{M t^{(\nu-1)(1-\mu)}}{\varGamma(\nu(1-\mu)+ \mu)} \Vert x \Vert . $$

• (iii) $\{P_{\mu}(t):t>0\}$ and $\{S_{\nu,\mu}(t):t >0\}$ are strongly continuous.

Lemma 2.5

If the assumption (H4) is satisfied, then $\| A P_{\mu}(t) x\|\leq\frac{ M_{1} t^{\mu-1}}{\varGamma(\mu)} \|x\|$.

Proof

We have

$$P_{\mu}(t)x= \mu t^{\mu-1} \int_{0}^{\infty}\mu\theta\varPsi_{\mu}(\theta )T \bigl(t^{\mu}\theta\bigr)\,d\theta, \qquad \int_{0}^{\infty}\theta\varPsi_{\mu}(\theta)\,d \theta=\frac{1}{\varGamma(1+\mu )}. $$

By using (H4),

$$\bigl\Vert A P_{\mu}(t)x \bigr\Vert = \biggl\Vert \mu t^{\mu-1} \int_{0}^{\infty}\mu\theta\varPsi_{\mu}(\theta)AT \bigl(t^{\mu}\theta\bigr)x \,d\theta \biggr\Vert \leq\frac{\mu M_{1} t^{\mu -1}}{\varGamma(\mu+1)} \Vert x \Vert = \frac{ M_{1} t^{\mu-1}}{\varGamma(\mu)} \Vert x \Vert . $$

 □

Definition 2.3

We say $x\in\bar{C}$ is a mild solution to (1.1) if it satisfies

$$\begin{aligned} x(t) =& S_{\nu,\mu}(t)\bigl[x_{0}-g(x)\bigr]+ \int_{0}^{t} \bigl[P_{\mu}(t-s)\sigma- AP_{\mu}(t-s)\bigr] B u(s)\,ds \\ &{}+ \int_{0}^{t} P_{\mu}(t-s)f_{1} \bigl(s,x(s)\bigr) \,ds+ \int_{0}^{t} P_{\mu}(t-s)f_{2} \bigl(s,x(s)\bigr)\,d\omega(s) \\ &{}+ \int_{0}^{t} P_{\mu}(t-s) G\bigl(s,x(s) \bigr) \,dB^{H}(s)+ \int_{0}^{t} P_{\mu}(t-s) \int _{V} h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv). \end{aligned}$$

3 Approximate boundary controllability

In this section, we discuss the approximate controllability for the system (1.1), so we introduce the following linear Hilfer fractional differential system with control on the boundary:

$$ \textstyle\begin{cases} D^{\nu,\mu}_{0+} x(t) = \sigma x(t) , \quad t \in J = (0,b], \\ \tau x(t) = B_{1} u(t),\quad t \in J, \\ I^{(1-\nu)(1-\mu)}_{0+}x(0)=x_{0}. \end{cases} $$

(3.1)

It is convenient at this point to introduce the operators associated with (3.1) as

$$\begin{aligned} \varGamma_{0}^{b} =& \int_{0}^{b} (b-s)^{2(\mu-1)} \bigl[T_{\mu}(b-s)\sigma- A T_{\mu}(b-s)\bigr] \\ &{}\times BB^{*} \bigl[T_{\mu}(b-s)\sigma - A T_{\mu}(b-s) \bigr]^{*} \,ds, \quad \frac{1}{2}< \mu< 1, \end{aligned}$$

and

$$R\bigl(b,\varGamma_{0}^{b}\bigr)= \bigl(b I + \varGamma_{0}^{b}\bigr)^{-1}, \quad b>0, $$

where $B^{*}$ and $[T_{\mu}(b-s)\sigma - A T_{\mu}(b-s) ]^{*}$ denote the adjoint of B and $[T_{\mu}(b-s)\sigma - A T_{\mu}(b-s) ]$, respectively.

Let $x(b;x_{0}, u)$ be the state value of (1.1) at terminal state b, corresponding to the control u and the initial value $x_{0}$. Denote by $R(b,x_{0}) = \{ x(b;x_{0},u):u \in L_{2}(J, U) \}$ the reachable set of system (1.1) at terminal time b, its closure in X is denoted $\overline{ R(b,x_{0})}$.

Definition 3.1

(see [19])

The system (1.1) is said to be approximately controllable on the interval J if $\overline{ R(b,x_{0})} = L_{2}(\varOmega, X)$.

Lemma 3.1

(see [19])

The fractional linear control system (3.1) is approximately controllable on J if and only if $\lambda (\lambda I + \varGamma^{b}_{0})^{-1}\rightarrow0$ as $\lambda\rightarrow 0^{+}$.

Lemma 3.2

For any $\bar{x}_{b} \in L_{2}(\varOmega, X)$ there exist $\bar{\psi}\in L_{2}(\varOmega; L_{2}(J;L_{Q}))$ and $\bar{\varphi}\in L_{2}(\varOmega; L_{2}(J;L^{0}_{2}))$ such that

$$\bar{x}_{b}= E \bar{x}_{b} + \int_{0}^{b} \bar{\psi}(s) \,d\omega(s)+ \int_{0}^{b} \bar{\varphi}(s) \,dB^{H}(s). $$

Now for any $\kappa>0$ and $\bar{x}_{b} \in L_{2}(\varOmega,X)$, we define the control function in the following form:

$$\begin{aligned} u^{\kappa}(t) =&B^{*} \bigl[T_{\mu}(b-t)\sigma - A T_{\mu}(b-t) \bigr]^{*} \bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \biggl\{ E \bar{x}_{b} - S_{\nu,\mu}(b) \bigl[x_{0}-g(x)\bigr] \\ &{}- \int_{0}^{b} P_{\mu}(b-s)f_{1} \bigl(s,x(s)\bigr) \,ds- \int_{0}^{b} P_{\mu}(b-s)f_{2} \bigl(s,x(s)\bigr)\,d\omega(s) \\ &{}- \int_{0}^{b} P_{\mu}(b-s) G\bigl(s,x(s) \bigr) \,dB^{H}(s)- \int_{0}^{b} P_{\mu}(b-s) \int _{V} h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv)\biggr\} . \end{aligned}$$

Theorem 3.1

If the assumptions (H1)–(H9) are satisfied, then system (1.1) has a mild solution on J, provided that

$$\begin{aligned}& 36M^{2} \biggl[\frac{M^{2}_{2}}{\varGamma^{2}(\nu(1-\mu)+ \mu)}+\frac{ b^{\mu+2(1-\nu)(1-\mu)}[\delta_{1}+\delta_{2} \operatorname{Tr}(Q) + 2H b^{2H-1} \delta_{3}+\delta_{4}]}{\mu\varGamma^{2}(\mu)} \biggr] \\& \quad {} \times \biggl[1+ \frac{6 b^{2\mu-1} \|B \|^{2} \|B^{*}\|^{2}(M^{2} \|\sigma\|^{2}+ M_{1}^{2})^{2}}{\lambda^{2}(2\mu-1)\varGamma^{4}(\mu)} \biggr] < 1. \end{aligned}$$

(3.2)

Proof

For any $\kappa> 0$, consider the map $\varPhi_{\kappa}$ on C̄ defined by

$$\begin{aligned} (\varPhi_{\kappa}x) (t) =& S_{\nu,\mu}(t)\bigl[x_{0}-g(x) \bigr]+ \int_{0}^{t} \bigl[P_{\mu}(t-s)\sigma- AP_{\mu}(t-s)\bigr] B u^{\kappa}(s)\,ds \\ &{}+ \int_{0}^{t} P_{\mu}(t-s)f_{1} \bigl(s,x(s)\bigr) \,ds+ \int_{0}^{t} P_{\mu}(t-s)f_{2} \bigl(s,x(s)\bigr)\,d\omega(s) \\ &{}+ \int_{0}^{t} P_{\mu}(t-s) G\bigl(s,x(s) \bigr) \,dB^{H}(s) \\ &{}+ \int_{0}^{t} P_{\mu}(t-s) \int _{V} h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv), \quad t \in J. \end{aligned}$$

