Effects of fractional derivative and Wiener process on approximate boundary controllability of differential inclusion

Abstract

One of the core concepts of contemporary control theory is the idea that a dynamical system can be controlled. Several abstract settings have been developed to describe the distributed control systems in a domain in which the control is acted through the boundary. In this manuscript, we investigate the boundary approximate controllability (ABC) of stochastic differential inclusion (SDI) with Hilfer fractional derivative (HFD) and nonlocal condition by implementing the principle of stochastic analysis, the fixed-point theorem, fractional calculus and multi-valued map. Moreover, an example is offered to define the primary results.

1 Introduction

Many real-world phenomena, such as population growth, stock prices, weather prediction model and heat conduction in materials with memory are affected by random influences or noise. For example, Li et al. [1] studied the stochastic properties of thermoacoustic oscillations in an annular gas turbine combustion chamber driven by colored noise. Mohammed and El-Morshedy [2] discussed the influence of multiplicative noise on the stochastic exact solutions of the Nizhnik–Novikov–Veselov system. He et al. [3] studied the stochastic SIS model driven by random diffusion of air pollutants. Omar et al. [4] investigated the fractional stochastic modeling of COVID-19 under wide spread of vaccinations. Shone et al. [5] introduced applications of stochastic modeling in air traffic management.

In recent years, several authors have been interested in studying of fractional stochastic differential inclusions, which are a generalization of stochastic differential equations. For example, Sivasankar and Udhayakumar [6] studied the Hilfer fractional neutral stochastic Volterra integro-differential inclusions via almost sectorial operators. Yan and Zhang [7] discussed the existence of solutions to impulsive fractional partial neutral stochastic integro-differential inclusions with state-dependent delay. Boudaoui et al. [8] introduced the impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay. Balasubramaniam et al. [9] discussed the optimality of non-instantaneous impulsive fractional stochastic differential inclusion with fBm. Upadhyay and Kumar [10] studied the exponential nature and solvability of stochastic multi-term fractional differential inclusions with Clarke’s subdifferential. Raja and Vijayakumar [11] derived the existence results for Caputo fractional mixed Volterra–Fredholm-type integro-differential inclusions of order $r \in (1, 2)$ with sectorial operators.

Controllability is the capability of a control system to be directed from an arbitrary initial state to a likewise arbitrary final state through the permitted set of controls, for example, Raja et al. [12] discussed the existence and controllability results for fractional integro-differential inclusions of order $r \in (1, 2]$ with impulses. Johnson et al. [13] established optimal control results for impulsive fractional delay integro-differential equations of order $1< r< 2$ via sectorial operator. Kavitha and Vijayakumar [14] constructed the controllability results of non-densely defined Hilfer fractional neutral differential equations with finite delay. Ahmed and Ragusa [15] investigated the nonlocal controllability of Sobolev-type conformable fractional stochastic evolution inclusions with Clarke subdifferential. Wang and Ahmed [16] studied the null controllability of nonlocal Hilfer fractional stochastic differential equations. Sathiyaraj et al. [17] established the controllability and optimal control for a class of time-delayed fractional stochastic integro-differential systems. Dhayal and Malik [18] studied the existence and controllability of impulsive fractional stochastic differential equations driven by Rosenblatt process with Poisson jumps. Yan and Zhou [19] presented the optimization of exact controllability for fractional impulsive partial stochastic differential systems via analytic sectorial operators. Zhang and Guo [20] studied the controllability of stochastic game-based control systems. Sivasankar et al. [21] introduced a new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. Bedi et al. [22] studied the Hilfer fractional evolution equations by the properties of controllability and stability.

Approximate controllability means that system can be steered to arbitrary small neighborhood of final state. Many authors studied the approximate controllability for different nonlinear stochastic system, for example, Raja et al. [23] investigated the approximate controllability results for fractional integro-differential systems of order $1< r < 2$ with sectorial operators. Kavitha and Vijayakumar [24] studied the approximate controllability for Hilfer fractional neutral evolution hemivariational inequality. Bedi et al. [25] discussed the existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators. Tamilalagan and Balasubramaniam [26] investigated the approximate controllability of fractional stochastic differential equations driven by mixed fractional Brownian motion via resolvent operators. Ahmed et al. [27] discussed the approximate controllability of noninstantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion. Mahmudov [28] discussed the mean square finite-approximate controllability of semilinear stochastic differential equations with non-Lipschitz coefficients. Sakthivel et al. [29] established the approximate controllability of nonlinear fractional dynamical systems. Zhang et al. [30] studied the approximate controllability of impulsive fractional stochastic differential equations with state-dependent delay. Furthermore, few authors studied the approximate boundary controllability, for example, Carthel et al. [31] discussed the exact and the approximate boundary controllability for the heat equation. Wang [32] investigated the approximate boundary controllability for semilinear delay differential equations. Palanisamy and Chinnathambi [33] studied the approximate boundary controllability of Sobolev-type stochastic differential systems. Olive [34] introduced the boundary approximate controllability of some linear parabolic systems. Li et al. [35] studied the approximate boundary null controllability and approximate boundary synchronization for a coupled system of wave equations with Neumann boundary controls. Ahmed et al. [36] established sufficient conditions for approximate boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fractional Brownian motion and Poisson jumps. Through the fixed-point theorem techniques, many authors proved the existence solution and controllability for different problems, for example, Klamka et al. [37] discussed the Banach fixed-point theorem in semilinear controllability problems. Ma et al. [38] studied the approximate controllability of Atangana–Baleanu fractional neutral delay integro-differential stochastic systems with nonlocal conditions by using the fixed point theorem and Dunford–Pettis theorem. Priya and Kaliraj [39] introduced application of fixed point technique of Rothe’s-type to interpret the controllability criteria of neutral, nonlinear, fractional-order impulsive system.

On the other hand, we noted that the approximate controllability results associated with the nonlocal Hilfer fractional stochastic differential inclusions are very rare and have not yet been processed. By presenting this work, we aim to fill this gap in the literature. Therefore, our results are entirely new and contribute to giving a valuable idea on this subject.

Now, consider the nonlocal SDI with HFD and control on the boundary, as:

$$\begin{aligned} \left \{ \textstyle\begin{array}{ll} D_{0+}^{\mathfrak{I} ,\mathfrak{h} } m(t) \in Gm(t)+ \sigma (t,m(t))+ \wp (t,m(t)) \frac{d\omega (t)}{dt} \\ + \partial \mathfrak{L}(t,m(t)),~ t \in \mathfrak{D}= (0,\alpha ], \\ \delta m(t)= \varrho _{1} \mathcal{X}(t),~ t \in \mathfrak{D}= (0, \alpha ], \\ I^{(1-\mathfrak{h})(1-\mathfrak{I})}_{0+}m(0)+\mathfrak{H}(m)= \mathfrak{g}~ m_{0}, \end{array}\displaystyle \right . \end{aligned}$$

(1.1)

where

$$ D^{\mathfrak{I} ,\mathfrak{h} }_{0+} m(t)= I^{\mathfrak{I} (1- \mathfrak{h} )}_{0+} \frac{d}{dt}I^{(1-\mathfrak{I} )(1-\mathfrak{h} )}_{0+} m(t),~~ I^{\mathfrak{h}} m(t)= \frac{1}{\Gamma (\mathfrak{h} )} \int ^{t}_{0} \frac{m(\Im )}{(t-\Im )^{1-\mathfrak{h} }} d\Im ,~~ t> 0,~ \mathfrak{h} > 0, $$

is the Hilfer fractional derivative (HFD) with order $0 \leq \mathfrak{I} \leq 1$, $\frac{1}{2}< \mathfrak{h} < 1$ (see [40, 41]).

The separable Hilbert space is denoted by $\mathfrak{B}$. In the current paper, Y and $\mathfrak{W}$ are the Hilbert space. $G: \mathfrak{W} \rightarrow \mathfrak{B}$ is linear bounded operator and $\delta : \mathfrak{W} \rightarrow \mathfrak{B}$ is linear operator. $\varrho _{1}: Y \rightarrow \mathfrak{W}$ is bounded linear operator. $m(\cdot )$ has values in $\mathfrak{W}$. Suppose the operator $\Xi : \mathfrak{W}\rightarrow \mathfrak{W}$ is defined by $Dom(\Xi )=\{m \in Dom(G); Gm = 0 \}$, $\Xi ~ m=Gm$, $\forall ~ m \in Dom(\Xi )$.

In this paper, $(\Omega , \mathcal{S}, \{\mathcal{S}_{t}\}_{t \geq 0},P)$ is a given filtered probability space that satisfies the usual conditions of completeness and right continuity. Also, we define $\{\omega (t)\}_{t \geq 0}$ is $\mathfrak{B}$-valued Brownian motion whose normal filtration is $\{\mathcal{S}_{\mathfrak{O}}\}_{\mathfrak{O} \geq 0}$. The space of all bounded linear operators from $\mathfrak{B}$ into $\mathfrak{W}$ is denoted by $L(\mathfrak{B}, \mathfrak{W})$ with $\parallel . \parallel$. $\partial \mathfrak{L}(t,m(t))$ be Clarke’s subdifferential of $\mathfrak{L}(t,m(t))$. $\mathcal{X}(\cdot )$ is the control function in $L^{2}(\mathfrak{D},Y)$. The nonlinear functions σ, ℘ and $\mathfrak{H}$ are defined later.

2 Preliminaries

Assume, $\lambda := \mathfrak{C}(\mathfrak{D}, L^{2}(\Omega , \mathcal{S},P, \mathfrak{W})$ with $\| m \|_{\lambda}= \sup _{t \in \mathfrak{D}}E \| t^{(1-\mathfrak{I})(1- \mathfrak{h})}m(t) \|^{2})^{1/2}$ is the Banach space of all continuous functions. Here, let $\mathfrak{Z}_{\mathfrak{r}} = \{m\in \lambda :\|m\|^{2}_{\lambda} \leq \mathfrak{r}\}$, where $\mathfrak{r}> 0$.

