A New Study on the Approximate Controllability of Sobolev-Type Stochastic ABC-Fractional Impulsive Differential Inclusions with Clarke Sub-Differential and Poisson Jumps

Abstract

This paper undertakes a rigorous analytical exposition of the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions, incorporating the Atangana–Baleanu fractional derivative in the Caputo configuration under the influence of Wiener process and Poissonian discontinuities. The system’s analytical landscape is further enriched by the incorporation of Clarke sub-differentials, facilitating the treatment of nonsmooth, nonconvex, and multivalued dynamics. The inherent complexity arising from the confluence of fractional memory, stochastic perturbations, and impulsive phenomena necessitates the deployment of a sophisticated apparatus from variational analysis, measurable selection theory, and multivalued fixed point frameworks within infinite-dimensional Banach spaces. This study delineates rigorous sufficient conditions, ensuring controllability under such hybrid influences, thereby generalizing classical paradigms to encompass nonlocal and discontinuous dynamical regimes. A precisely articulated exemplar is included to validate the theoretical constructs and demonstrate the operational efficacy of the proposed analytical methodology.

1. Introduction

The study of modern control theory has increasingly focused on dynamical systems that exhibit memory effects, stochastic perturbations, and nonsmooth nonlinear behavior. Such complexities naturally arise in applications, including cyber-physical systems, neural networks, and high-frequency financial markets. In these settings, traditional smooth and deterministic models are often inadequate as they cannot capture phenomena such as abrupt state changes, persistent memory effects, or random fluctuations.

Fractional calculus provides a powerful mathematical framework for modeling such effects [1,2,3]. In particular, the Atangana–Baleanu fractional derivative (ABC-FD) in the Caputo sense, introduced in [4], has attracted attention due to its nonsingular, globally nonlocal kernel based on the Mittag–Leffler function. This derivative generalizes earlier fractional operators that were used to describe memory effects in non-Markovian systems, such as generalized Langevin equations and anomalous diffusion processes [5,6,7,8,9]. The works of Kilbas, Srivastava, Tomovski, and collaborators [10,11,12] have laid a comprehensive theoretical foundation for such operators, introducing fractional integrals and derivatives that encompass a wide variety of memory kernels. The ABC-FD operator is particularly attractive because it combines physical interpretability with analytical tractability, including transform-based solution techniques [13,14].

A rigorous setting for the analysis of these systems is provided by Sobolev-type stochastic differential inclusions with memory-dependent terms. Recent studies have explored the controllability of various classes of fractional stochastic systems [15,16,17,18,19]. For example, Arthi et al. [20] considered nonlinear fractional stochastic systems with state-dependent delays and impulses using Krasnosel’skii’s fixed point theorem. Bedi [21] extended this work to multivalued impulsive stochastic inclusions in Hilbert spaces. Ramadan et al. [22] studied null controllability for ABC-FD stochastic differential equations with non-instantaneous impulses and Poisson jumps. Similarly, Mohammed et al. [23] examined Sobolev-type ABC-FD stochastic inclusions with Clarke sub-differentials. Dhayal [24] investigated approximate controllability for fractional systems driven by Rosenblatt noise with non-instantaneous impulses. These results represent significant progress but do not fully cover the combination of features considered in this work.

On the stochastic side, Wiener processes remain a fundamental model for Gaussian noise with independent and stationary increments [25,26,27,28]. Their highly irregular sample paths make them well-suited for representing random fluctuations in complex systems. To model sudden, jump-like changes, Poisson random measures are used to capture non-Gaussian impulsive events, such as network attacks, synaptic discharges, or market shocks [29,30]. In addition, Clarke’s generalized gradient [31] provides an essential tool for treating nonsmooth right-hand sides and multivalued differential inclusions, which naturally appear when systems experience abrupt structural changes.

Despite these developments, only a limited number of contributions [32,33,34] have considered systems that combine memory effects from ABC-FD, stochastic perturbations from Wiener processes, Poisson jumps, and nonsmooth Clarke-type inclusions within a unified framework. Existing works often impose restrictive assumptions on the impulsive terms or control constraints, or they are limited to finite-dimensional settings. The present work aims to address these gaps by developing a comprehensive theory for the approximate controllability of such systems in infinite dimensional Banach spaces.