For $t \in J$, we have

$$\begin{aligned}& E \bigl\Vert u^{\kappa}(t) \bigr\Vert ^{2} \\& \quad \leq6 E \biggl\Vert B^{*} \bigl[T_{\mu}(b-t)\sigma - A T_{\mu}(b-t) \bigr]^{*} \bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \\& \qquad {}\times\biggl[ \bar{x}_{b} - \int_{0}^{b} \bar{\psi}(s) \,d\omega(s)- \int_{0}^{b} \bar{\varphi}(s) \,dB^{H}(s) \biggr] \biggr\Vert ^{2} \\& \qquad {}+6 E \bigl\Vert B^{*} \bigl[T_{\mu}(b-t)\sigma - A T_{\mu}(b-t) \bigr]^{*} \bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} S_{\nu ,\mu}(b)\bigl[x_{0}-g(x)\bigr] \bigr\Vert ^{2} \\& \qquad {}+6 E \biggl\Vert B^{*} \bigl[T_{\mu}(b-t)\sigma - A T_{\mu}(b-t) \bigr]^{*} \bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \int_{0}^{b} P_{\mu}(b-s)f_{1} \bigl(s,x(s)\bigr) \,ds \biggr\Vert ^{2} \\& \qquad {}+6 E \biggl\Vert B^{*} \bigl[T_{\mu}(b-t)\sigma - A T_{\mu}(b-t) \bigr]^{*} \bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \int_{0}^{b} P_{\mu}(b-s)f_{2} \bigl(s,x(s)\bigr)\,d\omega(s) \biggr\Vert ^{2} \\& \qquad {}+6 E \biggl\Vert B^{*} \bigl[T_{\mu}(b-t)\sigma - A T_{\mu}(b-t) \bigr]^{*} \bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \int_{0}^{b} P_{\mu}(b-s) G\bigl(s,x(s) \bigr) \,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {}+36 E \biggl\Vert B^{*} \bigl[T_{\mu}(b-t)\sigma - A T_{\mu}(b-t) \bigr]^{*} \bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \\& \qquad {}\times\int_{0}^{b} P_{\mu}(b-s) \int_{V} h\bigl(s,x(s),v\bigr)\tilde {N}(ds, dv) \biggr\Vert ^{2} \\& \quad \leq\frac{6 \Vert B \Vert ^{2}(M^{2} \Vert \sigma \Vert ^{2}+ M_{1}^{2})}{\lambda^{2}\varGamma ^{2}(\mu)}\biggl[E \Vert \bar{x}_{b} \Vert ^{2}+\operatorname{Tr}(Q) \int_{0}^{b} E \bigl\Vert \bar{\psi}(s) \bigr\Vert ^{2}_{Q} \,ds \\& \qquad {}+ 2H b^{2H-1} \int_{0}^{b} E \bigl\Vert \bar{\varphi}(s) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds+ \frac{M^{2} b^{2(\nu-1)(1-\mu)}[E \Vert x_{0} \Vert ^{2} +M^{2}_{2} r+ M^{2}_{3}]}{\varGamma ^{2}(\nu(1-\mu)+ \mu)} \\& \qquad {}+\frac{ M^{2} b^{\mu}}{\mu\varGamma^{2}(\mu)} \int_{0}^{b} (b-s)^{\mu-1}\rho _{r}(s) \,ds +\operatorname{Tr}(Q)\frac{ M^{2} b^{\mu}}{\mu\varGamma^{2}(\mu)} \int_{0}^{b} (b-s)^{\mu-1} h_{r}(s) \,ds \\& \qquad {}+ \frac{2 H M^{2} b^{2H-1+\mu}}{\mu\varGamma^{2}(\mu)} \int_{0}^{b} (b-s)^{\mu-1} k_{r}(s) \,ds +\frac{ M^{2} b^{\mu}}{\mu\varGamma^{2}(\mu)} \int _{0}^{b}(b-s)^{\mu-1} \chi_{q}(s)\,ds\biggr]. \end{aligned}$$

It will be shown that the operator $\varPhi_{\kappa}$ from C̄ into itself has a fixed point.

We claim that there exists a positive number r such that $\varPhi _{\kappa}(B_{r}) \subseteq B_{r}$. If it is not true, then, for each positive number r, there is a function $x_{r}(\cdot)\in B_{r}$, but $\varPhi(x_{r})\notin B_{r}$, that is, $\|\varPhi _{\kappa}x_{r}\|^{2}_{\bar{C}}>r$ for some $t=t(r)\in J$, where $t(r)$ means that t is dependent of r.

From our hypotheses together with Lemma 2.4, Lemma 2.5, the Hölder inequality and Burkholder–Gungy’s inequality, we obtain

$$\begin{aligned} r \leq& \Vert \varPhi_{\kappa}x_{r} \Vert ^{2}_{\bar{C}}=\sup_{t \in J}t^{2(1-\nu )(1-\mu)}E \bigl\Vert \varPhi_{\kappa}(x_{r}) (t) \bigr\Vert ^{2} \\ \leq& \frac{36 M ^{2} [E \Vert x_{0} \Vert ^{2} +M^{2}_{2} r+ M^{2}_{3}]}{\varGamma^{2}(\nu (1-\mu)+ \mu)} + \frac{216 b^{2\mu-1} \Vert B \Vert ^{4} (M^{2} \Vert \sigma \Vert ^{2}+ M_{1}^{2})}{\lambda^{2}(2\mu-1)\varGamma^{2}(\mu)} \biggl[b^{2(1-\nu)(1-\mu)}\biggl(E \Vert \bar{x}_{b} \Vert ^{2} \\ &{}+\operatorname{Tr}(Q) \int_{0}^{b} E \bigl\Vert \bar{\psi}(s) \bigr\Vert ^{2}_{Q} \,ds + 2H b^{2H-1} \int_{0}^{b} E \bigl\Vert \bar{\varphi}(s) \bigr\Vert ^{2}_{L^{0}_{2}} \,ds\biggr) \\ &{}+\frac{ M^{2} [E \Vert x_{0} \Vert ^{2} +M^{2}_{2} r+ M^{2}_{3}]}{\varGamma^{2}(\nu(1-\mu)+ \mu)}+ \frac{ M^{2} b^{\mu+2(1-\nu)(1-\mu)}}{\mu\varGamma^{2}(\mu )} \int_{0}^{b} (b-s)^{\mu-1} \rho_{r}(s) \,ds \\ &{}+\operatorname{Tr}(Q)\frac{ M^{2} b^{\mu+2(1-\nu)(1-\mu)}}{\mu\varGamma^{2}(\mu)} \int_{0}^{b} (b-s)^{\mu-1} h_{r}(s) \,ds \\ &{}+ \frac{2 H M^{2} b^{2H-1+\mu+2(1-\nu)(1-\mu)}}{\mu\varGamma^{2}(\mu )} \int_{0}^{b} (b-s)^{\mu-1} k_{r}(s) \,ds \\ &{}+\frac{ M^{2} b^{\mu+2(1-\nu )(1-\mu)}}{\mu\varGamma^{2}(\mu)} \int_{0}^{b}(b-s)^{\mu-1}\chi_{q}(s) \,ds\biggr] \\ &{} + \frac{36 M^{2} b^{\mu+2(1-\nu)(1-\mu)}}{\mu\varGamma ^{2}(\mu)} \int_{0}^{t} (t-s)^{\mu-1} \rho_{r}(s) \,ds \\ &{} + \frac {36M^{2} b^{\mu+2(1-\nu)(1-\mu)} \operatorname{Tr}(Q)}{\mu\varGamma^{2}(\mu)} \int_{0}^{t} (t-s)^{\mu-1} h_{r}(s) \,ds \\ &{}+ \frac{72 H M^{2} b^{2H-1+\mu +2(1-\nu)(1-\mu)}}{\mu\varGamma^{2}(\mu)} \int_{0}^{b} (t-s)^{\mu-1} k_{r}(s) \,ds \\ &{}+\frac{36 M^{2} b^{\mu+2(1-\nu)(1-\mu)}}{\mu \varGamma^{2}(\mu)} \int_{0}^{t}(t-s)^{\mu-1}\chi_{q}(s) \,ds. \end{aligned}$$

(3.3)

Dividing both sides of (3.3) by r and taking the lower limit $r\rightarrow+\infty$, we get

$$\begin{aligned}& 36M^{2} \biggl[\frac{M^{2}_{2}}{\varGamma^{2}(\nu(1-\mu)+ \mu)}+\frac{ b^{\mu+2(1-\nu)(1-\mu)}[\delta_{1}+\delta_{2} \operatorname{Tr}(Q) + 2H b^{2H-1} \delta_{3}+\delta_{4}]}{\mu\varGamma^{2}(\mu)} \biggr] \\& \quad {}\times \biggl[1+ \frac{6 b^{2\mu-1} \|B \|^{2} \|B^{*}\|^{2}(M^{2} \|\sigma\|^{2}+ M_{1}^{2})^{2}}{\lambda ^{2}(2\mu-1)\varGamma^{4}(\mu)} \biggr] \geq1. \end{aligned}$$

This contradicts (3.2). Hence, for positive r, $\varPhi_{\kappa}(B_{r})\subseteq B_{r}$ for positive number r.

In fact, the operator $\varPhi_{\kappa}$ maps $B_{r}$ into a compact subset of $B_{r}$. To prove this, we first show that the set $V_{r}(t)=\{(\varPhi _{\kappa}x)(t):x\in B_{r}\}$ is precompact in X, for every fixed $t\in J$. This is trivial for $t=0$, since $V_{r}(0) = \{x_{0}\}$. Let t, $0 < t\leq b$, be fixed.

