A note on approximate controllability of second-order impulsive stochastic Volterra-Fredholm integrodifferential system with infinite delay

1 Introduction

In the field of mathematical control theory, controllability is one of the key ideas. Both finite and infinite dimensional systems have a powerful impact on the notion of controllability. Exact controllability and approximate controllability are the two basic notions of controllability in infinite-dimensional settings. The notion of exact controllability rarely applies to infinite-dimensional control problems. So, it is crucial to investigate approximate controllability. Mahmudov and Denker (2000), Mahmudov (2001) examined several kinds of controllability and established the required and adequate conditions for the controllability notions. In recent days, studying the numerical solution of differential systems is seeking great attention from many researchers. Further, Al-Smadi et al. (2014); Arqub and Rashaideh (2018) discussed the numerical solution of differential equations for periodic boundary value problems. Momani et al. (2020), the authors analyzed convergence for Lienard’s equation involving the Atangana-Baleanu-Caputo model and established the piecewise optimal fractional reproducing kernel solution. Moreover, by using the Atangana-Baleanu fractional approach, Momani et al. (2020) introduced the reproducing kernel algorithm for the numerical solution of the Van der Pol damping model.

The theory of impulsive differential equations is currently receiving a lot of interest from researchers and has become a significant topic of study in recent years. Chang (2007) used the fixed point approach of Schauder along with the semigroup operator to provide an adequate requirement for the controllability of impulsive differential systems involving delay. Further, Sivasankaran et al. (2011) established the existence of global solutions for second-order impulsive differential systems via the Alternative fixed point approach of Leray–Schauder. Many biological processes use differential systems with impulsive conditions. Applications include thresholds, bursting rhythm models in biology and medicine, and frequency modulated systems. Consult the books (Bainov and Simeonov, 1993; Laksmikantham et al., 1989) for more information.

On the other hand, second-order differential systems have received much greater attention since they are utilized to analyze a variety of real-world issues. Sometimes it is advantageous to deal directly with second-order differential systems instead of converting them to first-order systems. It may be utilized to model a variety of physical processes. Hernández et al. (2009) investigated the existence of mild solutions for impulsive second-order neutral differential systems involving infinite delay via the concept of the cosine family of operators. By utilizing the fixed point approach of Bohnenblust-Karlin, the authors have established the approximate controllability of second-order differential systems in (Mahmudov et al., 2016). Recently, Vijayakumar et al. (2021b) examined the approximate controllability of second-order impulsive neutral differential systems consisting of Sobolev-type, impulses, neutral functions, and infinite delay via the fixed point theorems along with the cosine function of operators.

Stochastic differential systems are used in the development and analysis of mechanical, electrical, control engineering, and physical sciences, in addition to producing more realistic models. Ren and Sun (2002) verified the existence and uniqueness of mild solutions for second-order stochastic evolution systems consisting of neutral functions involving infinite delay via the successive approximation technique. Later, Sakthivel et al. (2010) investigated the approximate controllability of second-order stochastic equations involving impulsive effects by means of Hölder’s inequality, stochastic theory, and the fixed point approach. For more specifics, refer to the book (Mao, 1997) and the research papers (Mahmudov and Mckibben, 2006; Ren et al., 2011; Revathi et al., 2016; Yan, 2015).

Moreover, Volterra-Fredholm’s integrodifferential systems play an important role in the study of physics and biology. Chang and Chalishajar (2008) formulated the requirements for the controllability of Volterra-Fredholm-type integrodifferential systems by utilizing the fixed point theorem of Bohnenblust-Karlin along with the semigroup operator. Further, Muthukumar and Balasubramaniam (2011) used the Banach fixed point approach to discuss the approximate controllability of stochastic Volterra-Fredholm type integrodifferential equations. Recently, Vijayakumar et al. (2021a) verified the approximate controllability outcomes for fractional Volterra-Fredholm integrodifferential systems consisting of the Sobolev type recently via fractional calculus, the cosine family of operators, and the fixed point technique.