Motivated by these lacunae, the present work delves into the approximate controllability of a class of Sobolev-type stochastic impulsive differential inclusions with nonlocal ABC-FDs in the Caputo framework. The system is perturbed by both Wiener noise and Poissonian jumps, and it incorporates nonsmooth multivalued dynamics, constituting a novel synthesis of fractional memory, stochasticity, impulsivity, and nonconvexity in a unified infinite-dimensional formalism.

$$\begin{array}{ccc}\hfill {}^{\mathcal{ABC}}D_{0+}^{\upsilon }\mathrm{\Theta }\left\{x\left(t\right)-y(t,x(t\left)\right)\right\}& \in & Z\left\{x\left(t\right)-y(t,x(t\left)\right)\right\}+\mho U\left(t\right)+\zeta (t,x\left(t\right)){\displaystyle \frac{dW\left(t\right)}{dt}}\hfill \\ & & +\partial F(t,x\left(t\right))+{\int }_{\mathrm{\Pi }}H(t,x\left(t\right),\mathbb{P}\left(t\right))\aleph (dt,d\mathbb{P}),t\in \mathcal{J}:=(0,\rho ],t\ne t_{\kappa },\hfill \\ \hfill \mathrm{\Delta }x{\mid }_{t=t_{\kappa }}& =& {\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right),\kappa =1,2,3,\dots ,\gamma ,\hfill \\ \hfill x\left(0\right)+m\left(x\right)& =& x_0.\hfill \end{array}$$

(1)

The expression ${}^{\mathcal{ABC}}D_{0+}^{\upsilon }$ represents the ABC-FD with order ${\displaystyle \frac{1}{2}}<\upsilon <1$. $x(·)$ and acquires its values in X, the separable Hilbert space endowed with $\langle ·,·\rangle$ and $∥·∥$. Let Y be another separable Hilbert space with ${\langle ·,·\rangle }_Y$ and ${∥·∥}_Y$. Let ${\left\{W\left(t\right)\right\}}_{t\ge 0}$ be a Y-valued Wiener process with a covariance of ${\rm Y}\ge 0$. If $0=t_0<t_1<\dots <t_{\gamma }<t_{\gamma +1}=\rho$, $\mathrm{\Delta }x{\mid }_{t=t_{\kappa }}=x\left(t_{\kappa }^{+}\right)-x\left(t_{\kappa }^{-}\right)$. Z and $\mathrm{\Theta }$ are linear operators on X. Let $\partial F(t,x(t\left)\right)$ be the Clarke sub-differential of $F(t,x(t\left)\right)$. Let $(\mathrm{\Omega },\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathbb{P})$ be a complete filtered probability space satisfying the usual conditions, that is, the filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ is a right-continuous increasing family and $\left\{{\mathcal{F}}_0\right\}$ contains all $\mathbb{P}$-null sets. On this space, we consider a stationary Poisson point process ${\left\{P_t\right\}}_{t\ge 0}$ to be defined on $(\mathrm{\Pi },\mathfrak{X},\lambda (d\mathbb{P}\left)\right)$ and adapted to the filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$. The corresponding counting measure $\tilde{N}(t,d\mathbb{P})$ counts the number of jumps occurring in the set $\mathbb{P}\subseteq \mathfrak{X}$ up to time t, and $\mathbb{E}\left[\tilde{N}(t,𝚤)\right]=t\lambda (𝚤),\mathrm{for}𝚤\in \mathfrak{X}.$ The compensated Poisson random measure (or Poisson martingale measure) is then defined as $\aleph (t,d\mathbb{P}):=\tilde{N}(t,d\mathbb{P})-t\lambda \left(d\mathbb{P}\right),$ and this captures the stochastic fluctuations introduced by the jump component of the process. The classical initial condition (often referred to as a local condition) $x\left(0\right)=x_0$ is generalized to the following nonlocal condition: $x\left(0\right)+m\left(x\right)=x_0,m:C(\mathcal{J},X)\to X.$ Here, $m(·)$ is a mapping defined on the space of continuous functions over $\mathcal{J}$. Clearly, if $m\equiv 0$, then the condition reduces to the classical local form $x\left(0\right)=x_0$. The use of nonlocal conditions offers a significant modeling advantage as it allows the incorporation of information or measurements taken over an entire interval rather than at a single point, thereby yielding more realistic and accurate models. If $y:\mathcal{J}\times X\to X$, $H:\mathcal{J}\times X\times \mathrm{\Pi }\to X$ and $\zeta :\mathcal{J}\times X\to {\mathcal{L}}_{{\rm Y}}^2(Y,X)$, where ${\mathcal{L}}_{{\rm Y}}^2(Y,X)$ be the space of $\mathrm{{\rm Y}}$–Hilbert–Schmidt operators from Y into X. Throughout this work, $U(·)$ denotes the admissible control function associated with System (1). We assume that $U\in {\mathcal{L}}_2(\mathcal{J},\mathcal{U})$, where $\mathcal{U}$ is a separable Hilbert space of control values, and ${\mathcal{L}}_2(\mathcal{J},\mathcal{U})$ denotes the set of all square-integrable controls on $\mathcal{J}$. The control $U(·)$ acts on the state dynamics through a bounded linear control operator $\mho :\mathcal{U}\to X$, thereby steering the state trajectory toward the desired target set.

