In this section, we discuss the controllability criteria for the following higher-order fractional damped stochastic system with distributed delay:
$$\begin{aligned} &{^{C}_{0}D^{{\eta _{1}}}_{t}} z(t) - \mathfrak{B} {^{C}_{0}D^{{\eta _{2}}}_{t}} z(t)= \mathfrak{C}u(t)+ \int _{-h}^{0} d_{\lambda } \mathfrak{D}(t,{ \lambda })u(t+{\lambda })+\mathfrak{f}\bigl(t,z(t)\bigr) \\ &\hphantom{{^{C}_{0}D^{{\eta _{1}}}_{t}} z(t) - \mathfrak{B} {^{C}_{0}D^{{\eta _{2}}}_{t}} z(t)=}{} + \sigma \bigl(t,z(t)\bigr) \frac{dw(t)}{dt},\quad t \in [0,{ \mathcal{T}}], \end{aligned}$$
(18)
$$\begin{aligned} &z(0)=z_{0},\qquad z^{\prime }(0)=z_{1},\qquad \ldots,\qquad z^{\rho -1}=z_{\rho -1}, \end{aligned}$$
(19)
$$\begin{aligned} &u(t)=\vartheta (t), \quad {\lambda } \leq t\leq 0, \end{aligned}$$
(20)
where \(\rho -1<{\eta _{1}}\leq \rho \), \(\mu -1<{\eta _{2}} \leq \mu \), and \(\mu \leq \rho -1\), \(\mathfrak{B}\), \(\mathfrak{C}\), and \(\mathfrak{D}(t,{\lambda })\) are the same as defined in the previous section, λ is a negative constant, \(z\in \mathbb{R}^{n}\), \(u(t)\in \mathbb{R}^{m}\), \(\mathfrak{f}:\mathcal{J}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\), and \(\sigma :\mathcal{J}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n \times m}\). \(w(t)\) is a given m-dimensional Wiener process with the filtration \(\mathcal{F}_{t}\) generated by \(w({\omega })\). Then the solution of system (18)–(20) is given by
$$\begin{aligned} z(t)={}&\sum_{r=0}^{\rho -1}z^{r}(0)t^{r} E_{{\eta _{1}}-{\eta _{2}},1+r}\bigl( \mathfrak{B}t^{{\eta _{1}}-{\eta _{2}}}\bigr)-\sum _{r=0}^{\mu -1}z^{r}(0)t^{{ \eta _{1}}-{\eta _{2}}+r} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}-{ \eta _{2}}+1+r}\bigl(\mathfrak{B}t^{{\eta _{1}}-{\eta _{2}}}\bigr) \\ &{}+ \int _{-h}^{0} \biggl[ \int _{{\lambda }}^{0}\bigl(t-({\omega }-{\lambda }) \bigr)^{{ \eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({ \omega }-{ \lambda })\bigr)^{{\eta _{1}}-{\eta _{2}}}\bigr) \\ &{}\times \mathfrak{D}({ \omega }-{\lambda },{\lambda })\vartheta ({\omega }) \,d{\omega } \biggr]\,d\mathfrak{D}_{{\lambda }} \\ &{}+ \int _{0}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{f} \bigl({\omega },z({\omega })\bigr)\,d{\omega } \\ &{}+ \int _{0}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \biggl( \int _{0}^{\eta }\sigma (\theta )\,dw(\theta ) \biggr)d{ \omega } \\ &{}+ \int _{0}^{t+{\lambda }} [(t-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{ \eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{C} \\ &{}+ \int _{-h}^{0} \bigl(t-({\omega }-{\lambda }) \bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{ \eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({\omega }-{ \lambda })\bigr)^{{ \eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{D}({\omega }-{\lambda },{\lambda })\,d\mathfrak{D}_{{\lambda }} ]u({\omega })\,d{\omega } \\ &{}+ \int _{t+{\lambda }}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{ \eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{ \eta _{2}}}\bigr) \mathfrak{C}u({\omega })\,d{\omega }. \end{aligned}$$
(21)
Fix the control function
$$\begin{aligned} &u (t )= \textstyle\begin{cases} \mathbb{G}_{1}^{*}({\mathcal{T}},t)W^{-1}(\hat{\gamma }),\quad t \in [0,{\mathcal{T}}+{\lambda }], \\ \mathbb{G}_{2}^{*}({\mathcal{T}},t)W^{-1}(\hat{\gamma }), \quad t \in [{\mathcal{T}}+{\lambda },{\mathcal{T}}], \end{cases}\displaystyle \end{aligned}$$
(22)
$$\begin{aligned} &\mathbb{G}_{1}({\mathcal{T}},t)= \biggl[({\mathcal{T}}-{\omega })^{{ \eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({ \mathcal{T}}-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{C}, \\ &\mathbb{G}_{1}({\mathcal{T}},t)={}+ \int _{-h}^{0} \bigl({\mathcal{T}}-({\omega }-{ \lambda })\bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl({ \mathcal{T}}-({ \omega }-{\lambda })\bigr)^{{\eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{D}({ \omega }-{\lambda },{\lambda })\,d\mathfrak{D}_{{\lambda }} \biggr], \\ &\mathbb{G}_{2}({\mathcal{T}},t)= \bigl[({\mathcal{T}}-{\omega })^{{ \eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({ \mathcal{T}}-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{C} \bigr], \\ &\hat{\gamma }=1/2 \Biggl[z_{\mathcal{T}}-\sum_{r=0}^{\rho -1}z^{r}(0){ \mathcal{T}}^{r} E_{{\eta _{1}}-{\eta _{2}},1+r}\bigl(\mathfrak{B} { \mathcal{T}}^{{\eta _{1}}-{\eta _{2}}}\bigr) \\ &\hphantom{\hat{\gamma }=}{}+\sum_{r=0}^{\mu -1}z^{r}(0){ \mathcal{T}}^{{\eta _{1}}-{\eta _{2}}+r} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}-{\eta _{2}}+1+r}\bigl(\mathfrak{B} { \mathcal{T}}^{{\eta _{1}}-{ \eta _{2}}}\bigr) \\ &\hphantom{\hat{\gamma }=}{}- \int _{-h}^{0} \biggl[ \int _{{\lambda }}^{0}\bigl({\mathcal{T}}-({\omega }-{ \lambda })\bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl( \mathfrak{B}\bigl({ \mathcal{T}}-({\omega }-{\lambda })\bigr)^{{\eta _{1}}-{ \eta _{2}}}\bigr)\mathfrak{D}({\omega }-{\lambda },{\lambda }) \\ &\hphantom{\hat{\gamma }=}{}\times \vartheta ({\omega }) \,d{\omega } \biggr]\,d\mathfrak{D}_{{ \lambda }}- \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{ \omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl( \mathfrak{B}({\mathcal{T}}-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \biggl( \int _{0}^{\eta }\sigma (\theta )\,dw(\theta ) \biggr)d{ \omega } \\ &\hphantom{\hat{\gamma }=}{}- \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{f}\bigl({\omega },z({ \omega })\bigr) \Biggr]. \end{aligned}$$
(23)
We assume the following hypotheses.