For $0 < \epsilon< t$ and arbitrary $\delta> 0$, take

$$\begin{aligned}& \bigl(\varPhi_{\kappa}^{\epsilon, \delta}x\bigr) (t) \\& \quad = \frac{\mu}{\varGamma(\nu (1-\mu))} \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\nu (1-\mu)-1}s^{\mu-1} \varPsi_{\mu}(\theta)T\bigl(s^{\mu}\theta\bigr) \bigl[x_{0}-g(x)\bigr] \,d\theta \,ds \\& \qquad {}+ \mu \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)\bigl[T\bigl((t-s)^{\mu}\theta\bigr)\sigma- AT \bigl((t-s)^{\mu}\theta\bigr)\bigr] B u^{\kappa}(s) \,d\theta \,ds \\& \qquad {}+ \mu \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu -1} \varPsi_{\mu}(\theta) T\bigl((t-s)^{\mu}\theta\bigr) f_{1}\bigl(s,x(s)\bigr) \,d\theta \,ds \\& \qquad {}+ \mu \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta (t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta\bigr) f_{2} \bigl(s,x(s)\bigr) \,d\theta \,d\omega(s) \\& \qquad {}+ \mu \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu -1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta\bigr) G\bigl(s,x(s) \bigr) \,d\theta \,dB^{H}(s) \\& \qquad {}+ \mu \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\int_{V} \theta (t-s)^{\mu-1}\varPsi_{\mu}( \theta)T\bigl((t-s)^{\mu}\theta\bigr) h\bigl(s,x(s),v\bigr)\tilde {N}(ds, dv) \\& \quad = \frac{\mu T(\epsilon^{\mu}\delta)}{\varGamma(\nu(1-\mu))} \int _{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\nu(1-\mu )-1}s^{\mu-1} \varPsi_{\mu}(\theta)T\bigl(s^{\mu}\theta-\epsilon^{\mu}\delta\bigr) \bigl[x_{0}-g(x)\bigr] \,d\theta \,ds \\& \qquad {} + \mu T\bigl(\epsilon^{\mu}\delta\bigr) \int _{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)\bigl[T\bigl((t-s)^{\mu}\theta- \epsilon^{\mu}\delta\bigr)\sigma \\& \qquad {}- AT\bigl((t-s)^{\mu}\theta- \epsilon^{\mu}\delta\bigr) \bigr] B u^{\kappa}(s) \,d\theta \,ds \\& \qquad {}+ \mu T\bigl(\epsilon^{\mu}\delta\bigr) \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta) T\bigl((t-s)^{\mu}\theta- \epsilon^{\mu}\delta\bigr) f_{1}\bigl(s,x(s)\bigr) \,d\theta \,ds \\& \qquad {}+ \mu T\bigl(\epsilon^{\mu}\delta\bigr) \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta- T\bigl(\epsilon ^{\mu}\delta\bigr)\bigr) f_{2}\bigl(s,x(s)\bigr) \,d\theta \,d \omega(s) \\& \qquad {}+ \mu T\bigl(\epsilon^{\mu}\delta\bigr) \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta- \epsilon ^{\mu}\delta\bigr) G\bigl(s,x(s)\bigr)\,d\theta \,dB^{H}(s) \\& \qquad {}+ \mu T\bigl(\epsilon^{\mu}\delta\bigr) \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\int_{V} \theta(t-s)^{\mu-1}\varPsi_{\mu}( \theta)T\bigl((t-s)^{\mu}\theta- \epsilon^{\mu}\delta\bigr) \\& \qquad {}\times h \bigl(s,x(s),v\bigr)\tilde{N}(ds, dv). \end{aligned}$$

Since $u^{\kappa}(s)$ is bounded and $T(\epsilon^{\mu}\delta)$, $\epsilon ^{\mu}\delta> 0$ is a compact operator, then the set $V_{r}^{\epsilon, \delta} (t) = \{ (\varPhi_{\kappa}^{\epsilon, \delta}x) (t) : x \in B_{r} \}$ is a precompact set in X for every ϵ, $0 < \epsilon< t$, and for all $\delta> 0$. Moreover, for every $x \in B_{r}$, we have

$$\begin{aligned}& \bigl\Vert \varPhi_{\kappa}x -\varPhi_{\kappa}^{\epsilon, \delta} x \bigr\Vert ^{2}_{\bar {C}} \\ & \quad =\sup_{t \in J} t^{2(1-\nu)(1-\mu)} E \bigl\Vert ( \varPhi_{\kappa}x) (t)-\bigl(\varPhi_{\kappa}^{\epsilon, \delta} x\bigr) (t) \bigr\Vert ^{2} \\ & \quad \leq \frac{36\mu^{2}\sup_{t \in J} t^{2(1-\nu)(1-\mu)}}{\varGamma ^{2}(\nu(1-\mu))} \\ & \qquad {}\times E \biggl\Vert \int_{0}^{t} \int_{0}^{\delta}\theta(t-s)^{\nu(1-\mu )-1} s^{\mu-1}\varPsi_{\mu}(\theta) T\bigl(s^{\mu}\theta\bigr) \bigl[ x_{0}-g(x)\bigr] \,d\theta \,ds \biggr\Vert ^{2} \\ & \qquad {} +36 \sup_{t \in J} t^{2(1-\nu)(1-\mu)}E \biggl\Vert \mu \int_{t-\epsilon }^{t} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta )\bigl[T\bigl((t-s)^{\mu}\theta\bigr)\sigma \\ & \qquad {}- AT\bigl((t-s)^{\mu}\theta\bigr)\bigr] B u^{\kappa}(s) \,d\theta \,ds \biggr\Vert ^{2} \\ & \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} E \biggl\Vert \mu \int_{t-\epsilon }^{t} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta) T\bigl((t-s)^{\mu}\theta\bigr) f_{1}\bigl(s,x(s)\bigr) \,d\theta \,ds \biggr\Vert ^{2} \\ & \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} \\ & \qquad {}\times E \biggl\Vert \mu \int_{t-\epsilon}^{t} \int _{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta\bigr) f_{2}\bigl(s,x(s)\bigr) \,d\theta \,d\omega(s) \biggr\Vert \\ & \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} \\ & \qquad {}\times E \biggl\Vert \mu \int_{t-\epsilon }^{t} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta )T\bigl((t-s)^{\mu}\theta\bigr) G\bigl(s,x(s) \bigr) \,d\theta \,dB^{H}(s) \biggr\Vert ^{2} \\ & \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} \\ & \qquad {}\times E \biggl\Vert \mu \int_{t-\epsilon }^{t} \int_{\delta}^{\infty}\int_{V} \theta(t-s)^{\mu-1}\varPsi_{\mu}( \theta)T\bigl((t-s)^{\mu}\theta\bigr) h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv) \biggr\Vert ^{2} \\ & \qquad {}+ \frac{36\mu^{2}\sup_{t \in J} t^{2(1-\nu)(1-\mu)}}{\varGamma^{2}(\nu (1-\mu))} \\ & \qquad {}\times E \biggl\Vert \int_{t-\epsilon}^{t} \int_{\delta}^{\infty}\theta (t-s)^{\nu(1-\mu)-1} s^{\mu-1}\varPsi_{\mu}(\theta) T\bigl(s^{\mu}\theta\bigr) \bigl[ x_{0}-g(x)\bigr] \,d\theta \,ds \biggr\Vert ^{2} \\ & \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} E \biggl\Vert \mu \int_{0}^{t} \int _{0}^{\delta}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)\bigl[T\bigl((t-s)^{\mu}\theta \bigr)\sigma \\ & \qquad {}- AT\bigl((t-s)^{\mu}\theta\bigr)\bigr] B u^{\kappa}(s) \,d\theta \,ds \biggr\Vert ^{2} \\ & \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} E \biggl\Vert \mu \int_{0}^{t} \int _{0}^{\delta}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta) T\bigl((t-s)^{\mu}\theta \bigr) f_{1}\bigl(s,x(s)\bigr) \,d\theta \,ds \biggr\Vert ^{2} \\ & \qquad {}+ 36 \sup_{t \in J} t^{2(1-\nu)(1-\mu)}E \biggl\Vert \mu \int_{0}^{t} \int _{0}^{\delta}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta\bigr) f_{2} \bigl(s,x(s)\bigr) \,d\theta \,d\omega(s) \biggr\Vert ^{2} \\ & \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} \\ & \qquad {}\times E \biggl\Vert \mu \int_{0}^{t} \int _{0}^{\delta}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta\bigr) G\bigl(s,x(s) \bigr) \,d\theta \,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} \\& \qquad {}\times E \biggl\Vert \mu \int_{0}^{t} \int _{0}^{\delta}\int_{V} \theta(t-s)^{\mu-1}\varPsi_{\mu}( \theta)T\bigl((t-s)^{\mu}\theta\bigr) h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv) \biggr\Vert ^{2}. \end{aligned}$$

We see that, for each $x\in B_{r}$, $\|\varPhi_{\kappa}x -\varPhi_{\kappa}^{\epsilon, \delta} x \|^{2}_{\bar{C}}\rightarrow0$ as $\epsilon \rightarrow0^{+}$ and $\delta\rightarrow0^{+}$. Therefore, there are precompact sets arbitrarily close to the set $V_{r}(t)$ and so $V_{r}(t)$ is precompact in X.