Definition 2.1

([42])

Assume $\mathfrak{J}^{*}$ is the dual space of the Banach space $\mathfrak{J}$. The Clarke’s generalized directional derivative of a locally Lipschitz function $\mathfrak{L}: \mathfrak{J} \rightarrow \mathfrak{R}$ at $\mathfrak{a}$ in the direction ς is represented by

$$\begin{aligned} \mathfrak{L}^{0}{(\mathfrak{a};\varsigma )}= \limsup _{{y\rightarrow{0^{+}}}~{ \mathfrak{v}\rightarrow \mathfrak{a}}} \frac{{ \mathfrak{L}(\mathfrak{v} + y \varsigma ) - \mathfrak{L}(\mathfrak{v})}}{y }, \end{aligned}$$

and

$$\begin{aligned} \partial \mathfrak{L}(\mathfrak{a})=\{\mathfrak{a}^{*} \in \mathfrak{J}^{*}: \mathfrak{L}^{0} {(\mathfrak{a};\varsigma )}\geq \left \langle {{\mathfrak{a}^{*}},\varsigma} \right \rangle ,~ \forall ~ \varsigma \in \mathfrak{J}\}, \end{aligned}$$

where $\partial \mathfrak{L}(\mathfrak{a})$ is the Clarke’s generalized gradient of $\mathfrak{L}$ at $\mathfrak{a}$.

We need the succeeding hypotheses:

$(H1)$ Domain G is subset from Domain δ.

$(H2)$ $\varrho : Y\rightarrow \mathfrak{W}$ be continuous linear operator, such that (s.t.) ∀ $\mathcal{X} \in Y$ $\exists ~ \varrho \mathcal{X}$ in Domain $G,~ \delta (\varrho \mathcal{X})= \varrho _{1} \mathcal{X}$ and $E \| \varrho \mathcal{X} \|^{2} \leq C_{1} E \| \varrho _{1} \mathcal{X} \|^{2}$, where $C_{1}$ is a constant.

$(H3)$ A compact semigroup of bounded operator $\{\mathcal{V}(t),t\geq 0\}$ in $\mathfrak{W}$ is formed by Ξ and $\sup _{t\in \mathfrak{D}}\|\mathcal{V}(t)\|\leq \Pi$, $\Pi >0$.

$(H4)$ ∀ $t \in \mathfrak{D}$ and $\mathcal{X} \in Y$, $\mathcal{V}(t)\varrho \mathcal{X} \in Dom(\Xi )$ and $\| \Xi \mathcal{V}(t)\| \leq \Pi _{1}$, $\Pi _{1} >0$.

$(H5)$ $\sigma : \mathfrak{D}\times \mathfrak{W} \rightarrow 2^{\mathfrak{W}}$ is locally Lipschitz continuous (LLC), ∀ $t \in \mathfrak{D},~m,~ m_{1}$, $m_{2} \in \mathfrak{W}$, $\exists ~C_{2} > 0$ s.t.

$$ E\|\sigma (t,m_{1}) - \sigma (t,m_{2}) \|^{2} \leq C_{2} (E\| m_{1} - m_{2} \|^{2} ,~E\|\sigma (t,m) \|^{2} \leq C_{2} (1+E\| m\|^{2}). $$

$(H6)$ $\wp : \mathfrak{D}\times \mathfrak{W} \rightarrow L_{\Theta}( \mathfrak{B},\mathfrak{W})$ is LLC, ∀ $t \in \mathfrak{D},~m,~ m_{1}$, $m_{2} \in \mathfrak{W}$, $\exists ~C_{3} > 0$ s.t.

$$ E\|\wp (t,m_{1}) - \wp (t,m_{2}) \|_{\Theta}^{2} \leq C_{3} (E\| m_{1} - m_{2} \|^{2} ,~E\|\wp (t,m) \|_{\Theta}^{2} \leq C_{3} (1+E\| m\|^{2}). $$

Here, $L_{\Theta }(\mathfrak{B},\mathfrak{W})$ be the space of all Θ-Hilbert-Schmidt operators from $\mathfrak{B}$ to $\mathfrak{W}$.

$(H7)$ $\mathfrak{L}: \mathfrak{D}\times \mathfrak{W} \rightarrow \mathfrak{R}$ verifies:

$(I)$ $\mathfrak{L}(\cdot ,m): \mathfrak{D}\rightarrow \mathfrak{R}$ be measurable ∀ $m \in \mathfrak{W}$,

$(II)$ $\mathfrak{L}(t,\cdot ): \mathfrak{W}\rightarrow \mathfrak{R}$ is LLC for a.e. $t \in \mathfrak{D}$,

$(III)$ there exists $\vartheta \in L^{1}(\mathfrak{D}, \mathfrak{R}^{+})$ and $C_{4}>0$ verifying

$$ E \|\partial \mathfrak{L}(t,m)\|^{2}= \sup \{ E \|\mathfrak{N}(t)\|^{2}: \mathfrak{N}(t) \in \partial \mathfrak{L}(t,m)\} \leq \vartheta (t)+C_{4} E \|m \|^{2}, $$

for all $m \in \mathfrak{W}$ a.e. $t\in \mathfrak{D}$ and $m \in \mathfrak{W}$.

$(H8)$ $\mathfrak{H}:C(\mathfrak{D},\mathfrak{W})\rightarrow \mathfrak{W}$ is continuous, for any $m,~m_{1}, ~ m_{2}\in C(\mathfrak{D},\mathfrak{W})~\exists ~C_{5} > 0$ s.t.

$$ E\| \mathfrak{H}(m_{1})-\mathfrak{H}(m_{2})\|^{2}\leq C_{5} E \| m_{1}-m_{2} \|^{2},~~E\| \mathfrak{H}(m)\|^{2}\leq C_{5}(1+ E \| m\|^{2}). $$

Consider $m(t)$ be the solution of (1.1) and $\mathfrak{x}(t)=m(t)- \varrho \mathcal{X}(t)$.

Hence (1.1) can be written in the form:

$$\begin{aligned} \left \{ \textstyle\begin{array}{ll} D_{0+}^{\mathfrak{I} ,\mathfrak{h} } \mathfrak{x}(t)\in \Xi \mathfrak{x}(t)+G\varrho \mathcal{X}(t)- \varrho D_{0+}^{\mathfrak{I} , \mathfrak{h} } \mathcal{X}(t) \\ +\sigma (t,m(t))+ \wp (t,m(t)) \frac{d\omega (t)}{dt}+ \partial \mathfrak{L}(t,m(t)),~ t \in \mathfrak{D}= (0,\alpha ], \\ I^{(1-\mathfrak{I})(1-\mathfrak{h})}_{0+}[\mathfrak{x}(0)+\varrho \mathcal{X}(0)]=I^{(1-\mathfrak{I})(1-\mathfrak{h})}_{0+}m(0)=m_{0}- \mathfrak{H}(m). \end{array}\displaystyle \right . \end{aligned}$$

Hence, the integral inclusion of (1.1) is given by

$$\begin{aligned} m(t) \in & \frac{m_{0}-\mathfrak{H}(m)}{\Gamma (\mathfrak{I}+ \mathfrak{h}-\mathfrak{I} \mathfrak{h})} t^{(\mathfrak{I}-1)(1-\mathfrak{h})}+ \frac{1}{\Gamma (\mathfrak{h})} \int _{0}^{t} (t-\Im )^{\mathfrak{h}-1} \Xi m(\Im )d\Im \\ &+\frac{1}{\Gamma (\mathfrak{h})} \int _{0}^{t} (t-\Im )^{ \mathfrak{h}-1}[G-\Xi ] \varrho \mathcal{X}(\Im )d\Im + \frac{1}{\Gamma (\mathfrak{h})} \int _{0}^{t} (t-\Im )^{\mathfrak{h}-1} \sigma (\Im ,m(\Im ))d \Im \\ &+ \frac{1}{\Gamma (\mathfrak{h})} \int _{0}^{t} (t-\Im )^{ \mathfrak{h}-1}\wp (\Im ,m(\Im ))d\omega (\Im ) \\ &+ \frac{1}{\Gamma (\mathfrak{h})} \int _{0}^{t} (t-\Im )^{ \mathfrak{h}-1} \partial \mathfrak{L}(\Im ,m(\Im ))d \Im . \end{aligned}$$

(2.1)

Lemma 2.1

([36])

If (2.1) verifies, then the mild solution of (1.1) is in the form:

$$\begin{aligned} m(t) =&\mathcal{V}_{\mathfrak{I} ,\mathfrak{h}}(t) [m_{0}- \mathfrak{H}(m)]+\int _{0}^{t} [P_{\mathfrak{h}} (t-\Im )G- \Xi P_{ \mathfrak{h}} (t-\Im )] \varrho \mathcal{X}(\Im )d\Im \\ &+ \int _{0}^{t} P_{\mathfrak{h}} (t-\Im )\sigma (\Im , m(\Im )) d \Im + \int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \mathfrak{N}(\Im ) d \Im \\ &+ \int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \wp (\Im , m(\Im )) d \omega (\Im ),~~t \in \mathfrak{D,} \end{aligned}$$

where

$$ \mathcal{V}_{\mathfrak{I},\mathfrak{h}}(t)= I^{\mathfrak{I}(1- \mathfrak{h})}_{0+} P_{\mathfrak{h}} (t),~~~~~~P_{\mathfrak{h}} (t)= t^{ \mathfrak{h}-1} T_{\mathfrak{h}} (t),~~~~ T_{\mathfrak{h}} (t)= \int _{0}^{ \infty}\mathfrak{h}\upsilon \Psi _{\mathfrak{h}}(\upsilon ) \mathcal{V}(t^{\mathfrak{h}} \upsilon ) d \upsilon , $$

with

$${\mathrm{\Psi }}_{\mathfrak{h}}(\upsilon )=\sum _{\mathbb{n}=1}^{\mathrm{\infty }}\frac{{(-\upsilon )}^{\mathbb{n}-1}}{(\mathbb{n}-1)!\mathrm{\Gamma }(1-\mathbb{n}\mathfrak{h})},\upsilon \in (0,\mathrm{\infty }).$$

Lemma 2.2

([43])

The operators $\mathcal{V}_{\mathfrak{I},\mathfrak{h}}$ and $P_{\mathfrak{h}}$ satisfy the following:

(i) $\{P_{\mathfrak{h}} (t):t>0\}$ is continuous.