-
(H1)
The linear fractional damped stochastic dynamical system with distributed delay (5)–(7) is controllable on \(\mathcal{J}\).
-
(H2)
The functions \(\mathfrak{f}\) and σ are continuous and satisfy the usual linear growth condition, that is, there exist positive real constants Ñ, L̃ such that
$$\begin{aligned} \bigl\Vert \mathfrak{f}(t,z) \bigr\Vert ^{2}\leq \tilde{N} \bigl(1+ \Vert z \Vert ^{2}\bigr),\qquad \bigl\Vert \sigma (t,z) \bigr\Vert ^{2}\leq \tilde{L} \bigl(1+ \Vert z \Vert ^{2} \bigr). \end{aligned}$$
-
(H3)
The functions \(\mathfrak{f}\), σ satisfy the following Lipschitz condition, and for every \(t\geq 0\) and \(z,y \in \mathbb{R}^{n}\), there exist positive real constants N, L such that
$$\begin{aligned} \bigl\Vert \mathfrak{f}(t,z)-\mathfrak{f}(t,y) \bigr\Vert ^{2}\leq N \Vert z-y \Vert ^{2},\qquad \bigl\Vert \sigma (t,z)-\sigma (t,y) \bigr\Vert ^{2}\leq L \Vert z-y \Vert ^{2} . \end{aligned}$$
For brevity, let us introduce the following notations:
$$\begin{aligned} &u_{1}= \bigl\Vert t^{r} E_{{\eta _{1}}-{\eta _{2}},1+r}\bigl( \mathfrak{B}t^{{\eta _{1}}-{ \eta _{2}}}\bigr) \bigr\Vert ^{2}, \\ &u_{2}= \bigl\Vert \mathfrak{B}t^{{\eta _{1}}-{\eta _{2}}+r} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}-{\eta _{2}}+1+r}\bigl( \mathfrak{B}t^{{ \eta _{1}}-{\eta _{2}}}\bigr) \bigr\Vert ^{2}, \\ &u=\|\vartheta ({\omega })\|^{2}, \\ &v= \int _{{\lambda }}^{0}\mathbb{E}\|\bigl(t-({ \omega }-{ \lambda })\bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({\omega }-{\lambda })\bigr)^{{\eta _{1}}-{ \eta _{2}}}\bigr)\mathfrak{D}({\omega }-{\lambda },{ \lambda }) \,d{\omega } \|^{2}, \\ &G= \int _{-h}^{0} v u \,d\mathfrak{D}_{{\lambda }},\qquad l= \bigl\Vert W^{-1} \bigr\Vert , \\ & u_{3}= \bigl\Vert E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \bigr\Vert ^{2}, \\ &M= \biggl\Vert \biggl[(t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl( \mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{C} \\ &\hphantom{M=}{} + \int _{-h}^{0} \bigl(t-({\omega }-{\lambda }) \bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{ \eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({\omega }-{ \lambda })\bigr)^{{ \eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{D}({\omega }-{\lambda },{\lambda })\,d\mathfrak{D}_{{\lambda }} \biggr] \biggr\Vert ^{2} \\ &\tilde{M}= \bigl\Vert (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl( \mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{C} \bigr\Vert ^{2}. \end{aligned}$$
(24)
Theorem 4.1
Assume that hypotheses (H1)–(H3) hold, then the nonlinear system (18)–(20) is controllable on \(\mathcal{J}\).
Proof
For arbitrary initial data, we can define a nonlinear operator Δ from I to I as follows:
$$\begin{aligned} (\Delta z) (t)={}&\sum_{r=0}^{\rho -1}z^{r}(0)t^{r} E_{{\eta _{1}}-{ \eta _{2}},1+r}\bigl(\mathfrak{B}t^{{\eta _{1}}-{\eta _{2}}}\bigr)-\sum _{r=0}^{ \mu -1}z^{r}(0)t^{{\eta _{1}}-{\eta _{2}}+r} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}-{\eta _{2}}+1+r}\bigl(\mathfrak{B}t^{{\eta _{1}}-{\eta _{2}}}\bigr) \\ &{}+ \int _{-h}^{0} \biggl[ \int _{{\lambda }}^{0} \bigl(t-({ \omega }-{\lambda }) \bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}\bigl(t- ({\omega }-{ \lambda })\bigr)^{{\eta _{1}}-{ \eta _{2}}}\bigr) \\ &{}\times \mathfrak{D}({\omega }-{\lambda },{\lambda }) \vartheta ({ \omega }) \,d{\omega } \biggr]\,d\mathfrak{D}_{{\lambda }} \\ &{}+ \int _{0}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{f} \bigl({\omega },z({\omega })\bigr)\,d{\omega } \\ &{}+ \int _{0}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \biggl( \int _{0}^{\eta }\sigma (\theta )\,dw(\theta ) \biggr)d{ \omega } \\ &{}+ \int _{0}^{t+{\lambda }} \biggl[(t-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{ \eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{C} \\ &{}+ \int _{-h}^{0} \bigl(t-({\omega }-{\lambda }) \bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{ \eta _{2}},{\eta _{1}} }\bigl(\mathfrak{B}\bigl(t-({\omega }-{ \lambda })\bigr)^{{ \eta _{1}}-{\eta _{2}}}\bigr) \\ & {}\times \mathfrak{D}({\omega }-{\lambda },{\lambda })\,d\mathfrak{D}_{{\lambda }} \biggr]u({\omega })\,d{\omega } \\ &{}+ \int _{t+{\lambda }}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{ \eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{ \eta _{2}}}\bigr) \mathfrak{C}u({\omega })\,d{\omega }, \end{aligned}$$
(25)
where \(u(t)\) is defined by (22).