Next we prove that the family $\{\varPhi_{\kappa}x: x \in B_{r} \}$ is an equicontinuous family of functions. Let $x \in B_{r}$ and $t_{1},t_{2} \in J$ such that $0< t_{1} < t_{2}$, then

$$\begin{aligned}& E \bigl\Vert (\varPhi_{\kappa}x) (t_{2})-(\varPhi_{\kappa}x) (t_{1}) \bigr\Vert ^{2} \\& \quad \leq 36E \bigl\Vert \bigl(S_{\nu, \mu}(t_{2})- S_{\nu, \mu}(t_{1})\bigr)\bigl[ x_{0} -g(x)\bigr] \bigr\Vert ^{2}_{\bar{C}} \\& \qquad {}+36 E \biggl\Vert \int_{t_{1}}^{t_{2}} \bigl[P_{\mu}(t_{2}-s) \sigma-AP_{\mu}(t_{2}-s)\bigr] B u^{\kappa}(s)\,ds \biggr\Vert ^{2}_{\bar{C}} \\& \qquad {}+36E \biggl\Vert \int_{t_{1}}^{t_{2}} P_{\mu}(t_{2}-s)f_{1} \bigl(s,x(s)\bigr) \,ds \biggr\Vert ^{2}_{\bar {C}} \\& \qquad {}+36E \biggl\Vert \int_{t_{1}}^{t_{2}} P_{\mu}(t_{2}-s)f_{2} \bigl(s,x(s)\bigr) \,d\omega(s) \biggr\Vert ^{2}_{\bar{C}} \\& \qquad {}+36 E \biggl\Vert \int_{t_{1}}^{t_{2}} P_{\mu}(t_{2}-s)G \bigl(s,x(s)\bigr) \,dB^{H}(s) \biggr\Vert ^{2}_{\bar {C}} \\& \qquad {}+36 E \biggl\Vert \int_{t_{1}}^{t_{2}} P_{\mu}(t_{2}-s) \int_{V} h\bigl(s,x(s),v\bigr)\tilde {N}(ds, dv) \biggr\Vert ^{2}_{\bar{C}} \\& \qquad {}+ 36 E \biggl\Vert \int_{0}^{t_{1}}\bigl[\bigl(P_{\mu}(t_{2}-s) \sigma-A P_{\mu}(t_{2}-s)\bigr) \\& \qquad {}-\bigl(P_{\mu}(t_{1}-s)\sigma- A P_{\mu}(t_{1}-s)\bigr)\bigr] B u^{\kappa}(s)\,ds \biggr\Vert ^{2}_{\bar {C}} \\& \qquad {}+36 E \biggl\Vert \int_{0}^{t_{1}} \bigl[P_{\mu}(t_{2}-s)-P_{\mu}(t_{1}-s) \bigr]f_{1}\bigl(s,x(s)\bigr) \,ds \biggr\Vert ^{2}_{\bar{C}} \\& \qquad {}+ 36 E \biggl\Vert \int_{0}^{t_{1}} \bigl[P_{\mu}(t_{2}-s)-P_{\mu}(t_{1}-s)\bigr]f_{2}\bigl(s,x(s)\bigr)\,d\omega(s) \biggr\Vert ^{2}_{\bar{C}} \\& \qquad {}+ 36 E \biggl\Vert \int_{0}^{t_{1}} \bigl[P_{\mu}(t_{2}-s)-P_{\mu}(t_{1}-s)\bigr]G\bigl(s,x(s)\bigr)\,dB^{H}(s) \biggr\Vert ^{2}_{\bar{C}} \\& \qquad {}+36 E \biggl\Vert \int_{0}^{t_{1}} \bigl[P_{\mu}(t_{2}-s)-P_{\mu}(t_{1}-s) \bigr] \int_{V} h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv) \biggr\Vert ^{2}_{\bar{C}}. \end{aligned}$$

From the above fact, we see that $E \|(\varPhi_{\kappa}x)(t_{2})-(\varPhi _{\kappa}x)(t_{1})\|^{2}$ tends to zero independently of $x\in B_{r}$ as $t_{2}\rightarrow t_{1}$. Thus, $\varPhi_{\kappa}(B_{r})$ is both equicontinuous and bounded. By the Arzela–Ascoli theorem $\varPhi_{\kappa}(B_{r})$ is precompact in X. Hence $\varPhi_{\kappa}$ is a completely continuous operator on X. From the Schauder fixed point theorem, $\varPhi_{\kappa}$ has a fixed point in $B_{r}$. Any fixed point of $\varPhi_{\kappa}$ is a mild solution of (1.1) on J. The proof is completed. □

Theorem 3.2

Assume that (H1)–(H9) are satisfied. Further, if the functions $f_{1}$, $f_{2}$, G and h are uniformly bounded, then the system (1.1) is approximately controllable on J.

Proof

Let $x_{\kappa}$ be a fixed point of $\varPhi_{\kappa}$. By using the stochastic Fubini theorem, it can easily be seen that

$$\begin{aligned} x^{\kappa}(b) =&\bar{x}_{b} - \lambda\bigl(\lambda I + \varGamma^{b}_{0}\bigr)^{-1} \biggl\{ E \bar{x}_{b} - S_{\nu, \mu}(b)\bigl[x^{\kappa}_{0}-g \bigl(x^{\kappa}\bigr)\bigr]- \int_{0}^{b} P_{\mu}(b-s)f_{1} \bigl(s,x^{\kappa}(s)\bigr) \,ds \\ &{}- \int_{0}^{b} P_{\mu}(b-s)f_{2} \bigl(s,x^{\kappa}(s)\bigr)\,d\omega(s) - \int_{0}^{b} P_{\mu}(b-s)G \bigl(s,x^{\kappa}(s)\bigr) \,dB^{H}(s) \\ &{}- \int_{0}^{b} P_{\mu}(b-s) \int_{V} h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv) \\ &{}+ \int_{0}^{b} \bar{\psi}(s) \,d\omega(s)+ \int_{0}^{b} \bar{\varphi}(s) \,dB^{H}(s) \biggr\} . \end{aligned}$$

(3.4)

It follows from the assumption on $f_{1}$, $f_{2}$, G and h that there exists $D > 0$ such that

$$\begin{aligned}& \bigl\Vert f_{1}\bigl(s,x^{\kappa}(s)\bigr) \bigr\Vert ^{2} \leq D, \qquad \bigl\Vert f_{2}\bigl(s,x^{\kappa}(s) \bigr) \bigr\Vert ^{2}_{Q} \leq D, \\& \bigl\Vert G \bigl(s,x^{\kappa}(s)\bigr) \bigr\Vert ^{2}_{L^{0}_{2}} \leq D, \qquad \bigl\Vert h\bigl(s,x^{\kappa}(s),v\bigr) \bigr\Vert ^{2} \leq D. \end{aligned}$$

Consequently, the sequences $\{ f_{1}(s,x^{\kappa}(s))\}$, $\{f_{2}(s,x^{\kappa}(s))\}$, $\{G(s,x^{\kappa}(s))\}$, $\{h(s,x^{\kappa}(s),v)\}$ are weakly compact in $L_{2}(J,X)$, $L_{2}(L_{Q}(K,X))$, $L_{2}(L^{0}_{2}(Y,X))$ and $L_{2}(J,X)$, so there are subsequences, still denoted by $\{ f_{1}(s,x^{\kappa}(s))\}$, $\{ f_{2}(s,x^{\kappa}(s))\}$, $\{G(s,x^{\kappa}(s))\}$, $\{h(s,x^{\kappa}(s),v)\}$, that are weakly convergent to $\{ f_{1} (s)\}$, $\{f_{2} (s)\}$, $\{G(s)\}$, $\{ h(s,v)\}$ in $L_{2}(J,X)$, $L_{2}(L_{Q}(K,X))$, $L_{2}(L^{0}_{2}(Y,X))$ and $L_{2}(J,X)$.

From Eq. (3.4), we have

$$\begin{aligned}& E \bigl\Vert x^{\kappa}(b)-\bar{x}_{b} \bigr\Vert ^{2} \\& \quad \leq 9 E \bigl\Vert \lambda\bigl(\lambda I + \varGamma^{b}_{0}\bigr)^{-1} \bigl(E \bar{x}_{b} - S_{\nu, \mu}(b)\bigl[x^{\kappa}_{0}-g \bigl(x^{\kappa}\bigr)\bigr]\bigr) \bigr\Vert ^{2} \\& \qquad {} + 9E \biggl\Vert \lambda\bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \int_{0}^{b} P_{\mu}(b-s) \bigl(f_{1}\bigl(s,x^{\kappa}(s)\bigr)- f_{1}(s)\bigr) \,ds \biggr\Vert ^{2} \\& \qquad {} + 9E \biggl\Vert \lambda\bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \int_{0}^{b} P_{\mu}(b-s) f_{1}(s) \,ds \biggr\Vert ^{2} \\& \qquad {} + 9E \biggl\Vert \lambda \bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \int_{0}^{b} P_{\mu}(b-s) \bigl(f_{2}\bigl(s,x^{\kappa}(s)\bigr)- f_{2}(s)\bigr) \,d\omega(s) \biggr\Vert ^{2} \\& \qquad {} +9E \biggl\Vert \lambda\bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \int_{0}^{b} P_{\mu}(b-s) f_{2}(s) \,d\omega(s) \biggr\Vert ^{2} \\& \qquad {} + 9E \biggl\Vert \lambda\bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \int_{0}^{b} P_{\mu}(b-s) \bigl(G \bigl(s,x^{\kappa}(s)\bigr)- G(s)\bigr) \,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {} + 9E \biggl\Vert \lambda\bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \int_{0}^{b} P_{\mu}(b-s) g(s) \,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {} + 9E \biggl\Vert \lambda\bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \int_{0}^{b} P_{\mu}(b-s) \biggl( \int_{V} h\bigl(s,x^{\kappa}(s),v\bigr)- \int_{V} h(s,v)\biggr)\tilde{N}(ds, dv) \biggr\Vert ^{2} \\& \qquad {} + 9E \biggl\Vert \lambda\bigl(\lambda I + \varGamma^{b}_{0} \bigr)^{-1} \int _{0}^{b} P_{\mu}(b-s) \int_{V} h(s,v)\tilde{N}(ds, dv) \biggr\Vert ^{2}. \end{aligned}$$

On the other hand, by Lemma 3.1, the operator $\lambda(\lambda I + \varGamma^{b}_{0})^{-1} \rightarrow0$ strongly as $\lambda\rightarrow0^{+}$ for all $0 \leq s \leq b$, and $\| \lambda(\lambda I + \varGamma ^{b}_{0})^{-1} \| \leq1$ together with the Lebesgue dominated convergence theorem and the compactness of $P_{\mu}(t)$ implies that $E\| x_{\kappa}(b) - \bar{x}_{b} \|^{2} \rightarrow0$ as $\lambda\rightarrow0^{+}$. This proves the approximate controllability of (1.1). □

4 Null boundary controllability

In this section, we investigate the sufficient conditions for exact null controllability for the system (1.1), so we consider the fractional stochastic linear system with fractional Brownian motion and control on the boundary in the form

$$ \textstyle\begin{cases} D^{\nu,\mu}_{0+} y(t) = \sigma\mu(t)+f_{1}(t)+ f_{2}(t)\frac{d\omega (t)}{dt}+G(t)\frac{dB^{H}(t)}{dt} ,\quad t \in J = (0,b], \\ \tau y(t) = B_{1} u(t), \quad t \in J, \\ I^{(1-\nu)(1-\mu)}_{0+}y(0)=y_{0}, \end{cases} $$

(4.1)

associated with the system (1.1).