(ii)

$$ \|P_{\mathfrak{h}}(t) m\|\leq \frac{\Pi t^{\mathfrak{h}-1}}{\Gamma (\mathfrak{h})}\|m\|,~\| \mathcal{V}_{\mathfrak{I},\mathfrak{h}}(t)m\|\leq \frac{\Pi t^{(\mathfrak{I} -1)(1-\mathfrak{h})}}{\Gamma (\mathfrak{I}(1-\mathfrak{h})+\mathfrak{h})} \|m\|,~t>0. $$

(iii) $P_{\mathfrak{h}}(t)$, $\mathcal{V}_{\mathfrak{I},\mathfrak{h}}(t)$ are strongly continuous, for $t>0$.

Lemma 2.3

([36])

$\| \Xi P_{\mathfrak{h}} (t) m \|\leq \frac{ \Pi _{1} t^{\mathfrak{h}-1}}{\Gamma (\mathfrak{h})} \|m\|$, if $(H4)$ is satisfied.

Here, $\digamma : L^{2}_{\mathcal{S}}(\mathfrak{D},\mathfrak{W}) \rightarrow 2^{L^{2}_{\mathcal{S}}(\mathfrak{D},\mathfrak{W})}$ is defined as:

$\digamma (m)= \{ \mathfrak{N} \in L^{2}_{\mathcal{S}}(\mathfrak{D}, \mathfrak{W}): \mathfrak{N}(t) \in \partial \mathfrak{L}(t,m(t))$ a.e. $t \in \mathfrak{D}$ for $m \in L^{2}_{\mathcal{S}}(\mathfrak{D},\mathfrak{W})\}$.

Lemma 2.4

([42])

∀ $m \in L^{2}_{\mathcal{S}}(\mathfrak{D},\mathfrak{W})$, $\digamma (m)$ is convex, weakly compact and nonempty, s.t. $(H7)$ is verified.

Lemma 2.5

([42])

If $m_{n}\rightarrow m$ in $L^{2}_{\mathcal{S}}(\mathfrak{D},\mathfrak{W})$, $\beta _{n}\rightarrow \beta$ weakly in $L^{2}_{\mathcal{S}}(\mathfrak{D},\mathfrak{W})$ and $\beta _{n} \in \digamma (m_{n})$, then $\beta \in \digamma (m)$, s.t., $(H7)$ is realized.

Theorem 2.1

([44])

Let $\mathcal{K}_{\varepsilon}: \mathfrak{Q} \rightarrow 2^{\mathfrak{Q}}$, $\mathfrak{Q}$ be upper semicontinuous multi-valued maps, locally convex Banach space and compact convex valued (CCV) s.t. $\mathcal{K}_{\varepsilon}(\mathfrak{L})$ is relatively compact (r.c.) set. Then, $\mathcal{K}_{\varepsilon}$ has a fixed point, if $\Psi = \{ m \in \mathfrak{Q} : \varsigma m \in \mathcal{K}_{ \varepsilon}(m), \varsigma > 1 \}$ is bounded.

3 Approximate boundary controllability

To discuss the ABC for (1.1), we consider the linear SDI with HFD and control on the boundary

$$\begin{aligned} \left \{ \textstyle\begin{array}{ll} D_{0+}^{\mathfrak{I} ,\mathfrak{h} } m(t) \in Gm(t)+ \sigma (t)+ \wp (t) \frac{d\omega (t)}{dt},~~~ t \in \mathfrak{D}, \\ \delta m(t)= \varrho _{1} \mathcal{X}(t),~ t \in \mathfrak{D}, \\ I^{(1-\mathfrak{h})(1-\mathfrak{I})}_{0+}m(0)=m_{0}. \end{array}\displaystyle \right . \end{aligned}$$

(3.1)

We present the operators associated with (3.1) as

$$\begin{aligned} L_{0}^{\alpha }=\int _{0}^{\alpha }(\alpha -\Im )^{\mathfrak{h}-1}[T_{ \mathfrak{h}} (\alpha -\Im )G- \Xi T_{\mathfrak{h}} (\alpha -\Im )] \varrho \varrho ^{*} [T_{\mathfrak{h}} (\alpha -\Im )G- \Xi T_{ \mathfrak{h}} (\alpha -\Im )]^{*}d\Im , \end{aligned}$$

and

$$\begin{aligned} N(\alpha , L_{0}^{\alpha}) =& (\alpha I+L_{0}^{\alpha})^{-1},~\alpha > 0, \end{aligned}$$

where the adjoint of ϱ and $[T_{\mathfrak{h}} (\alpha -\Im )G- \Xi T_{\mathfrak{h}} (\alpha - \Im )]$ are denoted by $\varrho ^{*}$ and $[T_{\mathfrak{h}} (\alpha -\Im )G- \Xi T_{\mathfrak{h}} (\alpha - \Im )]^{*}$, respectively.

Let the state value of (1.1) at terminal state α be $m(\alpha ;m_{0}, \mathcal{X})$, according to the control $\mathcal{X}$ and $m_{0}$. $N(\alpha ,m_{0}) = \{ m(\alpha ;m_{0},\mathcal{X}):\mathcal{X} \in L^{2}( \mathfrak{D},Y) \}$ is the reachable set of (1.1) at terminal time α and $\overline{N(\alpha ,m_{0})}$ is the closure in $\mathfrak{W}$.

Definition 3.1

([29])

(1.1) is approximately controllable on $\mathfrak{D}$ if $\overline{N(\alpha ,m_{0})} = L^{2}(\Omega , \mathfrak{W})$.

Lemma 3.1

([29])

If and only if $\zeta (\zeta I + L^{\alpha}_{0})^{-1}\rightarrow 0$ as $\zeta \rightarrow 0^{+}$, then (3.1) is approximately controllable on $\mathfrak{D}$.

Lemma 3.2

For any $\overline{ m}_{\alpha }\in L^{2}(\Omega , \mathfrak{W})$ ∃ $\bar{\Upsilon}\in L^{2}(\Omega ; L^{2}(\mathfrak{D};L_{\Theta}))$ s.t.

$$ \overline{ m}_{\alpha}= E ~\overline{ m}_{\alpha }+ \int _{0}^{ \alpha }\bar{\Upsilon}(\Im ) d\omega (\Im ). $$

Now, for any $m_{\alpha }\in L^{2}(\Omega ,\mathfrak{W})$, the function of control is introduced as

$$\begin{aligned} \mathcal{X}(t) =&\varrho ^{*} [T_{\mathfrak{h}} (\alpha -t)G- \Xi T_{ \mathfrak{h}} (\alpha -t)]^{*} (\zeta ~ I + L^{\alpha}_{0})^{-1} \{E~ \overline{ m}_{\alpha }- \mathcal{V}_{\mathfrak{I} ,\mathfrak{h}}( \alpha ) [m_{0}-\mathfrak{H}(m)] \\ &- \int _{0}^{\alpha }P_{\mathfrak{h}} (\alpha -\Im )\sigma (\Im , m( \Im )) d\Im - \int _{0}^{\alpha }P_{\mathfrak{h}} (\alpha -\Im ) \mathfrak{N}(\Im ) d\Im \\ &- \int _{0}^{\alpha }P_{\mathfrak{h}} (\alpha -\Im ) \wp (\Im , m( \Im )) d\omega (\Im )+ \int _{0}^{\alpha }\bar{\Upsilon}(\Im ) d \omega (\Im )\},~~t \in \mathfrak{D}. \end{aligned}$$

Through this paper, set $\sup \{E \| \bar{\Upsilon}(\Im )\|^{2}_{\Theta}:0 < \Im < t \leq \alpha \}=C_{6}$.

Theorem 3.1

If $(H1)$–$(H8)$ are verified, then there is mild solution of (1.1) on $\mathfrak{D}$ s.t.

$$\begin{aligned} \Re _{2} =&\Bigg\{ \frac{25 C_{5} \Pi ^{2} \alpha ^{2(\mathfrak{I}-1)(1-\mathfrak{h})}}{\Gamma ^{2}(\mathfrak{I}(1-\mathfrak{h})+\mathfrak{h})} + \frac{25 \Pi ^{2} \alpha ^{2\mathfrak{h}-1}}{(2\mathfrak{h}-1)\Gamma ^{2}(\mathfrak{h})} \bigg[(C_{2}+Tr(\Theta )C_{3}) +C_{4} \alpha \bigg]\Bigg\} \\ &\times \left \{ 1+ \frac{25 \|\varrho \|^{2} \| \varrho ^{*} \|^{2} \alpha ^{2\mathfrak{h}-1}[\|G\|^{2} \Pi ^{2}+ \Pi _{1}^{2}]^{2}}{\zeta ^{2}(2\mathfrak{h}-1)\Gamma ^{4}(\mathfrak{h}) } \right \}< 1. \end{aligned}$$

Proof

Consider the map $\mathcal{K}_{\varepsilon}: \lambda \rightarrow 2^{\lambda}$ as follows

$$\begin{aligned} \mathcal{K}_{\varepsilon }(m)=\left \{ \textstyle\begin{array}{ll} \mathfrak{U} \in \lambda : \mathfrak{U}(t)=\mathcal{V}_{\mathfrak{I} , \mathfrak{h}}(t) [m_{0}-\mathfrak{H}(m)]+\int _{0}^{t} [P_{ \mathfrak{h}} (t-\Im )G- \Xi P_{\mathfrak{h}} (t-\Im )] \varrho \mathcal{X}(\Im )d\Im \\ + \int _{0}^{t} P_{\mathfrak{h}} (t-\Im )\sigma (\Im , m(\Im )) d\Im + \int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \mathfrak{N}(\Im ) d\Im \\ + \int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \wp (\Im , m(\Im )) d \omega (\Im )~,\mathfrak{N} \in F(m). \end{array}\displaystyle \right \} \end{aligned}$$