By Theorem 3.2, control (22) transfers (21) from the initial state \(z_{0}\) to the final state \(z_{\mathcal{T}}\) provided that the operator Δ has a fixed point in I. So, if the operator Δ has a fixed point, then system (18)–(20) is controllable. As mentioned before, to prove the controllability of system (18)–(20), it is enough to show that Δ has a fixed point in I. To do this, we can employ the contraction mapping principle. In the following, we will divide the proof into two steps.
Firstly, we show that Δ maps I into itself. From (25) we have
$$\begin{aligned} &\sup_{0\leq t \leq {\mathcal{T}}}\mathbb{E} \bigl\Vert (\Delta z) (t) \bigr\Vert ^{2} \\ &\quad =7\sup_{0\leq t \leq {\mathcal{T}}}\mathbb{E} \Biggl\Vert \sum _{r=0}^{ \rho -1}z^{r}(0)t^{r} E_{{\eta _{1}}-{\eta _{2}},1+r}\bigl(\mathfrak{B}t^{{ \eta _{1}}-{\eta _{2}}}\bigr) \Biggr\Vert ^{2} \\ &\qquad {}+7\sup_{0\leq t \leq {\mathcal{T}}}\mathbb{E} \Biggl\Vert \sum _{r=0}^{ \mu -1}z^{r}(0)t^{{\eta _{1}}-{\eta _{2}}+r} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}-{\eta _{2}}+1+r}\bigl(\mathfrak{B}t^{{\eta _{1}}-{\eta _{2}}}\bigr) \Biggr\Vert ^{2} \\ &\qquad {}+7\sup_{0\leq t \leq {\mathcal{T}}}\mathbb{E} \biggl\Vert \int _{-h}^{0} \biggl[ \int _{{\lambda }}^{0}\bigl(t-({\omega }-{\lambda }) \bigr)^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({\omega }-{ \lambda })\bigr)^{{\eta _{1}}-{\eta _{2}}}\bigr) \\ &\qquad {}\times \mathfrak{D}({\omega }-{\lambda },{\lambda })\vartheta ({ \omega }) d{ \omega } \biggr]\,d\mathfrak{D}_{{\lambda }} \biggr\Vert ^{2} \\ &\qquad {}+7\sup_{0\leq t \leq {\mathcal{T}}}\mathbb{E}\biggl\| \int _{0}^{t} (t-{ \omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl( \mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{f}\bigl({ \omega },z({\omega })\bigr)\,d{\omega } \biggr\Vert ^{2} \\ &\qquad {}+7\sup_{0\leq t \leq {\mathcal{T}}}\mathbb{E} \biggl\Vert \int _{0}^{t} (t-{ \omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl( \mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \biggl( \int _{0}^{\eta }\sigma (\theta )\,dw(\theta ) \biggr)d{ \omega } \biggr\Vert ^{2} \\ &\qquad {}+7\sup_{0\leq t \leq {\mathcal{T}}}\mathbb{E} \biggl\| \int _{0}^{t+{ \lambda }} \biggl[(t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{C} \\ &\qquad {}+ \int _{-h}^{0} \bigl(t-({\omega }-{\lambda }) \bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{ \eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({\omega }-{ \lambda })\bigr)^{{ \eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{D}({\omega }-{\lambda },{\lambda })\,d\mathfrak{D}_{{\lambda }} \biggr]u({\omega })\,d{\omega } \biggr\Vert ^{2} \\ &\qquad {}+7\sup_{0\leq t \leq {\mathcal{T}}}\mathbb{E} \biggl\| \int _{t+{ \lambda }}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{C}u({\omega })\,d{\omega }\biggr\| ^{2} \\ &\quad \triangleq \sum_{b=1}^{7} \mathcal{R}_{b}. \end{aligned}$$
(26)
Using Holder’s inequality, Burkholder–Davis–Gundy’s inequality (here \(C_{1} =4\)), and (24), we have the following estimates:
$$\begin{aligned} &\mathcal{R}_{1}\leq 7 \sum_{r=0}^{\rho -1} \mathbb{E} \bigl\Vert z_{r}t^{r} E_{{\eta _{1}}-{\eta _{2}},1+r}\bigl( \mathfrak{B}t^{{\eta _{1}}-{\eta _{2}}}\bigr) \bigr\Vert ^{2} \leq 7u_{1}\sum_{r=0}^{\rho -1} \mathbb{E} \Vert z_{r} \Vert ^{2}, \end{aligned}$$
(27)
$$\begin{aligned} &\mathcal{R}_{2}\leq 7 \sum_{r=0}^{\mu -1} \mathbb{E} \bigl\Vert z_{r} \mathfrak{B}t^{{\eta _{1}}-{\eta _{2}}+r} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}-{\eta _{2}}+1+r}\bigl(\mathfrak{B}t^{{\eta _{1}}-{\eta _{2}}}\bigr) \bigr\Vert ^{2} \leq 7u_{2} \sum_{r=0}^{\mu -1} \mathbb{E} \Vert z_{r} \Vert ^{2}, \end{aligned}$$
(28)
$$\begin{aligned} &\mathcal{R}_{3}\leq 7\mathbb{E} \biggl\Vert \int _{-h}^{0} \biggl[ \int _{{ \lambda }}^{0}\bigl(t-({\omega }-{\lambda }) \bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{ \eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({\omega }-{ \lambda })\bigr)^{{ \eta _{1}}-{\eta _{2}}}\bigr) \\ &\hphantom{\mathcal{R}_{3}\leq}{}\times \mathfrak{D}({\omega }-{\lambda },{\lambda })\vartheta ({ \omega }) d{ \omega } \biggr]\,d\mathfrak{D}_{{\lambda }} \biggr\Vert ^{2} \leq 7G, \end{aligned}$$
(29)
$$\begin{aligned} &\mathcal{R}_{4} \leq 7 \mathbb{E}\| \int _{0}^{t} (t-{\omega })^{{ \eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{f}\bigl({\omega },z({ \omega })\bigr)\,d{\omega }\|^{2} \\ &\hphantom{\mathcal{R}_{4}}\leq 7 u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \tilde{N} {\mathcal{T}} \int _{0}^{{\mathcal{T}}}\bigl(1+\mathbb{E}\|z({ \omega }) \|^{2}\bigr)\,d{\omega } , \end{aligned}$$