To the best of our knowledge, this investigation represents a pioneering effort to holistically synthesize these heterogeneous sources of irregularity—fractional memory, impulsivity, stochasticity, and nonsmooth multivaluedness—within a single cohesive analytical schema.

The principal aim of this work is to rigorously investigate the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions characterized by the confluence of several analytical complexities, including the following:

1.

The incorporation of a nonlocal, nonsingular fractional-order dynamics governed by the Atangana–Baleanu derivative in the Caputo configuration;

2.

Stochastic perturbations modeled via infinite-dimensional Wiener processes;

3.

Impulsive state discontinuities manifesting at discrete temporal instances;

4.

Non-Gaussian jump discontinuities generated by Poisson random measures;

5.

Nonsmooth, nonconvex dynamics encapsulated through Clarke’s generalized subdifferential formalism.

The need for such an integrated framework is driven by the reality that many physical, engineering, and financial systems exhibit simultaneous memory effects, abrupt state changes, and random perturbations. While recent advances have treated each of these phenomena in isolation—see, e.g., refs. [20,21,22,23,24]—a comprehensive theory that unifies Atangana–Baleanu fractional operators, Poisson jumps, non-instantaneous impulses, and Clarke-type nonsmooth dynamics remains conspicuously absent. Our results not only close this theoretical gap, but also provide a constructive methodology for verifying controllability in systems where deterministic models or purely Caputo-based fractional formulations fail to capture the full complexity of observed behaviors. This yields a mathematically rigorous yet practically relevant framework that can be exploited in the modeling and control of, for example, neurodynamic networks with stochastic perturbations, energy systems with sudden switching events, and financial systems influenced by jump-diffusion processes.

From a methodological standpoint, our analytical treatment synergistically fuses elements of infinite-dimensional stochastic analysis, nonsmooth variational calculus, and multivalued fixed point theory in Banach spaces. We established verifiable sufficient criteria for the approximate controllability of the proposed class of systems, thereby extending classical controllability paradigms to accommodate systems exhibiting memory-driven, abruptly discontinuous, and probabilistically nonhomogeneous behaviors.

This contribution not only addresses salient lacunae in the extant literature, but also furnishes a robust theoretical foundation for subsequent explorations into the control of highly complex, real-world dynamical processes that transcend conventional smooth and deterministic modeling frameworks.

The remainder of this paper is organized as follows. Section 2 introduces the notations, definitions, and essential lemmas used throughout the paper. Section 3 establishes the existence of mild solutions under appropriate assumptions and develops the approximate controllability results using fixed point techniques and measurable selection theorems. Section 4 presents a detailed illustrative example that verifies the theoretical conditions and demonstrates applicability. Finally, Section 5 concludes this paper with remarks and potential directions for future research.

2. Preliminaries

This section commences with some exploratory ideas that are utilized all throughout the paper.

Definition 1 ([35]).