For $t \in \mathfrak{D}$, and from $(H1)$–$(H9)$, we have

$$\begin{aligned} &E \|\mathcal{X}(t)\|^{2} \\ &\leq 25 E \|\varrho ^{*} [T_{\mathfrak{h}} (\alpha -t)G- \Xi T_{ \mathfrak{h}} (\alpha -t)]^{*} (\zeta ~ I + L^{\alpha}_{0})^{-1} \bigg[ E~ \overline{ m}_{\alpha}+ \int _{0}^{\alpha }\bar{\Upsilon}( \Im ) d\omega (\Im )\bigg]\|^{2} \\ &+ 25 E \|\varrho ^{*} [T_{\mathfrak{h}} (\alpha -t)G- \Xi T_{ \mathfrak{h}} (\alpha -t)]^{*} (\zeta ~ I + L^{\alpha}_{0})^{-1} \mathcal{V}_{\mathfrak{I} ,\mathfrak{h}}(\alpha ) [m_{0}-\mathfrak{H}(m)] \|^{2} \\ &+ 25 E \|\varrho ^{*} [T_{\mathfrak{h}} (\alpha -t)G- \Xi T_{ \mathfrak{h}} (\alpha -t)]^{*} (\zeta ~ I + L^{\alpha}_{0})^{-1} \int _{0}^{\alpha }P_{\mathfrak{h}} (\alpha -\Im )\sigma (\Im , m( \Im )) d\Im \|^{2} \\ &+ 25 E \|\varrho ^{*} [T_{\mathfrak{h}} (\alpha -t)G- \Xi T_{ \mathfrak{h}} (\alpha -t)]^{*} (\zeta ~ I + L^{\alpha}_{0})^{-1} \int _{0}^{\alpha }P_{\mathfrak{h}} (\alpha -\Im ) \mathfrak{N}(\Im ) d \Im \|^{2} \\ &+ 25 E \|\varrho ^{*} [T_{\mathfrak{h}} (\alpha -t)G- \Xi T_{ \mathfrak{h}} (\alpha -t)]^{*} (\zeta ~ I + L^{\alpha}_{0})^{-1} \int _{0}^{\alpha }P_{\mathfrak{h}} (\alpha -\Im ) \wp (\Im , m(\Im )) d\omega (\Im )\|^{2} \\ &\leq \frac{25 \|\varrho ^{*}\|^{2} [\|G\|^{2} \Pi ^{2} + \Pi _{1}^{2}]}{\zeta ^{2}\Gamma ^{2}(\mathfrak{h}) } \bigg\{ E \| \overline{ m}_{\alpha}\|^{2}+ Tr(\Theta ) \alpha ~ C_{6} \\ &+ \frac{ \Pi ^{2} \alpha ^{2(\mathfrak{I}-1)(1-\mathfrak{h})}}{\Gamma ^{2}(\mathfrak{I}(1-\mathfrak{h})+\mathfrak{h})} \bigg[E \|m_{0} \|^{2} + C_{5} (1+\mathfrak{r})\bigg] \\ &+ \frac{ \Pi ^{2} \alpha ^{2\mathfrak{h}-1)}}{(2\mathfrak{h}-1)\Gamma ^{2}(\mathfrak{h})} \bigg[(C_{2}+Tr(\Theta )C_{3}) (1+\mathfrak{r})+ \|\mathfrak{N} \|_{L^{1}( \mathfrak{D},R^{+})}+C_{4} \alpha \mathfrak{r}\bigg] \bigg\} . \end{aligned}$$

We show that $\mathcal{K}_{\varepsilon}$ has a fixed point, so we classified the proof into 6 steps.

${\mathbf{{\mathfrak{S1}:}}}$ For all $m \in \lambda$, $\mathcal{K}_{\varepsilon}(m)$ be convex, weakly compact and nonempty.

We can use Lemma 2.4 to show that $\mathcal{K}_{\varepsilon }(m)$ is weakly compact and nonempty.

As $\digamma (m)$ has convex values; so, if $\kappa _{1},~\kappa _{2} \in \digamma (m)$ and for all $\ell \in (0,1)$, then $\ell \kappa _{1}+(1-\ell )\kappa _{2} \in \digamma (m)$. Then, $\mathcal{K}_{\varepsilon }(m)$ is convex.

${\mathbf{{\mathfrak{S2}:}}}$ We present $\mathcal{K}_{\varepsilon }$ to be bounded on subset from λ.

Obviously, $\mathfrak{L}_{\mathfrak{r}}$ be a BCC of λ.

Next, we explore, for $\tau > 0$ and $\mathfrak{U} \in \mathcal{K}_{\varepsilon }(m)$, $m \in \mathfrak{L}_{ \mathfrak{r}}$, then $E\| \mathfrak{U}(t)\|^{2} \leq \tau$.

If $\chi \in \mathcal{K}_{\varepsilon}(m)$, there exists a $\mathfrak{N} \in \digamma (m)$ s.t.

$$\begin{aligned} \chi (t) =& \mathcal{V}_{\mathfrak{I} ,\mathfrak{h}}(t) [m_{0}- \mathfrak{H}(m)]+\int _{0}^{t} [P_{\mathfrak{h}} (t-\Im )G- \Xi P_{ \mathfrak{h}} (t-\Im )] \varrho \mathcal{X}(\Im )d\Im \\ &+ \int _{0}^{t} P_{\mathfrak{h}} (t-\Im )\sigma (\Im , m(\Im )) d \Im + \int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \mathfrak{N}(\Im ) d \Im \\ &+ \int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \wp (\Im , m(\Im )) d \omega (\Im )~~t \in \mathfrak{D}. \end{aligned}$$

(3.2)

Then

$$\begin{aligned} \| \Phi (t) \|^{2}_{\lambda} \leq &25\sup _{t\in \mathfrak{D}} t^{2(1- \mathfrak{I})(1-\mathfrak{h})}\{ E \|\mathcal{V}_{\mathfrak{I}, \mathfrak{h}}(t) [m_{0}-\mathfrak{H}(m)] \|^{2} \\ &+E \|\int _{0}^{t} [P_{\mathfrak{h}} (t-\Im )G- \Xi P_{\mathfrak{h}} (t-\Im )] \varrho \mathcal{X}(\Im )d\Im \|^{2}\\ &+ E \| \int _{0}^{t} P_{ \mathfrak{h}} (t-\Im )\sigma (\Im , m(\Im )) d\Im \|^{2} \\ &+ E \|\int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \mathfrak{N}(\Im ) d \Im \|^{2}+ E \|\int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \wp (\Im , m( \Im )) d\omega (\Im ) \|^{2} \} \\ \leq & \Bigg\{ \frac{25 \Pi ^{2} }{\Gamma ^{2}(\mathfrak{I}(1-\mathfrak{h})+\mathfrak{h})} \bigg[E \|m_{0} \|^{2} + C_{5} (1+\mathfrak{r})\bigg] \\ &+ \frac{25 \Pi ^{2} \alpha ^{1-2\mathfrak{I}(1-\mathfrak{h})}}{(2\mathfrak{h}-1)\Gamma ^{2}(\mathfrak{h})} \bigg[(C_{2}+Tr(\Theta )C_{3}) (1+\mathfrak{r})+ \|\mathfrak{N} \|_{L^{1}( \mathfrak{D},R^{+})}+C_{4} \alpha \mathfrak{r} \bigg]\Bigg\} \\ &\times \left \{ 1+ \frac{25 \|\varrho \|^{2} \| \varrho ^{*} \|^{2} \alpha ^{2\mathfrak{h}-1}[\|G\|^{2} \Pi ^{2}+ \Pi _{1}^{2}]^{2}}{\zeta ^{2}(2\mathfrak{h}-1)\Gamma ^{4}(\mathfrak{h}) } \right \} \\ &+ \frac{625 \|\varrho \|^{2} \| \varrho ^{*} \|^{2} \alpha ^{2\mathfrak{h}-1}[\|G\|^{2} \Pi ^{2}+ \Pi _{1}^{2}]^{2}[ E \| \overline{ m}_{\alpha}\|^{2}+ Tr(\Theta ) \alpha ~ C_{6}]}{\zeta ^{2}(2\mathfrak{h}-1)\Gamma ^{4}(\mathfrak{h}) }:= \tau . \end{aligned}$$

Hence, $\mathcal{K}_{\varepsilon}( \mathfrak{L}_{\mathfrak{r}})$ be bounded in λ.

${\mathbf{{\mathfrak{S3}:}}}$ We discuss $\{ \mathcal{K}_{\varepsilon}(m): m \in \mathfrak{L}_{\mathfrak{r}} \}$ to be equicontinuous.

For any $m \in \mathfrak{L}_{\mathfrak{r}}$, $\chi \in \mathcal{K}_{\varepsilon }(m),~\exists$ a $\mathfrak{N} \in \digamma (m )$ s.t. (3.2) verifies ∀ $t \in \mathfrak{D}$.