(30)
$$\begin{aligned} &\mathcal{R}_{5} \leq 28 \mathbb{E} \biggl\Vert \int _{0}^{t} (t-{\omega })^{{ \eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \biggl( \int _{0}^{\eta }\sigma \bigl( \theta ,z(\theta ) \bigr)\,dw(\theta ) \biggr)\,d{\omega } \biggr\Vert ^{2} \\ &\hphantom{\mathcal{R}_{5}}\leq 28 u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} L_{\sigma } \tilde{L} \int _{0}^{{\mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(1+ \mathbb{E} \bigl\Vert z( \theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega }, \end{aligned}$$
(31)
$$\begin{aligned} &\mathcal{R}_{6} \leq 7 \mathbb{E}\biggl\| \int _{0}^{t+{\lambda }} \biggl[(t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl( \mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{C} \\ &\hphantom{\mathcal{R}_{6} \leq}{}+ \int _{-h}^{0} \bigl(t-({\omega }-{\lambda }) \bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{ \eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({\omega }-{ \lambda })\bigr)^{{ \eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{D}({\omega }-{\lambda },{\lambda })\,d\mathfrak{D}_{{\lambda }} \biggr]u({\omega })\,d{\omega } \biggr\| ^{2} \\ &\hphantom{\mathcal{R}_{6}}\leq 14\mathbb{E} \Biggl\Vert \int _{0}^{t+{\lambda }} \biggl[(t-{\omega })^{{ \eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{C} \\ &\hphantom{\mathcal{R}_{6} \leq}{}+ \int _{-h}^{0} \bigl(t-({\omega }-{\lambda }) \bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{ \eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({\omega }-{ \lambda })\bigr)^{{ \eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{D}({\omega }-{\lambda },{\lambda })\,d\mathfrak{D}_{{\lambda }} \biggr] \\ &\hphantom{\mathcal{R}_{6} \leq}{}\times \biggl[(t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl( \mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{C}+ \int _{-h}^{0} \bigl(t-({\omega }-{\lambda }) \bigr)^{{\eta _{1}}-1} \\ &\hphantom{\mathcal{R}_{6} \leq}{}\times E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({ \omega }-{\lambda }) \bigr)^{{\eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{D}({ \omega }-{\lambda },{\lambda })\,d\mathfrak{D}_{{\lambda }} \biggr]^{*} W^{-1}1/2 \Biggl[z_{\mathcal{T}}-\sum_{r=0}^{\rho -1}z^{r}(0){ \mathcal{T}}^{r} \\ &\hphantom{\mathcal{R}_{6} \leq}{}\times E_{{\eta _{1}}-{\eta _{2}},1+r}\bigl(\mathfrak{B} {\mathcal{T}}^{{ \eta _{1}}-{\eta _{2}}}\bigr)+ \sum_{r=0}^{\mu -1}z^{r}(0){ \mathcal{T}}^{{ \eta _{1}}-{\eta _{2}}+r} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}-{ \eta _{2}}+1+r}\bigl(\mathfrak{B} { \mathcal{T}}^{{\eta _{1}}-{\eta _{2}}}\bigr) \\ &\hphantom{\mathcal{R}_{6} \leq}{}- \int _{-h}^{0} \biggl[ \int _{{\lambda }}^{0}\bigl({\mathcal{T}}-({\omega }-{ \lambda })\bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl( \mathfrak{B}\bigl({ \mathcal{T}}-({\omega }-{\lambda })\bigr)^{{\eta _{1}}-{ \eta _{2}}}\bigr) \\ &\hphantom{\mathcal{R}_{6} \leq}{}\times \mathfrak{D}({\omega }-{\lambda },{\lambda })\vartheta ({ \omega }) \,d{\omega } \biggr]\,d\mathfrak{D}_{{\lambda }} \\ &\hphantom{\mathcal{R}_{6} \leq}{}- \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \biggl( \int _{0}^{\eta }\sigma ( \theta )\,dw(\theta ) \biggr)d{ \omega } \\ &\hphantom{\mathcal{R}_{6} \leq}{}- \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{f}\bigl({\omega },z({ \omega })\bigr) \Biggr]d{ \omega } \Biggr\Vert ^{2} \\ &\hphantom{\mathcal{R}_{6}}\leq 84 M^{2} l^{2} ({\mathcal{T}}+{\lambda }) \Biggl\{ 1/2 \Biggl[ \mathbb{E} \Vert z_{\mathcal{T}} \Vert ^{2}+u_{1} \sum_{r=0}^{\rho -1} \mathbb{E} \Vert z_{r} \Vert ^{2}+u_{2} \sum _{r=0}^{\mu -1}\mathbb{E} \Vert z_{r} \Vert ^{2} \\ &\hphantom{\mathcal{R}_{6} \leq}{}+G+u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \tilde{N} {\mathcal{T}} \int _{0}^{{\mathcal{T}}}\bigl(1+\mathbb{E}\|z({ \omega }) \Vert ^{2}\bigr)\,d{\omega } \\ &\hphantom{\mathcal{R}_{6} \leq}{}+4u_{3} L_{\sigma } \tilde{L} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(1+\mathbb{E} \bigl\Vert z( \theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega } \Biggr] \Biggr\} \\ &\hphantom{\mathcal{R}_{6} }\leq 42 M^{2} l^{2} ({\mathcal{T}}+{\lambda }) \Biggl[ \mathbb{E} \Vert z_{ \mathcal{T}} \Vert ^{2}+u_{1}\sum _{r=0}^{\rho -1} \mathbb{E} \Vert z_{r} \Vert ^{2}+u_{2} \sum _{r=0}^{\mu -1}\mathbb{E} \Vert z_{r} \Vert ^{2} \\ &\hphantom{\mathcal{R}_{6} \leq}+{}G+u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \tilde{N} {\mathcal{T}} \int _{0}^{{\mathcal{T}}}\bigl(1+\mathbb{E}\|z({ \omega }) \Vert ^{2}\bigr)\,d{\omega } \\ &\hphantom{\mathcal{R}_{6} \leq}{}+4u_{3} L_{\sigma } \tilde{L} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(1+\mathbb{E} \bigl\Vert z( \theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega } \Biggr], \end{aligned}$$