For $0<\upsilon <1$, the ABC-FD is characterized by the following definition:

$${}^{\mathcal{ABC}}D_{\tau +}^{\upsilon }\mathit{ȷ}\left(t\right)={\displaystyle \frac{\begin{array}{c}\hfill \mu \end{array}\left(\upsilon \right)}{1-\upsilon }}{\int }_{\tau }^t{\mathit{ȷ}}^{\prime }\left(s\right){\mathcal{E}}_{\upsilon }(-\theta {(t-s)}^{\upsilon })ds,$$

(2)

where the function $\theta ={\displaystyle \frac{\upsilon }{1-\upsilon }}$ and ${\mathcal{E}}_{\upsilon }\left(l\right)={\sum }_{n=0}^{\infty }{\displaystyle \frac{l^n}{\mathrm{\Gamma }(n\upsilon +1)}}$ denote the Mittag–Leffler function. Additionally, the normalization function, denoted by $\mu \left(\upsilon \right)$, is expressed as $\begin{array}{c}\hfill \mu \end{array}\left(\upsilon \right)=(1-\upsilon )+{\displaystyle \frac{\upsilon }{\mathrm{\Gamma }\left(\upsilon \right)}}$. It is defined in such a way that $\mu \left(0\right)=\mu \left(1\right)=1$. The expression for the fractional integral of Atangana–Baleanu (AB-FI) is given as

$${}^{\mathcal{AB}}I_{\tau +}^{\upsilon }\mathit{ȷ}\left(t\right)={\displaystyle \frac{1-\upsilon }{\begin{array}{c}\hfill \mu \end{array}\left(\upsilon \right)}}\mathit{ȷ}\left(t\right)+{\displaystyle \frac{\upsilon }{\begin{array}{c}\hfill \mu \end{array}\left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_{\tau }^t{(t-s)}^{\upsilon -1}\mu \left(s\right)ds.$$

(3)

Through out this paper, we fix the notation to be as follows:

• The functional construct ${\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X)$ designates a Hilbertian configuration space encompassing all ${\mathcal{F}}_t$–adapted measurable random processes with ${∥x∥}_{{\mathcal{L}}_{\mathcal{F}}^2}={\left({\int }_0^{\rho }\mathbb{E}{∥x∥}^{2}dt\right)}^{1/2}<\infty$.
• To formulate the concept of a mild solution to System (1), we begin by introducing the relevant Banach space

  $$\begin{array}{ccc}\hfill \mathcal{Y}(\mathcal{J},{\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X))& =& \{x:\mathcal{J}\to {\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X):x\in C((t_{\kappa },t_{\kappa +1}],{\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X)),\kappa =1,2,\dots ,\gamma ,\hfill \\ & & \exists x\left(t_{\kappa }^{+}\right),x\left(t_{\kappa }^{-}\right)withx\left(t_{\kappa }^{-}\right)=x\left(t_{\kappa }\right)\}\hfill \end{array}$$

  with ${∥x∥}_{\mathcal{Y}}={sup}_{t\in \mathcal{J}}{\left[\mathbb{E}{∥x\left(t\right)∥}^2\right]}^{1/2}$.
• Let ${\mathrm{\Sigma }}_{\varsigma }\subset \mathcal{Y}$ be stipulated by ${\mathrm{\Sigma }}_{\varsigma }=\left\{x\in {\mathcal{Y}:\parallel x\parallel }_{\mathcal{Y}}^2\le \varsigma ,\varsigma >0\right\},$ which is a bounded, closed, and convex set in $\mathcal{Y}$.
• Define $\mathbb{G}\left(x\right):{\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X)\to 2^{{\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X)}$ by

  $$\mathbb{G}\left(x\right)=\left\{F\in {\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X):F\left(t\right)\in \partial F(t,x\left(t\right)),\forall x\in {\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X)\right\}.$$

The linear operators $Z:\mathbf{D}\left(Z\right)\subset X\to X$ and $\mathrm{\Theta }:\mathbf{D}\left(\mathrm{\Theta }\right)\subset X\to X$ verify the following:

1.

Z and $\mathrm{\Theta }$ are closed operators.

2.