Inspired by the above articles, this manuscript is concerned with studying the approximate controllability of second-order stochastic Volterra-Fredholm integrodifferential system involving impulsive effects and infinite delay of the form

Let $z_0=\alpha \in L^2(\mathrm{\Omega },{\mathcal{B}}_{\vartheta })$ and $z_1$ be the $\mathcal{U}$ -valued ${\mathfrak{J}}_0$ -measurable random variable independent of the Wiener process $\left\{W\right(\varpi ):\varpi ⩾0\}$ with a finite second moment. The control function $u(·)$ is takes values in $L^2(\mathcal{V},\mathcal{X})$ and $\mathcal{X}$ is a separable Hilbert space. In addition, $\mathcal{B}:\mathcal{X}\to \mathcal{U}$ is a bounded linear operator. $J_{\iota },\overline{J_{\iota }}:\mathcal{U}\to \mathcal{N}\left(\mathcal{U}\right)$ are multi-valued maps with closed graph. In addition, consider $0={\varpi }_0<{\varpi }_1<{\varpi }_2<\cdots <{\varpi }_{ȷ}<{\varpi }_{ȷ+1}=\mathfrak{p}$ is the prefixed points. The jumps at the points ${\varpi }_j$ belongs to $(0,\mathfrak{p})$ are given by $\mathrm{\Delta }z{|}_{\varpi ={\varpi }_{\iota }}=z\left({\varpi }_{\iota }^{+}\right)-z\left({\varpi }_{\iota }^{-}\right),\mathrm{\Delta }{z}^{\prime }{|}_{\varpi ={\varpi }_{\iota }}=z^{\prime }\left({\varpi }_{\iota }^{+}\right)-z^{\prime }\left({\varpi }_{\iota }^{-}\right),\forall$ $\iota =1,2,\dots ,ȷ$ . Here $z\left({\varpi }_{\iota }^{+}\right),z\left({\varpi }_{\iota }^{-}\right),z^{\prime }\left({\varpi }_{\iota }^{+}\right)$ and $z^{\prime }\left({\varpi }_{\iota }^{-}\right)$ represent the right and left limits of $z\left(\varpi \right)$ at $\varpi ={\varpi }_{\iota }$ and $z^{\prime }\left(\varpi \right)$ at $\varpi ={\varpi }_{\iota }$ , respectively. For our convenience, we denote ${\mathcal{W}}_1\left(\varpi \right)={\int }_0^{\varpi }l(\varpi ,\mu ,z_{\mu })d\mu$ and ${\mathcal{W}}_2\left(\varpi \right)={\int }_0^{\mathfrak{p}}m(\varpi ,\mu ,z_{\mu })d\mu$ .

The manuscript presentation plan is in the following way: We quickly give a few key facts and terminologies linked with our study in Section 2, which is used in the whole analysis of our work. Section 3 is designated for consideration of the approximate controllability. Section 4, continues our investigation of the system (1.1)–(1.3) with nonlocal circumstances. We offer an application that is presented to illustrate the concept of the primary outcomes in Section 5.

2 Preliminaries

To discuss our primary outcomes, we now introduce some basic concepts, key terms, and facts.

Make the assumption that $(\mathrm{\Omega },\mathfrak{J},\mathbb{P})$ is a complete probability space fitted with ${\left\{{\mathfrak{J}}_{\varpi }\right\}}_{\varpi \in [0,\mathfrak{p}]}$ being a normal filtration. The expectation with respect to the measure $\mathbb{P}$ is represented as $E(·)$ . Consider the separable Hilbert spaces $\mathcal{U},\mathbb{V}$ and $\left\{W\right(\varpi ),\varpi ⩾0\}$ denote a Wiener process with the bounded liner covariance operator $\mathcal{Q}$ $\ni$ $\mathit{Tr}\left(\mathcal{Q}\right)<\infty$ . Let us consider that there is a system ${\left\{e_{\breve{\kappa }}\right\}}_{\breve{\kappa }=1}^{\infty }$ belongs to $\mathbb{V}$ , with complete orthonormal and a bounded sequence of number ${\left\{{\lambda }_{\breve{\kappa }}\right\}}_{\breve{\kappa }=1}^{\infty }⩾0$ $\ni$ $$\mathcal{Q}{e}_{\breve{\kappa }}={\lambda }_{\breve{\kappa }}{e}_{\breve{\kappa }},\breve{\kappa }=1,2,\dots$$ and a sequence ${\beta }_{\breve{\kappa }}$ of independent Brownian motions such that $$\langle W\left(\varpi \right),e\rangle ={\displaystyle \sum _{\breve{\kappa }=1}^{\infty }}\sqrt{{\lambda }_{\breve{\kappa }}}\langle e_{\breve{\kappa }},e\rangle {\beta }_{\breve{\kappa }}\left(\varpi \right),\varpi \in \mathcal{V},$$ and $\mathfrak{J}={\mathfrak{J}}_{\varpi }^W,{\mathfrak{J}}_{\varpi }^W$ is the $\sigma$ -algebra referring W.