Consider

$$L_{0}^{b} u= \int_{0}^{b} \bigl[P_{\mu}(b-s)\sigma B - A P_{\mu}(b-s)B\bigr] u(s)\,ds :L_{2}(J, U)\rightarrow X, $$

where $L_{0}^{b} u$ has a bounded inverse operator $(L_{0})^{-1}$ with values in $L_{2}(J, U)/\operatorname{ker}(L_{0}^{b})$, and

$$\begin{aligned}& N_{0}^{b}(y, f_{1}, f_{2}, G) \\& \quad = S_{\nu, \mu}(b) y+ \int_{0}^{b} P_{\mu}(b-s)f_{1}(s) \,ds+ \int_{0}^{b} P_{\mu}(b-s)f_{2}(s) \,d\omega(s) \\& \qquad {}+ \int_{0}^{b} P_{\mu}(b-s)G(s) \,dB^{H}(s):X \times L_{2}(J, X)\rightarrow X. \end{aligned}$$

Definition 4.1

(see [16])

The system (4.1) is said to be exactly null controllable on J if $\operatorname{Im} L_{0}^{b} \supset \operatorname{Im} N_{0}^{b}$ or there exists a $\gamma>0$ such that $\|(L_{0}^{b})^{*}y \|^{2}\geq\gamma\|(N_{0}^{b})^{*} y\|^{2}$ for all $y \in X$.

Lemma 4.1

(see [39])

Suppose that the linear system (4.1) is exactly null controllable on J. Then the linear operator $(L_{0})^{-1} N^{b}_{0}:X\times L_{2}(J, X)\rightarrow L_{2}(J,U)$ is bounded and the control

$$\begin{aligned} u(t) =&-(L_{0})^{-1} \biggl[S_{\nu, \mu}(b) y_{0}+ \int_{0}^{b} P_{\mu}(b-s)f_{1}(s) \,ds+ \int_{0}^{b} P_{\mu}(b-s)f_{2}(s) \,d\omega(s) \\ &{}+ \int_{0}^{b} P_{\mu}(b-s)G(s) \,dB^{H}(s) \biggr](t) \end{aligned}$$

transfers the system (4.1) from $y_{0}$ to 0, where $L_{0}$ is the restriction of $L_{0}^{b}$ to $[\ker L_{0}^{b}]^{\perp}$.

To prove the null controllability for the system (1.1), we need in addition the hypothesis:

• (H10) The linear system (4.1) is exactly null controllable on J.

Theorem 4.1

If the hypotheses (H1)–(H10) are satisfied, then the boundary control system (1.1) is exactly null controllable on J provided that

$$\begin{aligned}& 36M^{2} \biggl[\frac{M^{2}_{2}}{\varGamma^{2}(\nu(1-\mu)+ \mu)}+\frac{ b^{\mu+2(1-\nu)(1-\mu)}[\delta_{1}+\delta_{2} \operatorname{Tr}(Q) + 2H b^{2H-1} \delta_{3}+\delta_{4}]}{\mu\varGamma^{2}(\mu)} \biggr] \\& \quad {}\times \biggl[1+ \frac{ b^{2\mu-1}\|(L_{0})^{-1}\|^{2} \|B \|^{2}(M^{2} \|\sigma\|^{2}+ M_{1}^{2})}{(2\mu-1)\varGamma^{2}(\mu)} \biggr] < 1. \end{aligned}$$

(4.2)

Proof

For an arbitrary $x(\cdot)$ define the operator Φ on C̄ as follows:

$$\begin{aligned} (\varPhi x) (t) =& S_{\nu, \mu}(t) \bigl(x_{0}-g(x)\bigr)+ \int_{0}^{t} \bigl[P_{\mu}(t-s)\sigma- AP_{\mu}(t-s)\bigr] B u(s)\,ds \\ &{}+ \int_{0}^{t} P_{\mu}(t-s)f_{1} \bigl(s,x(s)\bigr) \,ds+ \int_{0}^{t} P_{\mu}(t-s)f_{2} \bigl(s,x(s)\bigr)\,d\omega(s) \\ &{}+ \int_{0}^{t} P_{\mu}(t-s)G\bigl(s,x(s)\bigr) \,dB^{H}(s) \\ &{}+ \int_{0}^{t} P_{\mu}(t-s) \int_{V} h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv) , \quad t \in J, \end{aligned}$$

where

$$\begin{aligned} u(t) =&-(L_{0})^{-1}\biggl\{ S_{\nu, \mu}(b) \bigl( x_{0}-g(x)\bigr)+ \int_{0}^{b} P_{\mu}(b-s)f_{1} \bigl(s,x(s)\bigr)\,ds \\ &{}+ \int_{0}^{b} P_{\mu}(b-s) f_{2} \bigl(s,x(s)\bigr)\,d\omega(s) \,ds+ \int_{0}^{b} P_{\mu}(b-s) g\bigl(s,x(s) \bigr) \,dB^{H}(s) \\ &{}+ \int_{0}^{b} P_{\mu}(b-s) \int_{V} h\bigl(s,x(s),v\bigr)\tilde{N}(ds, dv) \biggr\} (t). \end{aligned}$$

It will be shown that the operator Φ from C̄ into itself has a fixed point.

We claim that there exists a positive number r such that $\varPhi(B_{r}) \subseteq B_{r}$. If it is not true, then, for each positive number r, there is a function $x_{r}(\cdot)\in B_{r}$, but $\varPhi(x_{r})\notin B_{r}$, that is, $\|\varPhi x_{r}\|^{2}_{\bar{C}}>r$ for some $t=t(r)\in J$, where $t(r)$ means that t is dependent of r.

From our hypotheses together with Lemma 2.4, Lemma 2.5, the Hölder inequality and Burkholder–Gungy’s inequality, we obtain

$$\begin{aligned} r \leq& \Vert \varPhi x_{r} \Vert ^{2}_{\bar{C}}= \sup_{t \in J}t^{2(1-\nu)(1-\mu )}E \bigl\Vert \varPhi(x_{r}) (t) \bigr\Vert ^{2} \\ \leq& \frac{36 M ^{2} [E \Vert x_{0} \Vert ^{2} +M^{2}_{2} r+ M^{2}_{3}]}{\varGamma^{2}(\nu (1-\mu)+ \mu)} + \frac{36 b^{2\mu-1} \Vert (L_{0})^{-1} \Vert ^{2} \Vert B \Vert ^{2} (M^{2} \Vert \sigma \Vert ^{2}+ M_{1}^{2})}{(2\mu-1)\varGamma^{2}(\mu)} \\ &{}\times\biggl\{ \frac{ M^{2} [E \Vert x_{0} \Vert ^{2} +M^{2}_{2} r+ M^{2}_{3}]}{\varGamma^{2}(\nu(1-\mu)+ \mu )}+ \frac{ M^{2}b^{\mu+2(1-\nu)(1-\mu)}}{\mu\varGamma^{2}(\mu)} \int _{0}^{b} (b-s)^{\mu-1} \rho_{r}(s) \,ds \\ &{}+\operatorname{Tr}(Q)\frac{ M^{2} b^{\mu+2(1-\nu)(1-\mu)}}{\mu\varGamma^{2}(\mu)} \int_{0}^{b} (b-s)^{\mu-1} h_{r}(s) \,ds \\ &{}+ \frac{2 H M^{2} b^{2H-1+\mu+2(1-\nu)(1-\mu)}}{\mu\varGamma^{2}(\mu )} \int_{0}^{b} (b-s)^{\mu-1} k_{r}(s) \,ds \\ &{}+\frac{ M^{2} b^{\mu+2(1-\nu )(1-\mu)}}{\mu\varGamma^{2}(\mu)} \int_{0}^{t}(t-s)^{\mu-1}\chi_{q}(s) \,ds \biggr\} \\ &{} + \frac{36 M^{2} b^{\mu+2(1-\nu)(1-\mu)}}{\mu\varGamma ^{2}(\mu)} \int_{0}^{t} (t-s)^{\mu-1} \rho_{r}(s) \,ds \\ &{}+ \frac{36M^{2} b^{\mu +2(1-\nu)(1-\mu)} \operatorname{Tr}(Q)}{\mu\varGamma^{2}(\mu)} \int_{0}^{t} (t-s)^{\mu -1} h_{r}(s) \,ds \\ &{}+ \frac{72 H M^{2} b^{2H-1+\mu+2(1-\nu )(1-\mu)}}{\mu\varGamma^{2}(\mu)} \int_{0}^{b} (t-s)^{\mu-1} k_{r}(s) \,ds \\ &{}+\frac{36 M^{2} b^{\mu+2(1-\nu)(1-\mu)}}{\mu\varGamma ^{2}(\mu)} \int_{0}^{t}(t-s)^{\mu-1}\chi_{q}(s) \,ds. \end{aligned}$$