(32)
$$\begin{aligned} &\mathcal{R}_{7} \leq 7 \mathbb{E} \Biggl\Vert \biggl[ \int _{t+{\lambda }}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl( \mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{C} \biggr] \\ &\hphantom{\mathcal{R}_{7} \leq}{}\times \biggl[ \int _{t+{\lambda }}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{ \eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{C} \biggr]^{*} \\ &\hphantom{\mathcal{R}_{7} \leq}{}\times W^{-1}1/2 \Biggl[z_{\mathcal{T}}-\sum _{r=0}^{\rho -1}z^{r}(0){ \mathcal{T}}^{r} E_{{\eta _{1}}-{\eta _{2}},1+r}\bigl(\mathfrak{B} { \mathcal{T}}^{{\eta _{1}}-{\eta _{2}}}\bigr) \\ &\hphantom{\mathcal{R}_{7} \leq}{}+\sum_{r=0}^{\mu -1}z^{r}(0){ \mathcal{T}}^{{\eta _{1}}-{\eta _{2}}+r} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}-{\eta _{2}}+1+r}\bigl(\mathfrak{B} { \mathcal{T}}^{{\eta _{1}}-{\eta _{2}}}\bigr) \\ &\hphantom{\mathcal{R}_{7} \leq}{}- \int _{-h}^{0} \biggl[ \int _{{\lambda }}^{0}\bigl({\mathcal{T}}-({\omega }-{ \lambda })\bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl( \mathfrak{B}\bigl({ \mathcal{T}}-({\omega }-{\lambda })\bigr)^{{\eta _{1}}-{ \eta _{2}}}\bigr) \\ &\hphantom{\mathcal{R}_{7} \leq}{}\mathfrak{D}({\omega }-{\lambda },{\lambda })\vartheta ({ \omega }) \,d{\omega } \biggr]\,d\mathfrak{D}_{{\lambda }} \\ &\hphantom{\mathcal{R}_{7} \leq}{}- \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \biggl( \int _{0}^{\eta }\sigma ( \theta )\,dw(\theta ) \biggr)d{ \omega } \\ &\hphantom{\mathcal{R}_{7} \leq}{}- \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{\omega })^{{\eta _{1}} -1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{f}\bigl({\omega },z({ \omega })\bigr) \Biggr]d{ \omega } \Biggr\Vert ^{2} \\ &\hphantom{\mathcal{R}_{7}}\leq 21 \tilde{M}^{2} l^{2} \Biggl[\mathbb{E} \Vert z_{\mathcal{T}} \Vert ^{2}+u_{1} \sum _{r=0}^{\rho -1} \mathbb{E} \Vert z_{r} \Vert ^{2}+u_{2} \sum_{r=0}^{\mu -1} \mathbb{E} \Vert z_{r} \Vert ^{2} \\ &\hphantom{\mathcal{R}_{7} \leq}{}+G+u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \tilde{N} {\mathcal{T}} \int _{0}^{{\mathcal{T}}}\bigl(1+\mathbb{E}\|z({ \omega }) \Vert ^{2}\bigr)\,d{\omega } \\ &\hphantom{\mathcal{R}_{7} \leq}{}+4u_{3} L_{\sigma } \tilde{L} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(1+\mathbb{E} \bigl\Vert z( \theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega } \Biggr] \end{aligned}$$
(33)
From (27)–(33), we have
$$\begin{aligned} \sup_{0\leq t \leq {\mathcal{T}}}\mathbb{E} \bigl\Vert (\Delta z) (t) \bigr\Vert ^{2} \leq {}&7u_{1}\sum_{r=0}^{\rho -1} \mathbb{E} \Vert z_{r} \Vert ^{2}+7u_{2} \sum _{r=0}^{\mu -1}\mathbb{E} \Vert z_{r} \Vert ^{2} \\ &{}+7G+7 u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \tilde{N} {\mathcal{T}} \int _{0}^{{\mathcal{T}}}\bigl(1+\mathbb{E}\|z({ \omega }) \Vert ^{2}\bigr)\,d{\omega } \\ &{}+28 u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} L_{ \sigma } \tilde{L} \int _{0}^{{\mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(1+ \mathbb{E} \bigl\Vert z( \theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega } \\ &{}+42 M^{2} l^{2} ({\mathcal{T}}+{\lambda }) \Biggl[ \mathbb{E} \Vert z_{ \mathcal{T}} \Vert ^{2}+u_{1}\sum _{r=0}^{\rho -1} \mathbb{E} \Vert z_{r} \Vert ^{2}+u_{2} \sum _{r=0}^{\mu -1}\mathbb{E} \Vert z_{r} \Vert ^{2} \\ &{}+G+u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \tilde{N} {\mathcal{T}} \int _{0}^{{\mathcal{T}}}\bigl(1+\mathbb{E}\|z({ \omega }) \Vert ^{2}\bigr)\,d{\omega } \\ &{}+4u_{3} L_{\sigma } \tilde{L} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(1+\mathbb{E} \bigl\Vert z( \theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega } \Biggr] \\ &{}+21 \tilde{M}^{2} l^{2} [\mathbb{E} \Vert z_{\mathcal{T}} \Vert ^{2}+u_{1} \sum _{r=0}^{\rho -1} \mathbb{E} \Vert z_{r} \Vert ^{2}+u_{2} \sum_{r=0}^{\mu -1} \mathbb{E} \Vert