$\mathbf{D}\left(\mathrm{\Theta }\right)\subset \mathbf{D}\left(Z\right)$ and $\mathrm{\Theta }$ are bijective.

3.

${\mathrm{\Theta }}^{-1}:X\to \mathbf{D}\left(\mathrm{\Theta }\right)$ is continuous.

4.

In this context, 1 in conjunction with 2, synergistically invokes alongside the closed graph theorem and collectively substantiates the boundedness of the linear transformation $Z{\mathrm{\Theta }}^{-1}$. Denote $∥\mathrm{\Theta }∥={\mathfrak{J}}_{\mathrm{\Theta }}$, ${∥\mho ∥}^2={\varsigma }_{\mho }$, and $∥{\mathrm{\Theta }}^{-1}∥={\tilde{\mathfrak{J}}}_{\mathrm{\Theta }}$.

Definition 2 ([31]).

The Clarke generalized directional derivative of $F:\mathcal{N}\to \mathbb{R}$ at x in the direction ν is characterized by

$$F^0(x;\nu )=\underset{\delta \to 0^{+},y\to x}{lim\; sup}{\displaystyle \frac{F(y+\delta \nu )-F\left(y\right)}{\delta }},$$

and

$$\partial F\left(x\right)=\left\{x^{*}\in {\mathcal{N}}^{*}:F^0(x,\nu )\ge \langle x^{*},\nu \rangle ,\forall \nu \in \mathcal{N}\right\}$$

is the Clarke sub-differential of F at $x\in \mathcal{N}.$

Lemma 1 ([29]).

$\mathfrak{N}:\mathcal{J}\times X\to S\left(X\right)$ is measurable (measurable) to t, ∀$x\in X$, and it is upper semi-continuous to x, ∀$t\in \mathcal{J}$ and ∀$x\in C(\mathcal{J},X)$, where ${\mathbb{S}}_{\mathfrak{N},x}:=\{F\in {\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X):F\left(x\right)\in \mathfrak{N}(t,x\left(t\right))\}$ for a.e. $t\in \mathcal{J}$ is nonempty. Let $P:{\mathcal{L}}^1(\mathcal{J},X)\to C(\mathcal{J},X)$, then

$$\begin{array}{c}\hfill P\circ {\mathbb{S}}_{\mathfrak{N},x}:C(\mathcal{J},X)\to S\left(C(\mathcal{J},X)\right),\\ \hfill x\mapsto P\circ {\mathbb{S}}_{\mathfrak{N},x}\left(\mathfrak{G}\right)=P\left({\mathbb{S}}_{\mathfrak{N},x}\right)\end{array}$$

is a closed graph operator in $C(\mathcal{J},X)\times C(\mathcal{J},X)$, where

$$S\left(X\right)=\left\{\mathfrak{X}\in \mathcal{P}\left(X\right):\mathfrak{X}isbounded,closed,andconvex\right\}.$$

Assessing the associated linear system is a crucial phase to figuring out the approximate controllability of (1):

$$\begin{array}{ccc}\hfill {}^{\mathcal{ABC}}D_{0+}^{\upsilon }\mathrm{\Theta }x\left(t\right)& \in & Zx\left(t\right)+\mho U\left(t\right),t\in \mathcal{J}\hfill \\ \hfill x\left(0\right)& =& x_0.\hfill \end{array}$$

(4)

Let $x(\rho ,x_0,U)$ denote the state of System (1) at $\rho$, which correlates with the original state $x_0\in X$ and control U. If $\mathcal{R}(\rho ,x_0)=\left\{x(\rho ,x_0,U):U\in {\mathcal{L}}_2(\mathcal{J},\mathcal{U})\right\},$ then its closure is delimited by the set $\overline{\mathcal{R}(\rho ,x_0)}$ in X.

Definition 3 ([36]). 

On $[0,\rho ]$, (1) is alleged to approximately controllable if $\overline{\mathcal{R}(\rho ,x_0)}={\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X).$

Remark 1.