(4.3)

Dividing both sides of (4.3) by r and taking the lower limit $r\rightarrow+\infty$, we get

$$\begin{aligned}& 36M^{2} \biggl[\frac{M^{2}_{2}}{\varGamma^{2}(\nu(1-\mu)+ \mu)}+\frac{ b^{\mu+2(1-\nu)(1-\mu)}[\delta_{1}+\delta_{2} \operatorname{Tr}(Q) + 2H b^{2H-1} \delta_{3}+\delta_{4}]}{\mu\varGamma^{2}(\mu)} \biggr] \\& \quad {}\times \biggl[1+ \frac{ b^{2\mu-1}\|(L_{0})^{-1}\|^{2} \|B \|^{2}(M^{2} \|\sigma\|^{2}+ M_{1}^{2})}{(2\mu-1)\varGamma^{2}(\mu)} \biggr] \geq1. \end{aligned}$$

This contradicts (4.2). Hence, for positive r, $\varPhi(B_{r})\subseteq B_{r}$ for positive number r.

In fact, the operator Φ maps $B_{r}$ into a compact subset of $B_{r}$. To prove this, we first show that the set $V_{r}(t)=\{(\varPhi x)(t):x\in B_{r}\}$ is precompact in X, for every fixed $t\in J$. This is trivial for $t=0$, since $V_{r}(0) = \{x_{0}\}$. Let t, $0 < t\leq b$, be fixed.

For $0 < \epsilon< t$ and arbitrary $\delta> 0$, take

$$\begin{aligned}& \bigl(\varPhi^{\epsilon, \delta}x\bigr) (t) \\& \quad = \frac{\mu}{\varGamma(\nu(1-\mu ))} \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\nu(1-\mu )-1}s^{\mu-1} \varPsi_{\mu}(\theta)T\bigl(s^{\mu}\theta\bigr) \bigl[x_{0}-g(x)\bigr] \,d\theta \,ds \\& \qquad {}+ \mu \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta (t-s)^{\mu-1} \varPsi_{\mu}(\theta)\bigl[T\bigl((t-s)^{\mu}\theta\bigr)\sigma- AT \bigl((t-s)^{\mu}\theta\bigr)\bigr] B u(s) \,d\theta \,ds \\& \qquad {}+ \mu \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu -1} \varPsi_{\mu}(\theta) T\bigl((t-s)^{\mu}\theta\bigr) f_{1}\bigl(s,x(s)\bigr) \,d\theta \,ds \\& \qquad {}+ \mu \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta (t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta\bigr) f_{2} \bigl(s,x(s)\bigr) \,d\theta \,d\omega(s) \\& \qquad {}+ \mu \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu -1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta\bigr) G\bigl(s,x(s) \bigr) \,d\theta \,dB^{H}(s) \\& \qquad {}+ \mu \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\int_{V} \theta (t-s)^{\mu-1}\varPsi_{\mu}( \theta)T\bigl((t-s)^{\mu}\theta\bigr) h\bigl(s,x(s),v\bigr) \,d\theta \tilde{N}(ds, dv) \\& \quad = \frac{\mu T(\epsilon^{\mu}\delta)}{\varGamma(\nu(1-\mu))} \int _{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\nu(1-\mu )-1}s^{\mu-1} \varPsi_{\mu}(\theta)T\bigl(s^{\mu}\theta-\epsilon^{\mu}\delta\bigr) \bigl[x_{0}-g(x)\bigr] \,d\theta \,ds \\& \qquad {} + \mu T\bigl(\epsilon^{\mu}\delta\bigr) \int _{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)\bigl[T\bigl((t-s)^{\mu}\theta- \epsilon^{\mu}\delta\bigr)\sigma \\& \qquad {}- AT\bigl((t-s)^{\mu}\theta- \epsilon^{\mu}\delta\bigr) \bigr] B u(s) \,d\theta \,ds \\& \qquad {}+ \mu T\bigl(\epsilon^{\mu}\delta\bigr) \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta) T\bigl((t-s)^{\mu}\theta- \epsilon^{\mu}\delta\bigr) f_{1}\bigl(s,x(s)\bigr) \,d\theta \,ds \\& \qquad {}+ \mu T\bigl(\epsilon^{\mu}\delta\bigr) \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta- T\bigl(\epsilon ^{\mu}\delta\bigr)\bigr) f_{2}\bigl(s,x(s)\bigr) \,d\theta \,d \omega(s) \\& \qquad {}+ \mu T\bigl(\epsilon^{\mu}\delta\bigr) \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta- \epsilon ^{\mu}\delta\bigr) G\bigl(s,x(s)\bigr)\,d\theta \,dB^{H}(s) \\& \qquad {}+ \mu T\bigl(\epsilon^{\mu}\delta\bigr) \int_{0}^{t-\epsilon} \int_{\delta}^{\infty}\int_{V} \theta(t-s)^{\mu-1}\varPsi_{\mu}( \theta)T\bigl((t-s)^{\mu}\theta- \epsilon^{\mu}\delta\bigr) \\& \qquad {}\times h \bigl(s,x(s),v\bigr) \,d\theta\tilde{N}(ds, dv). \end{aligned}$$

Since $T(\epsilon^{\mu}\delta)$, $\epsilon^{\mu}\delta> 0$ is a compact operator, the set $V_{r}^{\epsilon, \delta} (t) = \{ (\varPhi^{\epsilon , \delta}x)(t) : x \in B_{r} \}$ is a precompact set in X for every ϵ, $0 < \epsilon< t$, and for all $\delta> 0$. Moreover, for every $x \in B_{r}$, we have