z_{r} \Vert ^{2} \\ &{}+G+u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \tilde{N} {\mathcal{T}} \int _{0}^{{\mathcal{T}}}\bigl(1+\mathbb{E}\|z({ \omega }) \Vert ^{2}\bigr)\,d{\omega } \\ &{}+4u_{3} L_{\sigma } \tilde{L} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(1+\mathbb{E} \bigl\Vert z( \theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega } ] \\ \leq {}& C(1+{\mathcal{T}}) \biggl[ \int _{0}^{{\mathcal{T}}} \bigl(1+ \mathbb{E} \bigl\Vert z({ \omega }) \bigr\Vert ^{2} \bigr)\,d{\omega } \biggr] \\ \leq {}& C \Bigl(1+{\mathcal{T}}\sup_{0\leq t \leq {\mathcal{T}}} \mathbb{E} \bigl\Vert z({\omega }) \bigr\Vert ^{2} \Bigr) \end{aligned}$$
(34)
for all \(t\in [0,{\mathcal{T}}]\), where C is constant. This implies that Δ maps I into itself.
Secondly, we prove that Δ is a contraction mapping on I, for any \(z,y \in I\),
$$\begin{aligned}& \mathbb{E} \bigl\Vert (\Delta z) (t)-(\Delta y) (t) \bigr\Vert ^{2} \\& \quad \leq 6 \sup_{0\leq t \leq {\mathcal{T}}}\mathbb{E}\biggl\| \mathbb{G}_{1} ({\mathcal{T}},t) \mathbb{G}_{1}^{{\mathcal{T}}} ({\mathcal{T}},t)W^{-1} \\& \qquad {}\times \biggl[ \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{\omega })^{{ \eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({ \mathcal{T}}-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \\& \qquad {}\times \biggl( \int _{0}^{ \eta } \bigl[\sigma \bigl(\theta ,z(\theta ) \bigr)-\sigma \bigl(\theta ,y(\theta )\bigr) \bigr]\,dw(\theta ) \biggr)\,d{\omega } \\& \qquad {}+ \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \bigl(\mathfrak{f}\bigl({\omega },z({ \omega })\bigr)- \mathfrak{f}\bigl({\omega },y({\omega })\bigr) \bigr)\,d{\omega } \biggr] \\& \qquad {} +\mathbb{G}_{2} ({\mathcal{T}},t) \mathbb{G}_{2}^{{\mathcal{T}}} ({ \mathcal{T}},t)W^{-1} \\& \qquad {} \times \biggl[ \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{\omega })^{{ \eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({ \mathcal{T}}-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \\& \qquad {}\times \biggl( \int _{0}^{ \eta } \bigl[\sigma \bigl(\theta ,z(\theta ) \bigr)-\sigma \bigl(\theta ,y(\theta )\bigr) \bigr]\,dw(\theta ) \biggr)\,d{\omega } \\& \qquad {} + \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \bigl(\mathfrak{f}\bigl({\omega },z({ \omega })\bigr)- \mathfrak{f}\bigl({\omega },y({\omega })\bigr) \bigr)\,d{\omega } \biggr] \\& \qquad {} + \int _{0}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \biggl( \int _{0}^{\eta } \bigl[\sigma \bigl(\theta ,z(\theta ) \bigr)-\sigma \bigl(\theta ,y( \theta )\bigr) \bigr]\,dw(\theta ) \biggr)\,d{\omega } \\& \qquad {} + \int _{0}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \bigl( \mathfrak{f}\bigl({\omega },z({\omega })\bigr)-\mathfrak{f}\bigl({\omega },y({ \omega })\bigr) \bigr)\,d{\omega }\biggr\| ^{2} \\& \quad \leq 6M^{2} l^{2} \biggl\{ \biggl\Vert \int _{0}^{\mathcal{T}} ({ \mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{\omega })^{{\eta _{1}}-{ \eta _{2}}}\bigr) \\& \qquad {}\times \biggl[ \int _{0}^{\eta } \bigl(\sigma \bigl(\theta ,z(\theta ) \bigr)- \sigma \bigl(\theta ,y(\theta )\bigr) \bigr)\,dw(\theta ) \biggr]\,d{\omega } \biggr\Vert ^{2} \\& \qquad {} + \biggl\Vert \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \bigl[\mathfrak{f}\bigl({\omega },z({ \omega })\bigr)- \mathfrak{f}\bigl({\omega },y({\omega })\bigr) \bigr]\,d{\omega } \biggr\Vert ^{2} \biggr\} \\& \qquad {} +6\tilde{M}^{2} l^{2} \biggl\{ \biggl\Vert \int _{0}^{\mathcal{T}} ({ \mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{\omega })^{{\eta _{1}}-{ \eta _{2}}}\bigr) \\& \qquad {}\times \biggl[ \int _{0}^{\eta } \bigl(\sigma \bigl(\theta ,z(\theta ) \bigr)- \sigma \bigl(\theta ,y(\theta )\bigr) \bigr)\,dw(\theta ) \biggr]\,d{\omega } \biggr\Vert ^{2} \\& \qquad {} + \biggl\Vert \int _{0}^{\mathcal{T}} ({\mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{ \omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \bigl[\mathfrak{f}\bigl({\omega },z({ \omega })\bigr)- \mathfrak{f}\bigl({\omega },y({\omega })\bigr) \bigr]\,d{\omega } \biggr\Vert ^{2} \biggr\} \\& \qquad {} +6 \mathbb{E} \biggl\Vert \int _{0}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{ \eta _{1}}-{\eta _{2}}}\bigr) \\& \qquad {}\times \biggl[ \int _{0}^{\eta } \bigl(\sigma \bigl(\theta ,z( \theta )\bigr)-\sigma \bigl(\theta ,y(\theta )\bigr) \bigr)\,dw(\theta ) \biggr]\,d{\omega } \biggr\Vert ^{2} \\& \qquad {}+6 \mathbb{E} \biggl\Vert \int _{0}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{ \eta _{1}}-{\eta _{2}}}\bigr) \bigl[ \mathfrak{f}\bigl({\omega },z({\omega })\bigr)- \mathfrak{f}\bigl({\omega },y({ \omega })\bigr) \bigr]\,d{\omega } \biggr\Vert ^{2} \\& \quad \triangleq 6M^{2} l^{2} \sum_{b=1}^{2} \mathcal{S}_{b} +6\tilde{M}^{2} l^{2} \sum _{b=3}^{4} \mathcal{S}_{b} + \sum_{b=5}^{6} \mathcal{S}_{b}. \end{aligned}$$