In the framework of System (1), controllability refers to the ability to steer the state trajectory $x(·)$ from its initial value $x\left(0\right)$ to any preassigned terminal state $x_{\rho }\in X$ at time $\rho >0$ by means of a suitable admissible control $U(·)$ belonging to a prescribed control space. Formally, this means that, for every $x_{\rho }\in X$ there exists $U(·)$, such that the solution of (1) satisfies $x\left(\rho \right)=x_{\rho }$. In contrast, approximate controllability only requires that, for each $x_{\rho }\in X$ and each $\epsilon >0$, there exists a control $U(·)$ such that $\parallel x\left(\rho \right)-x_{\rho }\parallel <\epsilon$. This weaker notion is of particular relevance in infinite-dimensional stochastic systems, where exact controllability is typically impossible due to compactness of solution operators or spectral properties of the underlying generator. Hence, approximate controllability becomes the natural and practically achievable goal in such settings.

Remark 2.

In $\mathcal{J}$, (4) is approximately controllable iffy $\iota \varrho (\iota ,{\mathrm{\Delta }}_0^{\rho })\to 0\mathit{strongly}\mathit{as}\iota \to 0^{+}$, where $\varrho (\iota ,{\mathrm{\Delta }}_0^{\rho })={(\iota I+{\mathrm{\Delta }}_0^{\rho })}^{-1},\iota >0$, and ${\mathrm{\Delta }}_0^{\rho }={\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(\rho -s)\mho {\mho }^{*}{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }^{*}(\rho -s)ds.$

Lemma 2 ([36]).

For any ${\tilde{x}}_{\rho }\in {\mathcal{L}}^2(\mathrm{\Omega },X)$, $\mathrm{\exists } \tilde{\eta }\left(t\right)\in {\mathcal{L}}^2(\mathrm{\Omega };{\mathcal{L}}^2(\mathcal{J},{\mathcal{L}}_{{\rm Y}}^2))$, such that

$${\tilde{x}}_{\rho }=\mathbb{E}{\tilde{x}}_{\rho }+{\int }_0^{\rho }\tilde{\eta }\left(s\right)dW\left(s\right).$$

3. Main Results

Definition 4.

An ${\mathcal{F}}_t$-adapted process $x\in \mathcal{Y}$ is called a mild solution of Problem (1) if the following applies:

• $x\left(0\right)\in X$, and $\mathrm{\Delta }x{\mid }_{t=t_{\kappa }}={\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right)$ for $\kappa =1,2,\dots ,\gamma$;
• For all $t\in \mathcal{J}$,

  $$\begin{array}{ccc}\hfill x\left(t\right)& =& {\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(t\right)\mathrm{\Theta }\left\{x_0-m\left(x\right)-y(0,x\left(0\right))\right\}+y(t,x\left(t\right))\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\left\{\mho U\left(s\right)+F\left(s\right)\right\}ds\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\int }_{\mathrm{\Pi }}{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\left\{\mho U\left(s\right)+F\left(s\right)\right\}ds\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\int }_{\mathrm{\Pi }}{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +\sum _{0<t_{\kappa }<t}{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }(t-t_{\kappa })\mathrm{\Theta }{\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right).\hfill \end{array}$$
  Here, $\Re ={\theta }^{*}{({\theta }^{*}I-Z{\mathrm{\Theta }}^{-1})}^{-1}$, $\wp =-{\delta }^{*}Z{({\theta }^{*}I-Z{\mathrm{\Theta }}^{-1})}^{-1}$, with ${\theta }^{*}={\displaystyle \frac{\mu \left(\upsilon \right)}{1-\upsilon }}$, ${\delta }^{*}={\displaystyle \frac{\upsilon }{1-\upsilon }}$, and

  $$\begin{array}{ccc}\hfill V_{\upsilon }\left(t\right)& =& {\mathcal{E}}_{\upsilon }(-\wp t^{\upsilon })={\displaystyle \frac{1}{2\pi i}}{\int }_{\chi }{e}^{st}{s}^{\upsilon -1}{(s^{\upsilon }I-\wp )}^{-1}ds,\hfill \\ \hfill Q_{\upsilon }\left(t\right)& =& t^{\upsilon -1}{\mathcal{E}}_{\upsilon ,\upsilon }(-\wp t^{\upsilon })={\displaystyle \frac{1}{2\pi i}}{\int }_{\chi }{e}^{st}{(s^{\upsilon }I-\wp )}^{-1}ds.\hfill \end{array}$$