For $0 < t_{1} < t_{2} <q$, then

$$\begin{aligned} &E \|\Phi (t_{2}) - \Phi (t_{1}) \|^{2}_{\lambda} \\ & \leq 25 E \| \bigg(\mathcal{V}_{\mathfrak{I},\mathfrak{h}}(t_{2})- \mathcal{V}_{\mathfrak{I},\mathfrak{h}}(t_{1})\bigg) [m_{0}- \mathfrak{H}(m)] \|^{2}_{\lambda} \\ & +25 E \| \int _{0}^{t_{2}} P_{\mathfrak{h}} (t_{2}-\Im ) \sigma ( \Im , m(\Im )) d\Im -\int _{0}^{t_{1}} P_{\mathfrak{h}} (t_{1}-\Im ) \sigma (\Im , m(\Im )) d\Im \|^{2}_{\lambda} \\ &+ 25 E \|\int _{0}^{t_{2}} P_{\mathfrak{h}} (t_{2}-\Im )\wp (\Im , m( \Im )) d\omega (\Im )- \int _{0}^{t_{1}} P_{\mathfrak{h}} (t_{1}-\Im ) \wp (\Im , m(\Im ))d\omega (\Im )\|^{2}_{\lambda} \\ &+ 25 E \| \int _{0}^{t_{2}} P_{\mathfrak{h}} (t_{2}-\Im ) \mathfrak{N}(\Im ) d\Im -\int _{0}^{t_{1}} P_{\mathfrak{h}} (t_{1}- \Im ) \mathfrak{N}(\Im ) d\Im \|^{2}_{\lambda} \\ &+ 25 E \| \int _{0}^{t_{2}} [P_{\mathfrak{h}} (t_{2}-\Im )G- \Xi P_{ \mathfrak{h}} (t_{2}-\Im )] \varrho \mathcal{X}(\Im ) d\Im \\ &-\int _{0}^{t_{1}} [P_{\mathfrak{h}} (t_{1}-\Im )G- \Xi P_{ \mathfrak{h}} (t_{1}-\Im )] \varrho \mathcal{X}(\Im ) d\Im \|^{2}_{ \lambda} \\ &= 25 E \| \bigg(\mathcal{V}_{\mathfrak{I},\mathfrak{h}}(t_{2})- \mathcal{V}_{\mathfrak{I},\mathfrak{h}}(t_{1})\bigg) [m_{0}- \mathfrak{H}(m)] \|^{2}_{\lambda} \\ & +25 E \| \int _{t_{1}}^{t_{2}} P_{\mathfrak{h}} (t_{2}-\Im ) \sigma (\Im , m(\Im )) d\Im \\ &+\int _{0}^{t_{1}} \bigg[ P_{\mathfrak{h}} (t_{2}-\Im )- P_{ \mathfrak{h}} (t_{1}-\Im )\bigg] \sigma (s, x(s)) d\Im \|^{2}_{ \lambda} \\ & +25 E \| \int _{t_{1}}^{t_{2}} P_{\mathfrak{h}} (t_{2}-\Im ) \wp ( \Im , m(\Im ))d\omega (\Im ) \\ &+\int _{0}^{t_{1}} \bigg[ P_{\mathfrak{h}} (t_{2}-\Im )- P_{ \mathfrak{h}} (t_{1}-\Im )\bigg] \wp (\Im , m(\Im ))d\omega (\Im )\|^{2}_{ \lambda} \\ & +25 E \| \int _{t_{1}}^{t_{2}} P_{\mathfrak{h}} (t_{2}-\Im ) \mathfrak{N}(\Im ) d\Im \\ &+\int _{0}^{t_{1}} \bigg[ P_{\mathfrak{h}} (t_{2}-\Im )- P_{ \mathfrak{h}} (t_{1}-\Im )\bigg] \mathfrak{N}(\Im ) d\Im \|^{2}_{ \lambda} \\ & +25 E \| \int _{t_{1}}^{t_{2}} [P_{\mathfrak{h}} (t_{2}-\Im )G- \Xi P_{\mathfrak{h}} (t_{2}-\Im )] \varrho \mathcal{X}(\Im ) d\Im \\ &+\int _{0}^{t_{1}} \bigg[ P_{\mathfrak{h}} (t_{2}-\Im )G- \Xi P_{ \mathfrak{h}} (t_{2}-\Im )- P_{\mathfrak{h}} (t_{1}-\Im )G+ \Xi P_{ \mathfrak{h}} (t_{1}-\Im )\bigg] \varrho \mathcal{X}(\Im ) d\Im \|^{2}_{ \lambda}. \end{aligned}$$

From the compactness of $\mathcal{V}(t)(t > 0)$,

$$ E \|\Phi (t_{2}) - \Phi (t_{1}) \|^{2}_{\lambda}\rightarrow 0~~as~~t_{2} \rightarrow t_{1}.~~~~~~~~~~~~~~~~~~~~~~~~~~~ $$

Hence, $\mathcal{K}_{\varepsilon}(m)(t)$ be continuous in $\mathfrak{D}$. Also, for $t_{1}=0$, $t_{2} \in \mathfrak{D}$, one can show that $E \| \Phi (t_{2})- \Phi (0)\|^{2}_{\lambda}\rightarrow 0$ as $t_{2} \rightarrow 0$.

Hence, $\{ \mathcal{K}_{\varepsilon}(m)(t): m \in \mathfrak{L}_{\mathfrak{r}} \}$ is equicontinuous.

${\mathbf{{\mathfrak{S4}:}}}$ $\mathcal{K}_{\varepsilon}$ be completely continuous.

We explore, $\Upsilon (t)= \{\chi (t): \chi \in \mathcal{K}_{\varepsilon}( \mathfrak{L}_{\mathfrak{r}}) \}$ be r.c. in $\mathfrak{W}$ ∀ $t \in \mathcal{J}$, $\mathfrak{r}>0$.

Since, $\Upsilon (0)$ is r.c. in $\mathfrak{L}_{\mathfrak{r}}$. Assume $t \in (0,\alpha )$, fixed, $\eta \in (0, t)$ for $m \in \mathfrak{L}_{\mathfrak{r}}$, we define

$$\begin{aligned} \Phi ^{\eta ,\jmath} (t) =& \frac{\mathfrak{h}}{\Gamma (\mathfrak{I}(1-\mathfrak{h}))}\int _{0}^{t- \eta} \int _{\jmath}^{\infty }\upsilon (t-\Im )^{\mathfrak{I}(1- \mathfrak{h})-1}\Im ^{\mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon ) \mathcal{V}(\Im ^{\mathfrak{h}} \upsilon ) [m_{0}-\mathfrak{H}(m)] d \upsilon d\Im \\ &+\mathfrak{h} \int _{0}^{t-\eta}\int _{\jmath}^{\infty }\upsilon (t- \Im )^{\mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon ) [\mathcal{V}((t- \Im )^{\mathfrak{h}}\upsilon )G- \Xi \mathcal{V}((t-\Im )^{ \mathfrak{h}}\upsilon )] \varrho \mathcal{X}(\Im )d\upsilon d\Im \\ &+\mathfrak{h}\int _{0}^{t-\eta}\int _{\jmath}^{\infty }\upsilon (t- \Im )^{\mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon )\mathcal{V}((t- \Im )^{\mathfrak{h}}\upsilon )\sigma (\Im , m(\Im ))d\upsilon d\Im \\ &+\mathfrak{h}\int _{0}^{t-\eta}\int _{\jmath}^{\infty }\upsilon (t- \Im )^{\mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon )\mathcal{V}((t- \Im )^{\mathfrak{h}}\upsilon )\mathfrak{N}(\Im ) d\upsilon d\Im \\ &+\mathfrak{h}\int _{0}^{t-\eta}\int _{\jmath}^{\infty }\upsilon (t- \Im )^{\mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon )\mathcal{V}((t- \Im )^{\mathfrak{h}}\upsilon )\wp (\Im , m(\Im )) d\upsilon d\omega ( \Im ) \\ =& \frac{\mathfrak{h}\mathcal{V}(\eta ^{\mathfrak{h}} \jmath )}{\Gamma (\mathfrak{I}(1-\mathfrak{h}))} \int _{0}^{t-\eta} \int _{\jmath}^{\infty }\upsilon (t-\Im )^{ \mathfrak{I}(1-\mathfrak{h})-1}\Im ^{\mathfrak{h}-1}\Psi _{ \mathfrak{h}}(\upsilon )\mathcal{V}(\Im ^{\mathfrak{h}} \upsilon - \eta ^{\mathfrak{h}} \jmath )\\ &\times [m_{0}-\mathfrak{H}(m)] d\upsilon d\Im \\ &+ \mathfrak{h}\mathcal{V}(\eta ^{\mathfrak{h}} \jmath ) \int _{0}^{t- \eta}\int _{\jmath}^{\infty }\upsilon (t-\Im )^{\mathfrak{h}-1}\Psi _{ \mathfrak{h}}(\upsilon )\mathcal{V}((t-\Im )^{\mathfrak{h}}\upsilon - \eta ^{\mathfrak{h}} \jmath )\sigma (\Im , m(\Im ))d\upsilon d\Im \\ &+\mathfrak{h}\mathcal{V}(\eta ^{\mathfrak{h}} \jmath )\int _{0}^{t- \eta}\int _{\jmath}^{\infty }\upsilon (t-\Im )^{\mathfrak{h}-1}\Psi _{ \mathfrak{h}}(\upsilon )[\mathcal{V}((t-\Im )^{\mathfrak{h}}\upsilon - \eta ^{\mathfrak{h}} \jmath )G\\ &- \Xi \mathcal{V}((t-\Im )^{ \mathfrak{h}}\upsilon -\eta ^{\mathfrak{h}} \jmath )]\varrho \mathcal{X}(\Im )d\upsilon d\Im \\ &+\mathfrak{h}\mathcal{V}(\eta ^{\mathfrak{h}} \jmath )\int _{0}^{t- \eta}\int _{\jmath}^{\infty }\upsilon (t-\Im )^{\mathfrak{h}-1}\Psi _{ \mathfrak{h}}(\upsilon )\mathcal{V}((t-\Im )^{\mathfrak{h}}\upsilon - \eta ^{\mathfrak{h}} \jmath )\mathfrak{N}(\Im ) d\upsilon d\Im \\ &+\mathfrak{h}\mathcal{V}(\eta ^{\mathfrak{h}} \jmath )\int _{0}^{t- \eta}\int _{\jmath}^{\infty }\upsilon (t-\Im )^{\mathfrak{h}-1}\Psi _{ \mathfrak{h}}(\upsilon )\mathcal{V}((t-\Im )^{\mathfrak{h}}\upsilon - \eta ^{\mathfrak{h}} \jmath )\\ &\times\wp (\Im , m(\Im )) d\upsilon d\omega ( \Im ). \end{aligned}$$