(35)
Using the Lipschitz condition, we have the following estimates:
$$\begin{aligned} &\mathcal{S}_{1} \leq 24 M^{2} l^{2} \biggl\| \int _{0}^{\mathcal{T}} ({ \mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{\omega })^{{\eta _{1}}-{ \eta _{2}}}\bigr) \\ &\hphantom{\mathcal{S}_{1} \leq}{}\times \biggl[ \int _{0}^{\eta } \bigl(\sigma \bigl(\theta ,z(\theta ) \bigr)- \sigma \bigl(\theta ,y(\theta )\bigr) \bigr)\,dw(\theta ) \biggr]\,d{\omega } \biggr\| ^{2} \\ &\hphantom{\mathcal{S}_{1} }\leq 24 M^{2} l^{2} u_{3} L_{\sigma } L \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(\mathbb{E} \bigl\Vert z(\theta ) -y(\theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega }, \end{aligned}$$
(36)
$$\begin{aligned} &\mathcal{S}_{2} \leq 6M^{2} l^{2} \biggl\| \int _{0}^{\mathcal{T}} ({ \mathcal{T}}-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}({\mathcal{T}}-{\omega })^{{\eta _{1}}-{ \eta _{2}}}\bigr) \bigl[\mathfrak{f}\bigl({\omega },z({\omega })\bigr)- \mathfrak{f}\bigl({ \omega },y({\omega })\bigr) \bigr]\,d{\omega }\biggr\| ^{2} \\ &\hphantom{\mathcal{S}_{2}}\leq 6M^{2} l^{2} u_{3} {\mathcal{T}} N \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}}\mathbb{E}\bigl\| z({\omega })-y({\omega }) \bigr\| ^{2}\,d{\omega }, \end{aligned}$$
(37)
$$\begin{aligned} &\mathcal{S}_{3} \leq 24 \tilde{M}^{2} l^{2} u_{3} L_{\sigma } L \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(\mathbb{E} \bigl\Vert z(\theta ) -y(\theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega }, \end{aligned}$$
(38)
$$\begin{aligned} &\mathcal{S}_{4} \leq 6 \tilde{M}^{2} l^{2} u_{3} {\mathcal{T}} N \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}}\mathbb{E}\bigl\| z({\omega })-y({\omega }) \bigr\| ^{2}\,d{\omega }, \end{aligned}$$
(39)
$$\begin{aligned} &\mathcal{S}_{5} \leq 24 u_{3} L_{\sigma } L \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(\mathbb{E} \bigl\Vert z(\theta ) -y(\theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega }, \end{aligned}$$
(40)
$$\begin{aligned} &\mathcal{S}_{6} \leq 6 u_{3} {\mathcal{T}} N \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}}\mathbb{E}\bigl\| z({\omega })-y({\omega }) \bigr\| ^{2}\,d{\omega }. \end{aligned}$$
(41)
Together with inequalities (36)–(41), we get
$$\begin{aligned}& \mathbb{E} \bigl\Vert (\Delta z) (t)-(\Delta y) (t) \bigr\Vert ^{2} \\& \quad \leq 24 M^{2} l^{2} u_{3} L_{\sigma } L \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{\mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(\mathbb{E} \bigl\Vert z(\theta ) -y(\theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega } \\& \qquad {} +6M^{2} l^{2} u_{3} {\mathcal{T}} N \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}}\mathbb{E}\bigl\| z({\omega })-y({\omega }) \bigr\| ^{2}\,d{\omega } \\& \qquad {} +24 \tilde{M}^{2} l^{2} u_{3} L_{\sigma } L \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(\mathbb{E} \bigl\Vert z(\theta ) -y(\theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega } \\& \qquad {} +6 \tilde{M}^{2} l^{2} u_{3} {\mathcal{T}} N \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}}\mathbb{E}\bigl\| z({\omega })-y({\omega }) \bigr\| ^{2}\,d{\omega } \\& \qquad {} +24 u_{3} L_{\sigma } L \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}} \biggl( \int _{0}^{\eta }\bigl(\mathbb{E} \bigl\Vert z(\theta ) -y(\theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega } \\& \qquad {} +6 u_{3} {\mathcal{T}} N \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \int _{0}^{{ \mathcal{T}}}\mathbb{E}\bigl\| z({\omega })-y({\omega }) \bigr\| ^{2}\,d{\omega } \\& \quad \leq 24 u_{3} L_{\sigma } L \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \bigl(M^{2} l^{2}+ \tilde{M}^{2} l^{2}+1 \bigr) \int _{0}^{{\mathcal{T}}}\biggl( \int _{0}^{\eta }\bigl( \mathbb{E} \bigl\Vert z( \theta ) -y(\theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)\,d{\omega } \\& \qquad {} +6 u_{3} {\mathcal{T}} N \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \bigl(M^{2} l^{2}+ \tilde{M}^{2} l^{2}+1\bigr) \int _{0}^{{\mathcal{T}}}\mathbb{E}\bigl\| z({ \omega })-y({\omega }) \bigr\| ^{2}\,d{\omega } \\& \quad \leq 6 u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \bigl(M^{2} l^{2}+\tilde{M}^{2} l^{2}+1\bigr) \biggl[4L_{\sigma } L \int _{0}^{{\mathcal{T}}}\biggl( \int _{0}^{\eta }\bigl(\mathbb{E} \bigl\Vert z(\theta ) -y(\theta ) \bigr\Vert ^{2}\bigr)\,d\theta \biggr)d{ \omega } \\& \qquad {} +{\mathcal{T}}N \int _{0}^{{\mathcal{T}}}\mathbb{E}\bigl\| z({\omega })-y({ \omega }) \bigr\| ^{2}\,d{\omega }\biggr] \\& \quad \leq 6 u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} \bigl(M^{2} l^{2}+\tilde{M}^{2} l^{2}+1\bigr) (4L_{\sigma } L+{\mathcal{T}}N)\sup_{0 \leq t \leq {\mathcal{T}}}\mathbb{E} \bigl\Vert z(t)-y(t) \bigr\Vert ^{2}\,d{\omega }. \end{aligned}$$
(42)
Therefore we conclude that if \(6 u_{3} \frac{{\mathcal{T}}^{2{\eta _{1}}-1}}{2{\eta _{1}}-1} (M^{2} l^{2}+\tilde{M}^{2} l^{2}+1) (4L_{\sigma } L+{\mathcal{T}}N)\leq 1\), then Δ is a contraction mapping on I, which implies that the mapping Δ has a unique fixed point.
Hence we have
$$\begin{aligned} z(t)={}&\sum_{r=0}^{\rho -1}z^{r}(0)t^{r} E_{{\eta _{1}}-{\eta _{2}},1+r}\bigl( \mathfrak{B}t^{{\eta _{1}}-{\eta _{2}}}\bigr)-\sum _{r=0}^{\mu -1}z^{r}(0)t^{{ \eta _{1}}-{\eta _{2}}+r} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}-{ \eta _{2}}+1+r}\bigl(\mathfrak{B}t^{{\eta _{1}}-{\eta _{2}}}\bigr) \\ &{}+ \int _{-h}^{0} \biggl[ \int _{{\lambda }}^{0}\bigl(t-({\omega }-{\lambda }) \bigr)^{{ \eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({ \omega }-{ \lambda })\bigr)^{{\eta _{1}}-{\eta _{2}}}\bigr) \\ &{}\times \mathfrak{D}({ \omega }-{\lambda },{\lambda })\vartheta ({\omega }) \,d{\omega } \biggr]\,d\mathfrak{D}_{{\lambda }} \\ &{}+ \int _{0}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{ \eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \mathfrak{f} \bigl({\omega },z({\omega })\bigr)\,d{\omega } \\ &{}+ \int _{0}^{t} (t-{ \omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl( \mathfrak{B}(t-{\omega })^{{\eta _{1}}-{\eta _{2}}}\bigr) \\ &{} \times \biggl( \int _{0}^{\eta }\sigma (\theta )\,dw(\theta ) \biggr)d{ \omega } + \int _{0}^{t+{\lambda }} \biggl[(t-{\omega })^{{\eta _{1}}-1} E_{{ \eta _{1}}-{\eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{ \eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{C} \\ &{}+ \int _{-h}^{0} \bigl(t-({\omega }-{\lambda }) \bigr)^{{\eta _{1}}-1} E_{{\eta _{1}}-{ \eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}\bigl(t-({\omega }-{ \lambda })\bigr)^{{ \eta _{1}}-{\eta _{2}}}\bigr)\mathfrak{D}({\omega }-{\lambda },{\lambda })\,d\mathfrak{D}_{{\lambda }} \biggr]u({\omega })\,d{\omega } \\ &{}+ \int _{t+{\lambda }}^{t} (t-{\omega })^{{\eta _{1}}-1} E_{{\eta _{1}}-{ \eta _{2}},{\eta _{1}}}\bigl(\mathfrak{B}(t-{\omega })^{{\eta _{1}}-{ \eta _{2}}}\bigr) \mathfrak{C}u({\omega })\,d{\omega }. \end{aligned}$$
(43)
Thus \(z(t)\) is the solution of system (18)–(20), and it is easy to verify that \(z({\mathcal{T}})=z_{\mathcal{T}}\). Further the control function \(u(t)\) steers system (18)–(20) from the initial state to the final state \(z_{\mathcal{T}}\) on \(\mathcal{J}\). Hence system (18)–(20) is controllable on \(\mathcal{J}\). □
Remark 4.2
If \(\eta _{1} \in (1,2]\), \(\eta _{2} \in (0,1]\), \(\rho =2\), and \(\sigma =0\), then system (5)–(7) reduces to the system which was discussed in [3]. When \(\eta _{1} \in (1,2]\), \(\eta _{2} \in (0,1]\), \(\rho =2\), and \(C=\sigma =0\), system (18)–(20) reduces to the system studied in [4]. Controllability of the linear system is obtained by the Gramian matrix. Further, under the assumption that the linear control system is controllable and by using the successive approximation technique, the controllability of nonlinear systems was obtained. If we choose \(\eta _{1} \in (0,1]\), \(\eta _{2}=0\), \(\mathfrak{D}=0\), and \(\rho =1\) in (18)–(20), Theorem 3.3 in [31] can be regarded as a special case of our result.
Remark 4.3
It should be noted that the results in [21] have been derived for linear fractional-order systems, and the results in [6, 18] have been obtained for nonlinear fractional-order systems. However, in comparison with [13, 32], the results proposed in this paper are more general ones, as the results presented are applicable for higher-order fractional damped stochastic systems with distributed delays.