The space of all bounded operators from $\mathbb{V}$ into $\mathcal{U}$ with the norm $\Vert ·\Vert$ is represented by $L(\mathbb{V},\mathcal{U})$ . Consider $L_2^0=L_2\left({\mathcal{Q}}^{1/2}\mathbb{V}\text{;}\mathcal{U}\right)$ is the space containing all Hilbert–Schmidt operators with the norm $$\langle \theta ,{\theta }^{\ast }{\rangle }_{L_2^0}=\Vert \theta {\mathcal{Q}}^{1/2}{\Vert }^2=\mathit{Tr}\left(\theta \mathcal{Q}{\theta }^{\ast }\right)$$ and $\theta$ is known as a $\mathcal{Q}$ -Hilbert–Schmidt operator from ${\mathcal{Q}}^{1/2}\mathbb{V}$ $\to$ $\mathcal{U}$ . Consider $L_2(\mathrm{\Omega },{\mathfrak{J}}_{\mathfrak{p}},\mathcal{U})$ is the space containing ${\mathfrak{J}}_{\mathfrak{p}}$ -measurable square integrable random variables along with values in $\mathcal{U}$ . Let $L_2^{\mathfrak{J}}(\mathcal{V},\mathcal{U})$ represents the Banach space of all ${\mathfrak{J}}_{\varpi }$ -adapted, $\mathcal{U}$ -valued measurable square integrable systems on $\mathcal{V}\times \mathrm{\Omega }$ . Suppose that the Banach space of continuous function from $[0,\mathfrak{p}]$ into $L_2(\mathrm{\Omega },{\mathfrak{J}}_{\varpi },\mathcal{U})$ is $\mathcal{C}\left(\right[0,\mathfrak{p}\left]\text{;}{L}_2\right(\mathrm{\Omega },{\mathfrak{J}}_{\varpi },\mathcal{U}\left)\right)$ equipped with ${\sup }_{\varpi \in \mathcal{V}}E\Vert z\left(\varpi \right){\Vert }_{\mathcal{U}}^2<\infty$ . The family of all ${\mathfrak{J}}_0$ -measurable is represented by $L_2^0(\mathrm{\Omega },\mathcal{U}),\mathcal{U}$ -valued random variables $z\left(0\right)$ .