Since $\mathcal{V}(\eta ^{\mathfrak{h}} \jmath ),~\eta ^{\mathfrak{h}} \jmath > 0$ is compact operator. Therefore, $\Upsilon ^{\eta ,\jmath} (t)= \{ \Phi ^{\eta ,\jmath}(t): \Phi ^{ \eta ,\jmath} \in \mathcal{K}_{\varepsilon}( \mathfrak{L}_{ \mathfrak{r}}) \}$ is r.c. in $\mathfrak{W}$. So

$$\begin{aligned} &E\| \Phi (t)-\Phi ^{\eta ,\jmath}(t)\|^{2}_{\lambda}= \sup _{t\in I} t^{2(1-\mathfrak{I})(1-\mathfrak{h})}E\| \Phi (t)-\Phi ^{\eta ,\jmath}(t) \|^{2} \\ &\leq \frac{25 \mathfrak{h}^{2}}{\Gamma ^{2}(\mathfrak{I}(1-\mathfrak{h}))} \sup _{t\in I} t^{2(1-\mathfrak{I})(1-\mathfrak{h})} \\ &\times E \| \int _{0}^{t} \int _{0}^{\jmath }\upsilon (t-\Im )^{ \mathfrak{I}(1-\mathfrak{h})-1}\Im ^{\mathfrak{h}-1}\Psi _{ \mathfrak{h}}(\upsilon )\mathcal{V}(\Im ^{\mathfrak{h}} \upsilon ) [m_{0}- \mathfrak{H}(m)] d\upsilon d\Im \|^{2} \\ &+ \frac{25 \mathfrak{h}^{2}}{\Gamma ^{2}(\mathfrak{I}(1-\mathfrak{h}))} \sup _{t\in I} t^{2(1-\mathfrak{I})(1-\mathfrak{h})} \\ &\times E \| \int _{t-\eta}^{t} \int _{\jmath}^{\infty }\upsilon (t- \Im )^{\mathfrak{I}(1-\mathfrak{h})-1}\Im ^{\mathfrak{h}-1}\Psi _{ \mathfrak{h}}(\upsilon )\mathcal{V}(\Im ^{\mathfrak{h}} \upsilon )[m_{0}- \mathfrak{H}(m)] d\upsilon d\Im \|^{2} \\ &+25 \mathfrak{h}^{2} \sup _{t\in I} t^{2(1-\mathfrak{I})(1- \mathfrak{h})} E \| \int _{0}^{t} \int _{0}^{\jmath }\upsilon (t-\Im )^{ \mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon )\mathcal{V}((t-\Im )^{ \mathfrak{h}}\upsilon )\sigma (\Im , m(\Im ))d\upsilon d\Im \|^{2} \\ &+25 \mathfrak{h}^{2} \sup _{t\in I} t^{2(1-\mathfrak{I})(1- \mathfrak{h})} E \| \int _{t-\eta}^{t} \int _{\jmath}^{\infty } \upsilon (t-\Im )^{\mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon ) \mathcal{V}((t-\Im )^{\mathfrak{h}}\upsilon )\sigma (\Im , m(\Im ))d \upsilon d\Im \|^{2} \\ &+25\mathfrak{h}^{2} \sup _{t\in I} t^{2(1-\mathfrak{I})(1- \mathfrak{h})} E \| \int _{0}^{t} \int _{0}^{\jmath }\upsilon (t-\Im )^{ \mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon ) \\ &\times [\mathcal{V}((t-\Im )^{\mathfrak{h}}\upsilon -\eta ^{ \mathfrak{h}} \jmath )G- \Xi \mathcal{V}((t-\Im )^{\mathfrak{h}} \upsilon -\eta ^{\mathfrak{h}} \jmath )]\varrho \mathcal{X}(\Im )d \upsilon d\Im \|^{2} \\ &+25\mathfrak{h}^{2} \sup _{t\in I} t^{2(1-\mathfrak{I})(1- \mathfrak{h})} E \| \int _{t-\eta}^{t} \int _{\jmath}^{\infty } \upsilon (t-\Im )^{\mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon ) \\ &\times [\mathcal{V}((t-\Im )^{\mathfrak{h}}\upsilon -\eta ^{ \mathfrak{h}} \jmath )G- \Xi \mathcal{V}((t-\Im )^{\mathfrak{h}} \upsilon -\eta ^{\mathfrak{h}} \jmath )]\varrho \mathcal{X}(\Im )d \upsilon d\Im \|^{2} \\ &+25\mathfrak{h}^{2} \sup _{t\in I} t^{2(1-\mathfrak{I})(1- \mathfrak{h})} E \| \int _{0}^{t} \int _{0}^{\jmath }\upsilon (t-\Im )^{ \mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon )\mathcal{V}((t-\Im )^{ \mathfrak{h}}\upsilon )\mathfrak{N}(\Im )d\upsilon d\Im \|^{2} \\ &+25\mathfrak{h}^{2} \sup _{t\in I} t^{2(1-\mathfrak{I})(1- \mathfrak{h})} E \| \int _{t-\eta}^{t} \int _{\jmath}^{\infty } \upsilon (t-\Im )^{\mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon ) \mathcal{V}((t-\Im )^{\mathfrak{h}}\upsilon )\mathfrak{N}(\Im )d \upsilon d\Im \|^{2} \\ &+25\mathfrak{h}^{2} \sup _{t\in I} t^{2(1-\mathfrak{I})(1- \mathfrak{h})} E \| \int _{0}^{t} \int _{0}^{\jmath }\upsilon (t-\Im )^{ \mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon )\mathcal{V}((t-\Im )^{ \mathfrak{h}}\upsilon )\wp (\Im , m(\Im ))d\upsilon d\omega (\Im )\|^{2} \\ &+25\mathfrak{h}^{2} \sup _{t\in I} t^{2(1-\mathfrak{I})(1- \mathfrak{h})} E \| \int _{t-\eta}^{t} \int _{\jmath}^{\infty } \upsilon (t-\Im )^{\mathfrak{h}-1}\Psi _{\mathfrak{h}}(\upsilon ) \mathcal{V}((t-\Im )^{\mathfrak{h}}\upsilon )\\ &\times\wp (\Im , m(\Im ))d \upsilon d\omega (\Im )\|^{2}\rightarrow 0, \end{aligned}$$

when $\eta \rightarrow 0^{+}$ and $\jmath \rightarrow 0^{+}$.

Hence, $\Upsilon (t)$ is r.c. in $\mathfrak{W}$. From Arzela–Ascoli theorem and ${\mathbf{{\mathfrak{S3}}}}$, we conclude that $\mathcal{K}_{\varepsilon}$ is completely continuous.

${\mathbf{{\mathfrak{S5}:}}}$ Claim: $\mathcal{K}_{\varepsilon}$ has a closed graph.

Let $m_{n} \rightarrow m_{*}$ in λ, and $\Phi _{n}\rightarrow \Phi _{*}$ in λ. Our aim is to show that $\Phi _{*} \in \mathcal{K}_{\varepsilon}(m_{*})$.

Let $\Phi _{n} \in \mathcal{K}_{\varepsilon}(m_{n})$, then ∃ a $\mathfrak{N}_{n} \in \digamma (m_{n})$ with

$$\begin{aligned} \Phi _{n} (t) =& \mathcal{V}_{\mathfrak{I} ,\mathfrak{h}}(t) [m_{0}- \mathfrak{H}(m_{n})]+\int _{0}^{t} P_{\mathfrak{h}} (t-\Im )\sigma ( \Im , m_{n}(\Im )) d\Im \\ &+ \int _{0}^{t} [P_{\mathfrak{h}} (t-\Im )G- \Xi P_{\mathfrak{h}} (t- \Im )]\varrho \mathcal{X}(\Im ) d\Im + \int _{0}^{t} P_{\mathfrak{h}} (t- \Im ) \mathfrak{N}_{n}(\Im ) d\Im \\ &+ \int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \wp (\Im , m_{n}(\Im )) d \omega (\Im ). \end{aligned}$$

(3.3)

By the use of $(H1)$–$(H8)$,

$\{\mathfrak{H}(m_{n}), \sigma (\cdot , m_{n}), \mathfrak{N}_{n}, \wp (\cdot , m_{n})\}_{n \geq 1}\subseteq \mathfrak{W} \times \mathfrak{W} \times L^{2}_{\mathcal{S}}(\mathfrak{D},\mathfrak{W}) \times L_{\Theta}$ is bounded.

Let us consider the subsequence,

$$\begin{aligned} (\mathfrak{H}(m_{n}), \sigma (\cdot , m_{n}), \mathfrak{N}_{n}, \wp ( \cdot , m_{n}))\rightarrow (\mathfrak{H}(m_{*}), \sigma (\cdot , m_{*}), \mathfrak{N}_{*}, \wp (\cdot , m_{*})) \end{aligned}$$

(3.4)

weakly in $\mathfrak{W} \times \mathfrak{W} \times L^{2}_{\mathcal{S}}( \mathfrak{D},\mathfrak{W}) \times L_{\Theta}$.

From the compactness of $\mathcal{V}(t)$, (3.3) and (3.4), we get

$$\begin{aligned} \Phi _{n} (t) \rightarrow & \mathcal{V}_{\mathfrak{I} ,\mathfrak{h}}(t) [m_{0}-\mathfrak{H}(m_{*})]+\int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \sigma (\Im , m_{*}(\Im )) d\Im \\ &+ \int _{0}^{t} [P_{\mathfrak{h}} (t-\Im )G- \Xi P_{\mathfrak{h}} (t- \Im )] \varrho \mathcal{X}(\Im ) d\Im + \int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \mathfrak{N}_{*}(\Im ) d\Im \\ &+ \int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \wp (\Im , m_{*}(\Im )) d \omega (\Im ). \end{aligned}$$

(3.5)

We note that $\Phi _{n}\rightarrow \Phi _{*}$ in λ and $\mathfrak{N}_{n} \in \digamma (m_{n})$.

Lemma 2.5 and (3.5), give $\mathfrak{N}_{*} \in F(m_{*})$. Hence, $\Phi _{*} \in \mathcal{K}_{\varepsilon}(m_{*})$. Thus $\mathcal{K}_{\varepsilon}$ has a closed graph.

By the view of [42], $\mathcal{K}_{\varepsilon}$ is upper semicontinuous.