$$\begin{aligned}& \bigl\Vert \varPhi x -\varPhi^{\epsilon, \delta} x \bigr\Vert ^{2}_{\bar{C}} \\& \quad =\sup_{t \in J}t^{2(1-\nu)(1-\mu)} E \bigl\Vert (\varPhi x) (t)-\bigl(\varPhi^{\epsilon, \delta} x\bigr) (t) \bigr\Vert ^{2} \\& \quad \leq \frac{36\mu^{2}\sup_{t \in J} t^{2(1-\nu)(1-\mu)}}{\varGamma ^{2}(\nu(1-\mu))} \\& \qquad {}\times E \biggl\Vert \int_{0}^{t} \int_{0}^{\delta}\theta(t-s)^{\nu(1-\mu )-1} s^{\mu-1}\varPsi_{\mu}(\theta) T\bigl(s^{\mu}\theta\bigr) \bigl[ x_{0}-g(x)\bigr] \,d\theta \,ds \biggr\Vert ^{2} \\& \qquad {} +36 \Vert B \Vert ^{2} \bigl\Vert (L_{0})^{-1} \bigr\Vert ^{2} \sup_{t \in J} t^{2(1-\nu)(1-\mu )}E \biggl\Vert \mu \int_{t-\epsilon}^{t} \int_{\delta}^{\infty}\theta(t-s)^{\mu -1} \varPsi_{\mu}(\theta)\bigl[T\bigl((t-s)^{\mu}\theta\bigr)\sigma \\& \qquad {}- AT\bigl((t-s)^{\mu}\theta\bigr)\bigr] \biggl\{ S_{\nu, \mu}(b) \bigl[x_{0}-g(x)\bigr]+ \int_{0}^{b} P_{\mu}(b- \zeta)f_{1}\bigl(\zeta,x(\zeta)\bigr)\,d\zeta \\& \qquad {}+ \int_{0}^{b} P_{\mu}(b-\zeta) f_{2}\bigl(\zeta,x(\zeta)\bigr)\,d\omega(\zeta) + \int_{0}^{b} P_{\mu}(b-\zeta) G\bigl( \zeta,x(\zeta)\bigr) \,dB^{H}(\zeta) \\& \qquad {}+ \int_{0}^{b} P_{\mu}(b-s) \int_{V} h\bigl(\zeta,x(\zeta),v\bigr)\tilde{N}(d\zeta , dv) \biggr\} \,d\theta \,ds \biggr\Vert ^{2} \\& \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} E \biggl\Vert \mu \int_{t-\epsilon }^{t} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta) T\bigl((t-s)^{\mu}\theta\bigr) f_{1}\bigl(s,x(s)\bigr) \,d\theta \,ds \biggr\Vert ^{2} \\& \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} \\& \qquad {}\times E \biggl\Vert \mu \int_{t-\epsilon}^{t} \int _{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta\bigr) f_{2}\bigl(s,x(s)\bigr) \,d\theta \,d\omega(s) \biggr\Vert ^{2} \\& \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} \\& \qquad {}\times E \biggl\Vert \mu \int_{t-\epsilon }^{t} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta )T\bigl((t-s)^{\mu}\theta\bigr) G\bigl(s,x(s) \bigr) \,d\theta \,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} \\& \qquad {}\times E \biggl\Vert \mu \int_{t-\epsilon }^{t} \int_{\delta}^{\infty}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta )T\bigl((t-s)^{\mu}\theta\bigr) \,d\theta \int_{V} h\bigl(s,x(s),v\bigr) \tilde{N}(ds, dv) \biggr\Vert ^{2} \\& \qquad {}+36 \Vert B \Vert ^{2} \bigl\Vert (L_{0})^{-1} \bigr\Vert ^{2} \sup_{t \in J} t^{2(1-\nu)(1-\mu)} E \biggl\Vert \mu \int_{0}^{t} \int_{0}^{\delta}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)\bigl[T\bigl((t-s)^{\mu}\theta\bigr)\sigma \\& \qquad {}- AT\bigl((t-s)^{\mu}\theta\bigr)\bigr] \biggl\{ S_{\nu, \mu}(b) \bigl[x_{0}-g(x)\bigr]+ \int_{0}^{b} P_{\mu}(b-\zeta )f_{1}\bigl(\zeta,x(\zeta)\bigr)\,d\zeta \\& \qquad {}+ \int_{0}^{b} P_{\mu}(b-\zeta) f_{2}\bigl(\zeta,x(\zeta)\bigr)\,d\omega(\zeta) + \int_{0}^{b} P_{\mu}(b-\zeta) G\bigl( \zeta,x(\zeta)\bigr) \,dB^{H}(\zeta) \\& \qquad {}+ \int_{0}^{b} P_{\mu}(b-s) \int_{V} h\bigl(\zeta,x(\zeta),v\bigr)\tilde{N}(d\zeta , dv) \biggr\} \,d\theta \,ds \biggr\Vert ^{2} \\& \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} E \biggl\Vert \mu \int_{0}^{t} \int _{0}^{\delta}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta) T\bigl((t-s)^{\mu}\theta \bigr) f_{1}\bigl(s,x(s)\bigr) \,d\theta \,ds \biggr\Vert ^{2} \\& \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} E \biggl\Vert \mu \int_{0}^{t} \int _{0}^{\delta}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta\bigr) f_{2} \bigl(s,x(s)\bigr) \,d\theta \,d\omega(s) \biggr\Vert ^{2} \\& \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} \\& \qquad {}\times E \biggl\Vert \mu \int_{0}^{t} \int _{0}^{\delta}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta\bigr) G\bigl(s,x(s) \bigr) \,d\theta \,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {}+ 36\sup_{t \in J} t^{2(1-\nu)(1-\mu)} \\& \qquad {}\times E \biggl\Vert \mu \int_{0}^{t} \int _{0}^{\delta}\theta(t-s)^{\mu-1} \varPsi_{\mu}(\theta)T\bigl((t-s)^{\mu}\theta\bigr) \,d\theta \int_{V} h\bigl(s,x(s),v\bigr) \tilde{N}(ds, dv) \biggr\Vert ^{2}. \end{aligned}$$

We see that, for each $x\in B_{r}$, $\|\varPhi x -\varPhi^{\epsilon, \delta} x \|^{2}_{\bar{C}}\rightarrow0$ as $\epsilon\rightarrow0^{+}$ and $\delta \rightarrow0^{+}$. Therefore, there are precompact sets arbitrarily close to the set $V_{r}(t)$ and so $V_{r}(t)$ is precompact in X.

Next we prove that the family $\{\varPhi x: x \in B_{r} \}$ is an equicontinuous family of functions. Let $x \in B_{r}$ and $t_{1},t_{2} \in J$ such that $0< t_{1} < t_{2}$, then

$$\begin{aligned}& E \bigl\Vert (\varPhi x) (t_{2})-(\varPhi x) (t_{1}) \bigr\Vert ^{2} \\& \quad \leq 36E \bigl\Vert \bigl(S_{\nu, \mu}(t_{2})-S_{\nu, \mu}(t_{1}) \bigr)\bigl[x_{0}-g(x)\bigr] \bigr\Vert ^{2} \\& \qquad {}+36 \Vert B \Vert ^{2} \bigl\Vert (L_{0})^{-1} \bigr\Vert ^{2} E \biggl\Vert \int_{t_{1}}^{t_{2}} \bigl[P_{\mu}(t_{2}-s)\sigma-AP_{\mu}(t_{2}-s)\bigr] \biggl\{ S_{\nu, \mu}(b) \bigl[x_{0}-g(x)\bigr] \\& \qquad {}+ \int_{0}^{b} P_{\mu}(b-\zeta)f_{1} \bigl(\zeta,x(\zeta)\bigr)\,d\zeta+ \int_{0}^{b} P_{\mu}(b-\zeta) f_{2}\bigl(\zeta,x(\zeta)\bigr)\,d\omega(\zeta) \\& \qquad {}+ \int_{0}^{b} P_{\mu}(b-\zeta) G\bigl( \zeta,x(\zeta)\bigr) \,dB^{H}(\zeta)+ \int_{0}^{b} P_{\mu}(b-\zeta) \int_{V} h\bigl(\zeta,x(\zeta),v\bigr) \tilde{N}(d\zeta, dv) \biggr\} \,ds \biggr\Vert ^{2} \\& \qquad {}+36E \biggl\Vert \int_{t_{1}}^{t_{2}} P_{\mu}(t_{2}-s)f_{1} \bigl(s,x(s)\bigr) \,ds \biggr\Vert ^{2}+36E \biggl\Vert \int _{t_{1}}^{t_{2}} P_{\mu}(t_{2}-s)f_{2} \bigl(s,x(s)\bigr) \,d\omega(s) \biggr\Vert ^{2} \\& \qquad {}+36 E \biggl\Vert \int_{t_{1}}^{t_{2}} P_{\mu}(t_{2}-s)G \bigl(s,x(s)\bigr) \,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {}+36 E \biggl\Vert \int_{t_{1}}^{t_{2}} P_{\mu}(t_{2}-s) \int_{V} h\bigl(s,x(s),v\bigr) \tilde{N}(ds, dv) \biggr\Vert ^{2} \\& \qquad {}+ 36 \Vert B \Vert ^{2} \bigl\Vert (L_{0})^{-1} \bigr\Vert ^{2} E \biggl\Vert \int_{0}^{t_{1}}\bigl[P_{\mu}(t_{2}-s)\sigma-P_{\mu}(t_{1}-s) \sigma-AP_{\mu}(t_{2}-s)+AP_{\mu}(t_{1}-s) \bigr] \\& \qquad {}\times\biggl\{ S_{\nu, \mu}(b) \bigl[x_{0}-g(x)\bigr] + \int_{0}^{b} P_{\mu}(b- \zeta)f_{1}\bigl(\zeta,x(\zeta)\bigr)\,d\zeta \\& \qquad {}+ \int_{0}^{b} P_{\mu}(b-\zeta) f_{2}\bigl(\zeta,x(\zeta)\bigr)\,d\omega(\zeta) \\& \qquad {}+ \int_{0}^{b} P_{\mu}(b-\zeta) G\bigl( \zeta,x(\zeta)\bigr) \,dB^{H}(\zeta)+ \int_{0}^{b} P_{\mu}(b-\zeta) \int_{V} h\bigl(\zeta,x(\zeta),v\bigr) \tilde{N}(d\zeta, dv) \biggr\} \,ds \biggr\Vert ^{2} \\& \qquad {}+36 E \biggl\Vert \int_{0}^{t_{1}} \bigl[P_{\mu}(t_{2}-s)-P_{\mu}(t_{1}-s) \bigr]f_{1}\bigl(s,x(s)\bigr) \,ds \biggr\Vert ^{2} \\& \qquad {}+ 36 E \biggl\Vert \int_{0}^{t_{1}} \bigl[P_{\mu}(t_{2}-s)-P_{\mu}(t_{1}-s)\bigr]f_{2}\bigl(s,x(s)\bigr)\,d\omega(s) \biggr\Vert ^{2} \\& \qquad {}+ 36 E \biggl\Vert \int_{0}^{t_{1}} \bigl[P_{\mu}(t_{2}-s)-P_{\mu}(t_{1}-s)\bigr]G\bigl(s,x(s)\bigr)\,dB^{H}(s) \biggr\Vert ^{2} \\& \qquad {}+ 36 E \biggl\Vert \int_{0}^{t_{1}} \bigl[P_{\mu}(t_{2}-s)-P_{\mu}(t_{1}-s) \bigr] \int_{V} h\bigl(s,x(s),v\bigr) \tilde{N}(ds, dv) \biggr\Vert ^{2}. \end{aligned}$$

From the above fact, we see that $E \|(\varPhi x)(t_{2})-(\varPhi x)(t_{1})\|^{2}$ tends to zero independently of $x\in B_{r}$ as $t_{2}\rightarrow t_{1}$. Thus, $\varPhi(B_{r})$ is both equicontinuous and bounded. By the Arzela–Ascoli theorem $\varPhi(B_{r})$ is precompact in X. Hence Φ is a completely continuous operator on X. From the Schauder fixed point theorem, Φ has a fixed point in $B_{r}$. Any fixed point of Φ is a mild solution of (1.1) on J. Therefore the system (1.1) is exact null controllable on J. □