Provided that $\mathfrak{z}:\mathcal{V}\to \mathcal{U}$ is a measurable function, then it is Bochner integrable if $\Vert \mathfrak{z}\Vert$ is Lebesgue. Let $L^1(\mathcal{V},\mathcal{U})$ be the Banach space of measurable functions supplied along with $$\Vert \mathfrak{z}{\Vert }_{L^1}={\int }_{\mathcal{V}}\Vert \mathfrak{z}\left(\varpi \right)\Vert d\varpi \text{.}$$ Consider the subsequent representations: $$\begin{array}{c}\hfill \mathcal{N}\left(\mathcal{U}\right)=\{z\in \mathcal{N}(\mathcal{U}):\mathcal{U}\ne \varnothing \},\\ \hfill {\mathcal{N}}_{\mathit{cl}}\left(\mathcal{U}\right)=\{z\in \mathcal{N}(\mathcal{U}):z\text{is closed}\},\\ \hfill {\mathcal{N}}_{\mathit{bd}}\left(\mathcal{U}\right)=\{z\in \mathcal{N}(\mathcal{U}):z\text{is bounded}\},\\ \hfill {\mathcal{N}}_{\mathit{cv}}\left(\mathcal{U}\right)=\{z\in \mathcal{N}(\mathcal{U}):z\text{is convex}\},\\ \hfill {\mathcal{N}}_{\mathit{cp}}\left(\mathcal{U}\right)=\{z\in \mathcal{N}(\mathcal{U}):z\text{is compact}\}\text{.}\end{array}$$ Assume that ${\mathcal{U}}_d$ maps from $\mathcal{N}\left(\mathcal{U}\right)\times \mathcal{N}\left(\mathcal{U}\right)$ into $R^{+}\cup \left\{\infty \right\}$ presented as $${\mathcal{U}}_d(\mathbb{X},\mathbb{Y})=\max \left\{{\displaystyle \underset{a\in \mathbb{X}}{\sup }}d(a,\mathbb{Y}),{\displaystyle \underset{b\in \mathbb{Y}}{\sup }}d(\mathbb{X},b)\right\}\text{.}$$ In the above $d(\mathbb{X},b)={\inf }_{a\in \mathbb{X}}d(a,b),d(a,\mathbb{Y})={\inf }_{b\in \mathbb{Y}}d(a,b)$ .

Thus, $\left({\mathcal{N}}_{\mathit{bd},\mathit{cl}}\right(\mathcal{U}),{\mathcal{U}}_d)$ and $\left({\mathcal{N}}_{\mathit{cl}}\right(\mathcal{U}),{\mathcal{U}}_d)$ are metric space and generalized metric space respectively.

The abstract phase space ${\mathcal{B}}_{\vartheta }$ is now represented.

Consider a continuous function $\vartheta$ maps from $(-\infty ,0]$ into $(0,+\infty )$ with $\ell ={\int }_{-\infty }^0\vartheta \left(\varpi \right)d\varpi <+\infty$ . For any $c>0$ , we introduce $$\begin{array}{c}\hfill {\mathcal{B}}_{\vartheta }=\left\{\alpha :(-\infty ,0]\to \mathcal{U},{\left(E\Vert \alpha \left(\theta \right){\Vert }^2\right)}^{1/2}\text{is bounded and measurable function on}\right.\\ \hfill \left.[-c,0]\mathrm{and}{\int }_{-\infty }^0\vartheta \left(\mu \right){\displaystyle \underset{\theta \in [\mu ,0]}{\sup }}{\left(E\Vert \alpha \left(\theta \right){\Vert }^2\right)}^{1/2}d\mu <+\infty \right\}\text{.}\end{array}$$ Suppose that ${\mathcal{B}}_{\vartheta }$ is endowed with $$\Vert \alpha {\Vert }_{{\mathcal{B}}_{\vartheta }}={\int }_{-\infty }^0\vartheta \left(\mu \right){\displaystyle \underset{\mu ⩽\theta \le 0}{\sup }}{\left(E\Vert \alpha \left(\theta \right){\Vert }^2\right)}^{1/2}d\mu ,\alpha \in {\mathcal{B}}_{\vartheta },$$ next, $({\mathcal{B}}_{\vartheta },\Vert ·{\Vert }_{{\mathcal{B}}_{\vartheta }})$ denotes a Banach space (Li and Liu, 2007; Ren and Sun, 2002).