${\mathbf{{\mathfrak{S6}:}}}$

By the use of ${\mathbf{{\mathfrak{S1}:}}}$-${\mathbf{{\mathfrak{S5}:}}}$, we can deduce, $\mathcal{K}_{\varepsilon}$ is CCV, upper semicontinuous and $\mathcal{K}_{\varepsilon}( \mathfrak{L}_{\mathfrak{r}})$ is r.c.

By Theorem 2.1, we show that

$\Psi = \{ m \in \lambda : \varsigma m \in \mathcal{K}_{\varepsilon},~~ \varsigma >1 \}$ is bounded. ∀ $m \in \Psi$, there exists a $\mathfrak{N} \in \digamma (m)$ s.t.

$$\begin{aligned} m(t) =&\varsigma ^{-1} \mathcal{V}_{\mathfrak{I} ,\mathfrak{h}}(t) [m_{0}- \mathfrak{H}(m)] +\varsigma ^{-1}\int _{0}^{t} P_{\mathfrak{h}} (t-\Im )\sigma (\Im , m( \Im )) d\Im \\ &+\varsigma ^{-1} \int _{0}^{t} [P_{\mathfrak{h}} (t-\Im )G- \Xi P_{ \mathfrak{h}} (t-\Im )] \varrho \mathcal{X}(\Im ) d\Im + \varsigma ^{-1} \int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \mathfrak{N}(\Im ) d\Im \\ &+\varsigma ^{-1} \int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \wp (\Im , m( \Im )) d\omega (\Im ). \end{aligned}$$

(3.6)

Applying the hypotheses $(H1)-$-$(H8)$, we obtain

$$\begin{aligned} E\|m(t) \|^{2} \leq & 25 \{ E \| \mathcal{V}_{\mathfrak{I}, \mathfrak{h}}(t) [m_{0}-\mathfrak{H}(m)] \|^{2} \\ & + E \| \int _{0}^{t} P_{\mathfrak{h}} (t-\Im )\sigma (\Im , m(\Im )) d\Im \|^{2}\\ &+ E \| \int _{0}^{t} [P_{\mathfrak{h}} (t-\Im )G- \Xi P_{ \mathfrak{h}} (t-\Im )] \varrho \mathcal{X}(\Im ) d\Im \|^{2} \\ &+ E \|\int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \mathfrak{N}(\Im ) d \Im \|^{2}+ E \|\int _{0}^{t} P_{\mathfrak{h}} (t-\Im ) \wp (\Im , m( \Im )) d\omega (\Im ) \|^{2} \} \\ \leq & \Bigg\{ \frac{25 \Pi ^{2} \alpha ^{2(\mathfrak{I}-1)(1-\mathfrak{h})}}{\Gamma ^{2}(\mathfrak{I}(1-\mathfrak{h})+\mathfrak{h})} \bigg[(E \|m_{0} \|^{2} + C_{5}(1+ E \|m(t)\|^{2})\bigg] \\ &+ \frac{25 \Pi ^{2} \alpha ^{2\mathfrak{h}-1}}{(2\mathfrak{h}-1)\Gamma ^{2}(\mathfrak{h})} \\ &\times\bigg[(C_{2}+Tr(\Theta )C_{3}) (1+E \|m(t)\|^{2})+ \|\mathfrak{N} \|_{L^{1}( \mathfrak{D},R^{+})}+C_{4} \alpha E \|m(t)\|^{2} \bigg]\Bigg\} \\ &\times \left \{ 1+ \frac{25 \|\varrho \|^{2} \| \varrho ^{*} \|^{2} \alpha ^{2\mathfrak{h}-1}[\|G\|^{2} \Pi ^{2}+ \Pi _{1}^{2}]^{2}}{\zeta ^{2}(2\mathfrak{h}-1)\Gamma ^{4}(\mathfrak{h}) } \right \} \\ &+ \frac{625 \|\varrho \|^{2} \| \varrho ^{*} \|^{2} \alpha ^{2\mathfrak{h}-1}[\|G\|^{2} \Pi ^{2}+ \Pi _{1}^{2}]^{2}[ E \| \overline{ m}_{\alpha}\|^{2}+ Tr(\Theta ) \alpha ~ C_{6}]}{\zeta ^{2}(2\mathfrak{h}-1)\Gamma ^{4}(\mathfrak{h}) } \\ &\leq \Re _{1}+\Re _{2} E\|m(t)\|^{2}, \end{aligned}$$

(3.7)

where

$$\begin{aligned} \Re _{1} =&\Bigg\{ \frac{25 \Pi ^{2} \alpha ^{2(\mathfrak{I}-1)(1-\mathfrak{h})}(E \|m_{0} \|^{2}+ C_{5})}{\Gamma ^{2}(\mathfrak{I}(1-\mathfrak{h})+\mathfrak{h})} \\ &+ \frac{25 \Pi ^{2} \alpha ^{2\mathfrak{h}-1}}{(2\mathfrak{h}-1)\Gamma ^{2}(\mathfrak{h})} \bigg[(C_{2}+Tr(\Theta )C_{3}) + \|\mathfrak{N} \|_{L^{1}( \mathfrak{D},R^{+})} \bigg]\Bigg\} \\ &\times \left \{ 1+ \frac{25 \|\varrho \|^{2} \| \varrho ^{*} \|^{2} \alpha ^{2\mathfrak{h}-1}[\|G\|^{2} \Pi ^{2}+ \Pi _{1}^{2}]^{2}}{\zeta ^{2}(2\mathfrak{h}-1)\Gamma ^{4}(\mathfrak{h}) } \right \} \\ &+ \frac{625 \|\varrho \|^{2} \| \varrho ^{*} \|^{2} \alpha ^{2\mathfrak{h}-1}[\|G\|^{2} \Pi ^{2}+ \Pi _{1}^{2}]^{2}[ E \| \overline{ m}_{\alpha}\|^{2}+ Tr(\Theta ) \alpha ~ C_{6}]}{\zeta ^{2}(2\mathfrak{h}-1)\Gamma ^{4}(\mathfrak{h}) }, \end{aligned}$$

and

$$\begin{aligned} \Re _{2} =&\Bigg\{ \frac{25 C_{5} \Pi ^{2} \alpha ^{2(\mathfrak{I}-1)(1-\mathfrak{h})}}{\Gamma ^{2}(\mathfrak{I}(1-\mathfrak{h})+\mathfrak{h})} + \frac{25 \Pi ^{2} \alpha ^{2\mathfrak{h}-1}}{(2\mathfrak{h}-1)\Gamma ^{2}(\mathfrak{h})} \bigg[(C_{2}+Tr(\Theta )C_{3}) +C_{4} \alpha \bigg]\Bigg\} \\ &\times \left \{ 1+ \frac{25 \|\varrho \|^{2} \| \varrho ^{*} \|^{2} \alpha ^{2\mathfrak{h}-1}[\|G\|^{2} \Pi ^{2}+ \Pi _{1}^{2}]^{2}}{\zeta ^{2}(2\mathfrak{h}-1)\Gamma ^{4}(\mathfrak{h}) } \right \}. \end{aligned}$$

Since $\Re _{2} < 1$, from (3.7), we get

$$\begin{aligned} \|m\|^{2}_{\lambda}= \sup _{t \in \mathfrak{D}}E \|t^{(1-\mathfrak{I})(1- \mathfrak{h})}m(t) \|^{2} \leq \Re _{1}+\Re _{2} \|m\|^{2}_{\lambda}. \end{aligned}$$

Then, $\|m\|^{2}_{\lambda} \leq \frac{\Re _{1}}{1-\Re _{2}}$, consequently, Ψ is bounded. By Theorem 2.1, $\mathcal{K}_{\varepsilon}$ has a fixed point. Therefore, the inclusion system (1.1) has a mild solution on $\mathfrak{D}$. □

Theorem 3.2

If the functions σ, ℘, and $\mathfrak{N}$ are uniformly bounded and the assumptions $(H1)$–$(H8)$ are verified, then (1.1) is approximately controllable on $\mathfrak{D}$.

Proof

Let $m^{\xi}$ be a fixed point of $\mathcal{K}_{\varepsilon}$. Apply the stochastic Fubini theorem, then

$$\begin{aligned} m^{\xi}(\alpha ) &=\bar{m}_{\alpha }- \zeta (\zeta I + L^{\alpha}_{0})^{-1} \{E \bar {m}_{\alpha }- \mathcal{V}_{\mathfrak{I} ,\mathfrak{h}}( \alpha ) [m^{\xi}_{0}-\mathfrak{H}(m^{\xi})] \\ &- \int _{0}^{\alpha }P_{\mathfrak{h}} (\alpha -\Im )\sigma (\Im , m^{ \xi}(\Im )) d\Im - \int _{0}^{\alpha }P_{\mathfrak{h}} (\alpha -\Im ) \mathfrak{N}^{\xi}(\Im ) d\Im \\ &- \int _{0}^{\alpha }P_{\mathfrak{h}} (\alpha -\Im ) \wp (\Im , m^{ \xi}(\Im )) d\omega (\Im )+ \int _{0}^{\alpha }\bar{\Upsilon}(\Im ) d \omega (\Im )\}. \end{aligned}$$

(3.8)

From the hypothesis on σ, ℘, $\mathfrak{N}$ and the proof of S6 in last theorem, we get

$$\begin{aligned} E\|\sigma (t, m^{\xi}(t))\|^{2} +E\| \wp (t, m^{\xi}(t))\|^{2}_{ \Theta }+ E\|\mathfrak{N}^{\xi}(t)\|^{2} \leq (C_{2}+C_{3})(1+ \mathfrak{r})+C_{4} \mathfrak{r}+\vartheta (t):= \nu (t). \end{aligned}$$

Consequently, the sequence $\{ \sigma (\cdot , m^{\xi}(\cdot )),\wp (\cdot , m^{\xi}(\cdot )), \mathfrak{N}^{\xi}\}$ is bounded in $2^{\mathfrak{W}}\times L_{\Theta }(\mathfrak{B},\mathfrak{W}) \times L^{2}_{S}(\mathfrak{D},\mathfrak{W})$.