We stress that the displayed expression for the mild solution is to be understood as an integral inclusion. Concretely, a process $x(·)$ is a mild solution of Problem (1) if x is ${\mathcal{F}}_t$-adapted, satisfies the jump conditions, and there exists an ${\mathcal{F}}_t$-progressively measurable selection $f(·):t\mapsto f\left(t\right)\in \partial F\left(t,x\left(t\right)\right)$ for almost every $t\in \mathcal{J}$, such that the displayed variation-of-constants identity holds when $\partial F(t,x(t\left)\right)$ is replaced by the single-valued mapping $f\left(t\right)$. Equivalently, the multivalued term is interpreted in the sense of measurable selections (see, e.g., ref. [30] for background), such that each occurrence of the set-valued expression contributes a measurable integrand. Under the standing measurability and growth assumptions on $\partial F$, $\zeta$, and H, the stochastic integrals with respect to the Wiener process and the compensated Poisson random measure are well-defined, and the impulsive sum accounts for the jumps ${\mathcal{I}}_{\kappa }$. Thus, the notion of mild solution adopted here is consistent with the differential Inclusion (1) and with standard treatments of multivalued stochastic inclusions.

Let us adopt the subsequent assumptions.

$\left(A1\right)$

Assume that $Z\in {\ell }^{\upsilon }({\alpha }_0,l_0)$, then $∥V_{\upsilon }\left(t\right)∥\le \widehat{\mathfrak{L}}{e}^{lt}$ and $∥Q_{\upsilon }\left(t\right)∥\le \Re e^{lt}(1+t^{\upsilon -1})$, ∀$t>0,l>l_0$. Thus, ${\mathfrak{L}}^{*}={sup}_{t\ge 0}∥V_{\upsilon }\left(t\right)∥$, ${\mathfrak{L}}_1^{*}={sup}_{\mathfrak{t}\ge 0}\Re e^{lt}(1+t^{\upsilon -1})$. So, $∥V_{\upsilon }\left(t\right)∥\le {\mathfrak{L}}^{*}$ and $∥Q_{\upsilon }\left(t\right)∥\le {\mathfrak{L}}_1^{*}{t}^{\upsilon -1}$ (For more information, see [37]). Let $\exists \theta$ and $\psi$ such that $∥\Re ∥\le \theta$ and $∥\wp ∥\le \psi$, and $∥\iota \varrho (\iota ,{\mathrm{\Delta }}_0^{\rho })∥\le 1,\forall \iota >0$.

$\left(A2\right)$

$F:\mathcal{J}\times X\to \mathbb{R}$ accomplishes the following:

1.

$F(·,x)$ is measurable, ∀$x\in X$.

2.

$F(t,·)$ is locally Lipschitz continuous for a.e. $t\in \mathcal{J}$.

3.

${\Im }_F\in {\mathcal{L}}^1(\mathcal{J},{\mathbb{R}}^{+})$ and $k_F>0$ such that ∀$x\in X$ and a.e., $t\in \mathcal{J}$,

$$\mathbb{E}{∥\partial F(t,x(t\left)\right)∥}^2=sup\left\{{∥F∥}^2:F\in \partial F(t,x\left(t\right))\right\}\le {\Im }_F\left(t\right)+k_F{∥x∥}^2$$

$\left(A3\right)$

$m:C(\mathcal{J},X)\to X$, which is a completely continuous map, accomplishes the following:

1.

∃ constants $H_1>0$ and $H_2>0$ such that

$$\mathbb{E}{∥m\left(x\right)∥}^2\le H_1\mathbb{E}{∥x∥}^2+H_2,\forall x\in X.$$

$\left(A4\right)$

$\zeta :\mathcal{J}\times X\to {\mathcal{L}}_{{\rm Y}}^2(Y,X)$ accomplishes the following:

1.

$\zeta (·,x)$ is strongly measurable ∀$x\in X$, and $\zeta (t,·)$ is continuous ∀$t\in \mathcal{J}$.

2.