Consider $${\mathcal{B}}_{\vartheta }^{\prime }=\left\{z:(-\infty ,c]\to \mathcal{U}:z_j\in \mathcal{C}({\mathcal{V}}_{\iota },\mathcal{U}),\text{and there exists}z\left({\varpi }_{\iota }^{+}\right)\mathrm{and}z\left({\varpi }_{\iota }^{-}\right)\mathrm{with}z\left({\varpi }_{\iota }^{-}\right)=z\left({\varpi }_{\iota }\right),z_0=\alpha \in {\mathcal{B}}_{\vartheta },\iota =0,1,2,\dots ,ȷ\right\},$$ where $z_{\iota }$ is the restriction of z to ${\mathcal{V}}_{\iota }=({\varpi }_{\iota },{\varpi }_{\iota +1}],\iota =0,1,2,\dots ,ȷ$ . Set the seminorm $\Vert ·{\Vert }_{\mathfrak{p}}$ belongs to ${\mathcal{B}}_{\vartheta }^{\prime }$ given as $$\Vert z{\Vert }_{\mathfrak{p}}=\Vert z_0{\Vert }_{{\mathcal{B}}_{\vartheta }}+{\displaystyle \underset{\mu \in [0,\varpi ]}{\sup }}{(E\Vert z(\mu \left){\Vert }^2\right)}^{\frac{1}{2}},z\mathrm{in}{\mathcal{B}}_{\vartheta }^{\prime }\text{.}$$

From Deimling (1992, 1997), we present some facts related to multi-valued maps.

To provide approximate controllability, let us define the operators: $$\begin{array}{cc}\hfill & {\mathrm{\Upsilon }}_0^{\mathfrak{p}}={\int }_0^{\mathfrak{p}}\mathbb{S}(\mathfrak{p}-\mu ){\mathcal{BB}}^{\ast }{\mathbb{S}}^{\ast }(\mathfrak{p}-\mu )d\mu :\mathcal{U}\to \mathcal{U},\hfill \\ \hfill & \mathcal{R}(\eta ,{\mathrm{\Upsilon }}_0^{\mathfrak{p}})={(\eta I+{\mathrm{\Upsilon }}_0^{\mathfrak{p}})}^{-1}:\mathcal{U}\to \mathcal{U}\text{.}\hfill \end{array}$$ In the above ${\mathcal{B}}^{\ast }$ and ${\mathbb{S}}^{\ast }\left(\mathfrak{p}\right)$ represents adjoints of $\mathcal{B}$ and $\mathbb{S}\left(\mathfrak{p}\right)$ . The linear operator ${\mathrm{\Upsilon }}_0^{\mathfrak{p}}$ is bounded, as we can easily deduce.

To prove the approximate controllability of (1.1)–(1.3), we required the accompanying assumption:

• ${\mathbf{H}}_{\mathbf{0}}$ :

  $\eta \mathcal{R}(\eta ,{\mathrm{\Upsilon }}_0^{\mathfrak{p}})\to 0$ as $\eta \to 0^{+}$ belongs to the strong operator topology.

From Mahmudov and Denker (2000), the assumption ${\mathbf{H}}_{\mathbf{0}}$ is equivalent to the fact that the linear control problem

3 Approximate controllability outcomes

The approximate controllability of (1.1)–(1.3) is the primary subject of this section. Let ${\mathcal{V}}_0=(-\infty ,\mathfrak{p}]$ and before starting the main result, we provide the mild solution of (1.1)–(1.3).

The subsequent assumptions were made for the problem analysis (1.1)–(1.3):

• ${\mathbf{H}}_{\mathbf{1}}$ :

  A is the infinitesimal generator of a strongly continuous cosine family $\left\{\mathbb{C}\right(\varpi ):\varpi \in [0,\mathfrak{p}\left]\right\}$ on $\mathcal{U}$ and $\left\{\mathbb{S}\right(\varpi ):\varpi \in [0,\mathfrak{p}\left]\right\}$ fulfills $\Vert \mathbb{C}\left(\varpi \right){\Vert }^2⩽P_1,\Vert \mathbb{S}\left(\varpi \right){\Vert }^2⩽P_2$ $\forall \varpi ⩾0$ for some positive constants $P_1$ and $P_2$ .

• ${\mathbf{H}}_{\mathbf{2}}$ :

  The operator $\mathbb{C}\left(\varpi \right),\varpi ⩾0$ is compact.