Hence, there is subsequence, denoted by $\{ \sigma (\cdot , m^{\xi}(\cdot )),\wp (\cdot , m^{\xi}(\cdot )), \mathfrak{N}^{\xi}\}$ that converges weakly to $\{ \sigma (\cdot ),\wp (\cdot ),\mathfrak{N}^{\xi}\}$ in $2^{\mathfrak{W}} \times L_{\Theta }(\mathfrak{B},\mathfrak{W}) \times L^{2}_{S}(\mathfrak{D},\mathfrak{W})$.

From (3.8), we have

$$\begin{aligned} E \|m^{\xi}(\alpha )-\bar{m}_{\alpha }\|^{2} &\leq 25 E \|\zeta ( \zeta I + L^{\alpha}_{0})^{-1} \{E \bar {m}_{\alpha }- \mathcal{V}_{ \mathfrak{I} ,\mathfrak{h}}(\alpha ) [m^{\xi}_{0}-\mathfrak{H}(m^{\xi})] \|^{2} \\ &+25 E \|\zeta (\zeta I + L^{\alpha}_{0})^{-1} \int _{0}^{\alpha }P_{ \mathfrak{h}} (\alpha -\Im )[\sigma (\Im , m^{\xi}(\Im ))-\sigma ( \Im ) ]d\Im \|^{2} \\ &+25 E \|\zeta (\zeta I + L^{\alpha}_{0})^{-1} \int _{0}^{\alpha }P_{ \mathfrak{h}} (\alpha -\Im )\sigma (\Im )d\Im \|^{2} \\ &+ 25 E \|\zeta (\zeta I + L^{\alpha}_{0})^{-1} \int _{0}^{\alpha }P_{ \mathfrak{h}} (\alpha -\Im )[ \mathfrak{N}^{\xi}(\Im )-\mathfrak{N}( \Im )] d\Im \|^{2} \\ &+ 25 E \|\zeta (\zeta I + L^{\alpha}_{0})^{-1} \int _{0}^{\alpha }P_{ \mathfrak{h}} (\alpha -\Im )\mathfrak{N}(\Im )d\Im \|^{2} \\ &+ 25 E \|\zeta (\zeta I + L^{\alpha}_{0})^{-1} \int _{0}^{\alpha }P_{ \mathfrak{h}} (\alpha -\Im ) [\wp (\Im , m^{\xi}(\Im ))-\wp (\Im )] d \omega (\Im )\|^{2} \\ &+ 25 E \|\zeta (\zeta I + L^{\alpha}_{0})^{-1} \int _{0}^{\alpha }P_{ \mathfrak{h}} (\alpha -\Im ) \wp (\Im ) d\omega (\Im )\|^{2} \\ &+ 25 E\|\zeta (\zeta I + L^{\alpha}_{0})^{-1}\int _{0}^{\alpha } \bar{\Upsilon}(\Im ) d\omega (\Im )\|^{2}. \end{aligned}$$

On the other hand, by Lemma 3.1, the operator $\zeta (\zeta I + L^{\alpha}_{0})^{-1}\rightarrow 0$ strongly as $\zeta \rightarrow 0^{+}$ for all $0 \leq \Im \leq \alpha$, and $\| \zeta (\zeta I + L^{\alpha}_{0})^{-1} \| \leq 1$ together with the Lebesgue dominated convergence theorem and the compactness of $P_{\mathfrak{h}}(t)$, therefore, $E \|m^{\xi}(\alpha )-\bar{m}_{\alpha }\|^{2} \rightarrow 0$ as $\zeta \rightarrow 0^{+}$. Therefore, (1.1) is approximately controllable on $\mathfrak{D}$. □

4 Illustration

Let us consider the upcoming problem as:

$$\begin{aligned} \left \{ \textstyle\begin{array}{ll} D_{0+}^{\frac{4}{5},\frac{7}{9}}m(t,\aleph ) \in \frac{\partial ^{2}}{\partial \aleph ^{2}} (m(t,\aleph ) \\ +\frac{7}{100} ~(5)^{m(t,\aleph )}+\frac{1}{20} \cos (m(t,\aleph )) \frac{d\omega (t)}{dt} + \partial \mathfrak{L}(t,m(t,\aleph )) ,~ t \in \mathfrak{D}= (0,1],~ \aleph \in \mho \\ m(t,\aleph )=\mathcal{X}(t, \aleph ),~~t \in \mathfrak{D},~ \aleph \in \Game \\ I^{\frac{2}{45}}_{0+}m(0,\aleph )+\sum _{i=1}^{p} a_{i} ~m(t_{i}, \aleph )=m_{0}(\aleph ) ,~ \aleph \in \mho , \end{array}\displaystyle \right . \end{aligned}$$

(4.1)

where $D_{0+}^{\frac{4}{5},\frac{7}{9}}$ denotes the fractional derivative of Hilfer sense with order $\mathfrak{I}= \frac{4}{5}$, $\mathfrak{h}=\frac{7}{9}$, $0 < t_{0} < t_{1}<\cdots<t_{p} < 1$, $\mho \subset \mathfrak{R}$ and bounded, ω is a Brownian motion and sufficiently smooth boundary is ⅁. Here, $m(t)(\aleph )=m(t,\aleph )$, $\frac{7}{100} ~(5)^{m(t,\aleph )}= \sigma (t,m(t,\aleph ))$, $\frac{1}{20} \cos (m(t,\aleph ))= \wp (t,m(t,\aleph ))$, and $\mathfrak{L}(t,m(t))(\aleph ) = \mathfrak{L}(t,m(t,\aleph ))$.

Suppose $\mathfrak{W} = \mathfrak{B} = L^{2}( \mho )$, $Y= L^{2}(\Game )$, $\varrho _{1}=I$ (identity operator), and $G: Dom(G) \subset \mathfrak{W} \rightarrow \mathfrak{W}$ is introduced as $G= \frac{\partial ^{2}}{\partial \aleph ^{2}}$ with domain $= \{ m \in \mathfrak{W}, m, \frac{\partial m}{\partial \aleph}$ are absolutely continuous, $\frac{\partial ^{2} m}{\partial \aleph ^{2}} \in L^{2}( \mho ) \}$.

Then Δ can be written as

$$ \Delta m=\sum _{n=1}^{\infty }(-n^{2}) (m,m_{n})m_{n},~~~m \in D( \Delta ), $$

$m_{\iota}(\mu )=\sqrt{\frac{2}{\pi}} \sin \iota \mu$, $\iota =1,2,\ldots$ .

For $m \in \mathfrak{W}$ we have

$$ \mathcal{V}(t)m=\sum _{\iota =1}^{\infty }e^{ \frac{-\iota ^{2} t}{1+\iota ^{2}}}(m,m_{\iota})m_{\iota}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $$

Δ forms a compact semigroup $\{\mathcal{V}(t)\}_{t \geq 0}$ on $\mathfrak{W}$ with $\|\mathcal{V}(t) \|\leq e^{-t}\leq 1$.

$\mathcal{V}_{\frac{4}{5},\frac{7}{9} }(t)$ and $P_{\frac{7}{9}}(t)$ can be defined by

$$\begin{aligned}& \mathcal{V}_{\frac{4}{5},\frac{7}{9}}(t)m= \frac{7}{9\Gamma (\frac{8}{45})}\int _{0}^{t} \int _{0}^{\infty } \upsilon (t-\Im )^{\frac{-37}{45}} \Im ^{\frac{-2}{9}}\Psi _{ \frac{7}{9}}(\upsilon ) \mathcal{V}(\Im ^{\frac{7}{9}} \upsilon )~m d \upsilon d \Im ,\\& P_{\frac{7}{9}} (t) m= \frac{7}{9}\int _{0}^{\infty }\upsilon t^{ \frac{-2}{9}} \Psi _{\frac{7}{9}}(\upsilon ) \mathcal{V}(\Im ^{ \frac{7}{9}} \upsilon )m d \upsilon .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$

For $t > 0$, we have

$$ \|P_{\frac{7}{9}} (t) \|\leq \frac{\Pi t^{\frac{-2}{9}}}{\Gamma (\frac{7}{9})},~\|\mathcal{V}_{ \frac{4}{5},\frac{7}{9} }(t)\|\leq \frac{\Pi t^{\frac{-2}{45}}}{\Gamma (\frac{43}{45})}. $$

Therefore, (4.1) can be explored in the form of (1.1) and all assumptions of Theorem 3.1 and Theorem 3.2 are verified and

$$\begin{aligned} &\Bigg\{ \frac{25 C_{5} \Pi ^{2} \alpha ^{\frac{-4}{45}}}{\Gamma ^{2}(\frac{43}{45})} + \frac{25 \Pi ^{2} \alpha ^{\frac{5}{9}}}{\frac{5}{9}\Gamma ^{2}(\frac{7}{9})} \bigg[(C_{2}+Tr(\Theta )C_{3}) +C_{4} \alpha \bigg]\Bigg\} \\ &\times \left \{ 1+ \frac{25 \|\varrho \|^{2} \| \varrho ^{*} \|^{2} \alpha ^{\frac{5}{9}}[\|G\|^{2} \Pi ^{2}+ \Pi _{1}^{2}]^{2}}{\zeta ^{2}(\frac{5}{9})\Gamma ^{4}(\frac{7}{9}) } \right \}< 1. \end{aligned}$$

Thus, we obtain the approximately controllable for the system (4.1) on $(0, 1]$.

5 Conclusion

In this paper, we have dealt with the approximate boundary controllability of nonlocal Hilfer fractional stochastic differential inclusions with Clarke’s subdifferential. Firstly, by using stochastic analysis, nonsmooth analysis, theory of operator semigroups and fixed point theorems of multi-valued maps, we proved the existence of mild solutions for (1.1). Then we provided a sufficient conditions for the approximate boundary controllability of the nonlocal Hilfer fractional stochastic differential inclusions with Clarke’s subdifferential. Actually, our results covered a broader class of fractional stochastic inclusion. Finally, an example has been included to illustrate the applicability of the main results.