5 Applications

Let us consider the nonlocal Hilfer fractional stochastic partial differential system with fractional Brownian motion and Poisson jump in the following form:

$$ \textstyle\begin{cases} D_{0+}^{\nu, \frac{4}{7}}(x(t, \xi)) =\frac{\partial^{2}}{\partial \xi^{2} } x(t,\xi)+ F_{1}(t, x(t,\xi)) + F_{2}(t, x(t,\xi)) \frac {d\omega(t)}{dt} \\ \hphantom{D_{0+}^{\nu, \frac{4}{7}}(x(t, \xi)) ={}}{}+F_{3}(t, x(t,\xi)) \frac{dB^{H}(t)}{dt} \\ \hphantom{D_{0+}^{\nu, \frac{4}{7}}(x(t, \xi)) ={}}{}+ \int_{V} F_{4} (t, x(t, \xi), v) \tilde{N} (dt, dv),\quad t \in J=(0,b], \xi \in\varPi, \\ x(t,\xi)= u(t,\xi), \quad t \in J, \xi\in\varGamma, \\ I^{\frac{3(1-\nu)}{7}}_{0+}x(0,\xi)+\sum_{i=1}^{m} c_{i} x(t_{i}, \xi )=x_{0}(\xi), \quad \xi\in\varPi, \end{cases} $$

(5.1)

where $D_{0+}^{\nu,\frac{4}{7}}$ is the Hilfer fractional derivative, $0\leq\nu\leq1$, $\mu=\frac{4}{7}$, m is a positive integer, $0 < t_{1}< t_{2}<\cdots<t_{m}< b$, Π is a bounded and open subset of $R^{n}$ with sufficiently smooth boundary Γ, $\omega (t)$ is Wiener process, $B^{H}$ is a fractional Brownian motion with Hurst parameter $H \in(\frac{1}{2},1)$ and $u \in L_{2}(\varGamma)$.

The functions $x(t)(\xi)=x(t,\xi)$, $f_{1}(t,x(t))(\xi)= F_{1}(t, x(t,\xi))$, $f_{2}(t,x(t))(\xi)=F_{2}(t, x(t,\xi))$, $G(t,x(t))(\xi )=F_{3}(t, x(t,\xi))$, $h(t,x(t),v)(\xi)=F_{1}(t, x(t,\xi), v)$, and $g(x)(\xi)=\sum_{i=1}^{m} c_{i} x(t_{i}, \xi)$.

Let $X=Y=K=L_{2}(\varPi)$, $U=L_{2}(\varGamma)$ $B_{1}=I$, the identity operator and $\sigma x = \frac{\partial^{2} x}{\partial\xi^{2} }$ with domain $D(\sigma)= \{ x \in L_{2}(\varPi): \frac{\partial^{2}x}{\partial\xi^{2} } \in L_{2}(\varPi)\}$.

The operator θ is the trace operator such that $\theta x = x|_{\varGamma}$ is well defined and belongs to $H^{-1/2}(\varGamma)$ for each $x \in D(\sigma)$.

Define the operator $A:D(A) \subset X \rightarrow X$ given by $A x= \Delta x$ with domain $D(A) = H_{0}^{1}(\varPi)\cup H^{2}(\varPi)$ where $H^{k}(\varPi)$, $H^{\mu}(\varGamma)$ and $H_{0}^{1}(\varPi)$ are the usual Sobolev spaces on Π, Γ. Then A can be written as

$$Ax=\sum_{n=1}^{\infty}\bigl(-n^{2} \bigr) (x, x_{n})x_{n}, \quad x \in D(A), $$

where $x_{n}(y)=\sqrt{2} \sin ny$, $n=1, 2, 3,\ldots$ , is the orthogonal set of eigenvectors of A.

It is well known that A generates a compact semigroup $\{T(t), t \geq 0 \}$ in X and

$$T(t)x= \sum_{n=1}^{\infty}e^{\frac{-n^{2}}{1+n^{2}}t} (x, x_{n})x_{n}, \quad x \in X, $$

with

$$\bigl\Vert T(t) \bigr\Vert \leq e^{-t}\leq1. $$

Moreover, the two operators $S_{\nu, \frac{4}{7}}(t)$ and $P_{\frac {4}{7}}(t)$ can be defined by

$$\begin{aligned}& S_{\nu, \frac{4}{7}}(t) x=\frac{4}{7\varGamma(\frac{3\nu}{7})} \int _{0}^{t} \int_{0}^{\infty}\theta(t-s)^{\frac{3\nu}{7}-1} s^{\frac{3}{7}} \xi_{\frac{4}{7}}(\theta)T\bigl(t^{\frac{4}{7}} \theta\bigr) x \,d\theta \,ds, \\& P_{\frac{4}{7}}(t )x = \frac{4}{7} \int_{0}^{\infty}\theta(t-s)^{\frac {3}{7}} \xi_{\frac{4}{7}}(\theta)T\bigl((t-s)^{\frac{4}{7}} \theta\bigr) x \,d\theta. \end{aligned}$$

Clearly,

$$ \bigl\Vert P_{\frac{4}{7}}(t) \bigr\Vert \leq\frac{ t^{\frac{3}{7}}}{\varGamma(\frac {4}{7})}, \qquad \bigl\Vert S_{\nu,\frac{4}{7}}(t) \bigr\Vert \leq\frac{ t^{\frac{3}{7}(\nu -1)}}{\varGamma(\frac{3}{7}\nu+ \frac{4}{7})}. $$

Define the fractional Brownian motion in Y by

$$B^{H}(t)= \sum_{n= 1}^{\infty}\sqrt{ \lambda_{n}} \beta^{H}(t) e_{n}, $$

where $H \in(\frac{1}{2},1)$ and $\{\beta^{H}_{n} \}_{n \in N}$ is a sequence of one-dimensional fractional Brownian motions that are mutually independent.

Also, we define the linear operator $B : L_{2}(\varGamma) \rightarrow L_{2}(\varPi)$ by $Bu = v_{u}$, where $v_{u}$ is the unique solution to the Dirichlet boundary value problem,

$$\begin{aligned}& \Delta v_{u} = 0 \quad \mbox{in }\varPi, \\& v_{u} = u \quad \mbox{in }\varGamma. \end{aligned}$$

We consider the fractional linear system

$$ \textstyle\begin{cases} D_{0+}^{\nu, \frac{4}{7}}(x(t, \xi)) =\frac{\partial^{2}}{\partial \xi^{2} } x(t,\xi)+ F_{1}(t, \xi) + F_{2}(t, \xi) \frac{d\omega (t)}{dt} \\ \hphantom{D_{0+}^{\nu, \frac{4}{7}}(x(t, \xi)) ={}}{}+F_{3}(t, \xi) \frac{dB^{H}(t)}{dt}, \quad t \in J=(0,b], \xi\in\varPi, \\ x(t,\xi)= u(t,\xi),\quad t \in J, \xi\in\varGamma, \\ I^{\frac{3(1-\nu)}{7}}_{0+}x(0,\xi)=x_{0}(\xi),\quad \xi\in\varPi . \end{cases} $$

(5.2)

The system (5.2) has exact null controllability if there is a $\gamma >0$, such that

$$ \int_{0}^{b} \bigl\Vert B^{*}\bigl[P_{\frac{4}{7}}(b-s) \sigma- A P_{\frac {4}{7}}(b-s)\bigr]^{*}y \bigr\Vert ^{2}\,ds\geq \gamma\biggl[ \bigl\Vert S_{\nu,{\frac{4}{7}}}^{*}(b) y \bigr\Vert ^{2}+ \int_{0}^{b} \bigl\Vert P_{\frac{4}{7}}^{*}(b-s)y \bigr\Vert ^{2}\,ds\biggr], $$

or equivalently

$$ \int_{0}^{b} \bigl\Vert B \bigl[P_{\frac{4}{7}}(b-s) \sigma- A P_{\frac{4}{7}}(b-s)\bigr]y \bigr\Vert ^{2}\,ds\geq\gamma \biggl[ \bigl\Vert S_{\nu,{\frac{4}{7}} }(b) y \bigr\Vert ^{2}+ \int_{0}^{b} \bigl\Vert P_{\frac{4}{7}}(b-s)y \bigr\Vert ^{2} \,ds\biggr]. $$

Hence, the linear fractional system (5.2) is exactly null controllable on J. So the hypothesis (H10) is satisfied. Hence, all the hypotheses of Theorem 4.1 are satisfied and

$$\begin{aligned}& 36M^{2} \biggl[\frac{M^{2}_{2}}{\varGamma^{2}(\nu(1-\mu)+ \mu)}+\frac{ b^{\mu+2(1-\nu)(1-\mu)}[\delta_{1}+\delta_{2} \operatorname{Tr}(Q) + 2H b^{2H-1} \delta_{3}+\delta_{4}]}{\mu\varGamma^{2}(\mu)} \biggr] \\& \quad {}\times \biggl[1+ \frac{ b^{2\mu-1}\|(L_{0})^{-1}\|^{2} \|B \|^{2}(M^{2} \|\sigma\|^{2}+ M_{1}^{2})}{(2\mu-1)\varGamma^{2}(\mu)} \biggr] < 1. \end{aligned}$$

Then the system (5.1) is exactly null controllable on J.