• ${\mathbf{H}}_{\mathbf{3}}$ :

  The multi-valued maps with closed graph $J_{\iota },\overline{J_{\iota }}:\mathcal{U}\to {\mathcal{N}}_{\mathit{cl},\mathit{cp}}\left(\mathcal{U}\right),\iota =1,2,\dots ,ȷ$ and $\exists$ positive constants $P_{\iota },{\bar{P}}_{\iota }$ such that $$\begin{array}{c}\hfill {\mathcal{U}}_d^2\left(J_{\iota }\right(y),J_{\iota }(x\left)\right)⩽P_{\iota }\Vert y-x{\Vert }^2,y,x\in \mathcal{U},\\ \hfill {\mathcal{U}}_d^2\left(\overline{J_{\iota }}\right(y),\overline{J_{\iota }}(x\left)\right)⩽\overline{P_{\iota }}\Vert y-x{\Vert }^2,y,x\in \mathcal{U}\text{.}\end{array}$$

• ${\mathbf{H}}_{\mathbf{4}}$ :

  $G:\mathcal{V}\times {\mathcal{B}}_{\vartheta }\times \mathcal{U}\times \mathcal{U}\to \mathcal{N}\left(L_2^0\right)$ is $L^2$ -Caratheodory function fulfills:

  For every $z\in {\mathcal{B}}_{\vartheta },G(·,z,x,y)$ is measurable and the function $G(\varpi ,·,·,·)$ is u.s.c $\forall$ $\varpi \in \mathcal{V}$ . $\forall$ fixed $(z,x,y)$ in ${\mathcal{B}}_{\vartheta }\times \mathcal{U}\times \mathcal{U}$ , the set $${\mathbb{T}}_{G,z}=\left\{\mathfrak{z}\in L^2\left(\right[0,\mathfrak{p}],L_2^0):\mathfrak{z}\left(\varpi \right)\text{belongs to}G\left(\varpi ,z_{\varpi },{\mathcal{W}}_1\left(\varpi \right),{\mathcal{W}}_2\left(\varpi \right)\right)\right\},$$ for almost everywhere $\varpi \in \mathcal{V}$ and which is nonempty.

• ${\mathbf{H}}_{\mathbf{5}}$ :

  The functions $l,m$ maps from $D\times {\mathcal{B}}_{\vartheta }$ into $\mathcal{U}$ are continuous that fulfills the following conditions:

  1. $\exists$ positive constants $P_l,P_m$ such that $\forall$ $\varpi ,\mu \in \mathcal{V},\widehat{x},\widehat{y}\in {\mathcal{B}}_{\vartheta }$

     • $E{\mathcal{U}}_d^2\left({\int }_0^{\varpi }l(\varpi ,\mu ,\widehat{x})d\mu ,{\int }_0^{\varpi }l(\varpi ,\mu ,\widehat{y})d\mu \right)⩽P_l\Vert \widehat{x}-\widehat{y}{\Vert }_{{\mathcal{B}}_{\vartheta }}^2$ .

     • $E{\mathcal{U}}_d^2\left({\int }_0^{\mathfrak{p}}m(\varpi ,\mu ,\widehat{x})d\mu ,{\int }_0^{\mathfrak{p}}m(\varpi ,\mu ,\widehat{y})d\mu \right)⩽P_m\Vert \widehat{x}-\widehat{y}{\Vert }_{{\mathcal{B}}_{\vartheta }}^2$ .

  2. The continuous functions ${\breve{\mathcal{P}}}_l,{\breve{\mathcal{P}}}_m:\mathcal{V}\to [0,\infty )$ , such that

     • $E\Vert {\int }_0^{\varpi }l(\varpi ,\mu ,\widehat{x})d\mu {\Vert }^2⩽{\breve{\mathcal{P}}}_l\left(\varpi \right)\Vert \widehat{x}{\Vert }_{{\mathcal{B}}_{\vartheta }}^2,\varpi \in \mathcal{V},\widehat{x}\in {\mathcal{B}}_{\vartheta }$ .

     • $E\Vert {\int }_0^{\mathfrak{p}}m(\varpi ,\mu ,\widehat{x})d\mu {\Vert }^2⩽{\breve{\mathcal{P}}}_m\left(\varpi \right)\Vert \widehat{x}{\Vert }_{{\mathcal{B}}_{\vartheta }}^2,\varpi \in \mathcal{V},\widehat{x}\in {\mathcal{B}}_{\vartheta }$ .

• ${\mathbf{H}}_{\mathbf{6}}$ :

  1. $\exists$ a constant $P_G$ such that $$\begin{array}{c}\hfill E{\mathcal{U}}_d^2\left(G\right(\varpi ,{\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3),G(\varpi ,{\widehat{y}}_1,{\widehat{y}}_2,{\widehat{y}}_3\left)\right)\\ \hfill ⩽P_G\left[\Vert {\widehat{x}}_1-{\widehat{y}}_1{\Vert }_{{\mathcal{B}}_{\vartheta }}^2+E\Vert {\widehat{x}}_2-{\widehat{y}}_2{\Vert }^2+E\Vert {\widehat{x}}_3-{\widehat{y}}_3{\Vert }^2\right],\end{array}$$ where $\varpi \in \mathcal{V},{\widehat{x}}_1,{\widehat{y}}_1\in {\mathcal{B}}_{\vartheta },{\widehat{x}}_i,{\widehat{y}}_i\in \mathcal{U},i=2,3$ .

  2. $\exists$ an integrable function ${\mathfrak{m}}_{\mathfrak{z}}:\mathcal{V}\to [0,\infty )$ such that $$\begin{array}{c}\hfill E\Vert G(\varpi ,{\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3){\Vert }^2=\sup \left\{E\Vert \mathfrak{z}{\Vert }^2:\mathfrak{z}\in G(\varpi ,{\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3)\right\}\\ \hfill ⩽{\mathfrak{m}}_{\mathfrak{z}}\left(\varpi \right)\mathrm{\Theta }\left(\Vert {\widehat{x}}_1{\Vert }_{{\mathcal{B}}_{\vartheta }}^2+E\Vert {\widehat{x}}_2{\Vert }^2+E\Vert {\widehat{x}}_3{\Vert }^2\right),\end{array}$$ for a.e $\varpi \in \mathcal{V}$ and ${\widehat{x}}_1\in {\mathcal{B}}_{\vartheta },({\widehat{x}}_2,{\widehat{x}}_3)\in \mathcal{U}\times \mathcal{U}$ , where $\mathrm{\Theta }$ maps from $R^{+}$ into $(0,\infty )$ is a continuous nondecreasing function with $$\mathrm{\Theta }\left({\breve{\mathcal{P}}}_l\left(\varpi \right)\Vert \hat{x}{\Vert }_{{\mathcal{B}}_{\vartheta }}^2\right)⩽{\breve{\mathcal{P}}}_l\left(\varpi \right)\mathrm{\Theta }\left(\Vert \hat{x}{\Vert }_{{\mathcal{B}}_{\vartheta }}^2\right),\mathrm{\Theta }\left({\breve{\mathcal{P}}}_m\left(\varpi \right)\Vert \hat{x}{\Vert }_{{\mathcal{B}}_{\vartheta }}^2\right)⩽{\breve{\mathcal{P}}}_m\left(\varpi \right)\mathrm{\Theta }\left(\Vert \hat{x}{\Vert }_{{\mathcal{B}}_{\vartheta }}^2\right),$$ for each $\varpi \in \mathcal{V},\widehat{x}\in {\mathcal{B}}_{\vartheta }$ .

• ${\mathbf{H}}_{\mathbf{7}}$ :

  The following inequality holds $${\mathbb{P}}_4{\int }_0^{\mathfrak{p}}{\mathfrak{m}}_{\mathfrak{z}}\left(\mu \right)\left(1+{\breve{\mathcal{P}}}_l\left(\mu \right)+{\breve{\mathcal{P}}}_m\left(\mu \right)\right)d\mu ⩽{\int }_c^{\infty }\frac{1}{\mathrm{\Theta }\left(\wp \right)}d\wp ,$$ where

  (3.2)

  $$\begin{array}{cc}\hfill & \bar{P}=6\left\{P_{1}E\Vert \alpha \left(0\right){\Vert }^2+P_{2}E\Vert z_1{\Vert }^2+6{(\mathfrak{p}/\eta )}^2{P}_{\mathcal{B}}^2{P}_2^2\left[E\Vert E{\widehat{z}}_{\mathfrak{p}}+{\int }_0^{\mathfrak{p}}\widehat{\varphi }\left(\zeta \right)\mathit{dW}\left(\zeta \right){\Vert }^2\right.\right.\hfill \\ \hfill & \left.+P_{1}E\Vert \alpha \left(0\right){\Vert }^2+P_{2}E\Vert z_1{\Vert }^2+2ȷP_1{\displaystyle \sum _{\iota =1}^{ȷ}}E\Vert J_{\iota }\left(0\right){\Vert }^2+2ȷP_2{\displaystyle \sum _{\iota =1}^{ȷ}}E\Vert \overline{J_{\iota }}\left(0\right){\Vert }^2\right]\hfill \\ \hfill & \left.+2ȷP_1{\displaystyle \sum _{\iota =1}^{ȷ}}E\Vert J_{\iota }\left(0\right){\Vert }^2+2ȷP_2{\displaystyle \sum _{\iota =1}^{ȷ}}E\Vert \overline{J_{\iota }}\left(0\right){\Vert }^2\right\},\hfill \\ \hfill & {\mathbb{K}}_0=2{\ell }^2\left[{(6\mathfrak{p}/\eta )}^2{P}_{\mathcal{B}}^2{P}_2^2+6\right]\left[2ȷP_1{\displaystyle \sum _{\iota =1}^{ȷ}}{P}_{\iota }+2ȷP_2{\displaystyle \sum _{j=1}^k}\overline{P_{\iota }}\right],\hfill \end{array}$$

  (3.3)

  $$\begin{array}{cc}\hfill & {\mathbb{K}}_1=2{\ell }^2\left\{{(6\mathfrak{p}/\eta )}^2{P}_{\mathcal{B}}^2{P}_2^3\mathit{Tr}\left(\mathcal{Q}\right){\int }_0^{\mathfrak{p}}{\mathfrak{m}}_{\mathfrak{z}}\left(\zeta \right)\mathrm{\Theta }\left(\delta \left(\zeta \right)+{\breve{\mathcal{P}}}_l\left(\zeta \right)\delta \left(\zeta \right)+{\breve{\mathcal{P}}}_m\left(\zeta \right)\delta \left(\zeta \right)\right)d\zeta +\bar{P}\right\}\hfill \\ \hfill & +2\Vert \alpha {\Vert }_{{\mathcal{B}}_{\vartheta }}^2,\hfill \end{array}$$

  (3.4)

  $${\mathbb{K}}_2=12{\ell }^2{P}_2\mathit{Tr}\left(\mathcal{Q}\right),{\mathbb{P}}_3={\mathbb{K}}_1/(1-{\mathbb{K}}_0),{\mathbb{P}}_4={\mathbb{K}}_2/(1-{\mathbb{K}}_0),$$

  (3.5)

  $$\begin{array}{cc}& P_0=3ȷP_2\sum _{\iota =1}^{ȷ}{\bar{P}}_{\iota }+3ȷP_1\sum _{\iota =1}^{ȷ}{P}_{\iota }+\frac{9}{{\eta }^2}{\mathfrak{p}}^3{P}_2^3{P}_{\mathcal{B}}^2\mathit{Tr}\left(\mathcal{Q}\right){\ell }^2{P}_G\left(1+P_l+P_m\right)\\ & +\frac{9}{{\eta }^2}ȷP_1{P}_2^2{P}_{\mathcal{B}}^2{\mathfrak{p}}^2\sum _{\iota =1}^{ȷ}{P}_{\iota }+\frac{9}{{\eta }^2}ȷP_2^3{P}_{\mathcal{B}}^2{\mathfrak{p}}^2\sum _{\iota =1}^{ȷ}\overline{P_{\iota },}\end{array}$$

  (3.6)

  $$\begin{array}{cc}\hfill & \mathrm{and}\hfill \\ \hfill & P_{\mathcal{B}}=\Vert \mathcal{B}{\Vert }^2\text{.}\hfill \end{array}$$

To figure out the control function, the next lemma is required.