∃$k_{\zeta }>0$ such that

$$\mathbb{E}{∥\zeta (t,x)∥}_{{\mathcal{L}}_{{\rm Y}}^2}^2\le k_{\zeta }\left[1+\mathbb{E}{∥x∥}^2\right].$$

$\left(A5\right)$

$\exists k_y>0$ such that

$$\mathbb{E}{∥y(t,x(t\left)\right)∥}^2\le k_y\left[1+\mathbb{E}{∥x∥}^2\right].$$

$\left(A6\right)$

For every $t\in \mathcal{J}$ and $x\in X$, the mapping $\mathbb{P}\mapsto H(t,x,\mathbb{P})$ is continuous, and $t\mapsto H(t,x,\mathbb{P})$ is square–measurable. Moreover, there exists a constant $k_H>0$ such that

$${\int }_{\mathrm{\Pi }}{∥H(t,x,\mathbb{P})∥}^2\lambda \left(d\mathbb{P}\right)\le k_H\left[1+{∥x∥}^2\right].$$

$\left(A7\right)$

• 1. The operators ${\mathcal{I}}_{\kappa }:X\to X$ are completely continuous, and there exists a constant $d_{\kappa }>0$ such that

  $$\mathbb{E}{∥{\mathcal{I}}_{\kappa }\left(x\right)∥}^2\le d_{\kappa },\kappa =1,2,\dots ,\gamma ,\forall x\in X.$$
• 2. ∃ is a constant ${𝚤}^{*}>0$ such that

  $$\mathbb{E}{∥{\mathcal{I}}_{\kappa }\left(x\right)-{\mathcal{I}}_{\kappa }\left(\overline{x}\right)∥}^2\le {𝚤}^{*}\mathbb{E}{∥x-\overline{x}∥}^2$$
  ∀$x,\overline{x}\in X$ and $\kappa =1,2,\dots ,\gamma$.

Remark 3.

Let $\mathbb{F}={\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ denote the underlying filtration on the probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, satisfying the usual conditions (right-continuity and completeness). The assumptions imposed on H ensure the well-posedness of the Poisson stochastic integral appearing in the system. In particular, the progressive measurability of the mapping $t\mapsto H(t,x,\mathbb{P})$, together with the growth condition

$${\int }_{\mathrm{\Pi }}{\parallel H(t,x,\mathbb{P})\parallel }^2\lambda \left(d\mathbb{P}\right)\le k_H\left[1+{\parallel x\parallel }^2\right],$$

guarantees that, for each $x\in X$, the mapping $(t,\mathbb{P})\mapsto H(t,x(t),\mathbb{P})$ is $\mathbb{F}$-progressively measurable and square-integrable. Consequently, the stochastic integral with respect to the compensated Poisson random measure,

$${\int }_0^t{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P}),$$

is well-defined in the sense of stochastic integration in Hilbert spaces (see, e.g., [26]). Whenever $\mathbb{E}[·]$ is used, it denotes the expectation with respect to the probability measure $\mathbb{P}$.

Remark 4.

While Assumption $\left(A7\right)$ requires the impulsive mappings ${\mathcal{I}}_{\kappa }$ to be completely continuous and Lipschitz in the mean-square norm, this class includes a wide family of nonlinear integral or Nemytskii-type operators arising naturally in control and physical models. An explicit example satisfying $\left(A7\right)$ is provided in Section 4.

Lemma 3 ([38]).

Assume that conditions $\left(A1\right)$–$\left(A2\right)$ hold. Then, for every $x\in {\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X)$, the set-valued mapping $\mathbb{G}\left(x\right)$ is nonempty, convex, and weakly compact. Moreover, if $x_n\to x$ strongly in ${\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X)$ with $x_n\in \mathbb{G}\left(x_n\right)$ and $x_n\to x$ weakly in ${\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X)$, then $x\in \mathbb{G}\left(x\right)$.

Theorem 1 ([30]).

Define $\mathcal{C}$ to be a locally convex Banach space, and let ${\xi }_{\tau }:\mathcal{C}\to 2^{\mathcal{C}}$ to be an upper semicontinuous set-valued map with nonempty, compact, and convex values. Suppose there exists a closed neighborhood $M\subset \mathcal{C}$ of the origin such that ${\xi }_{\tau }\left(M\right)$ is relatively compact. Define

$$\mathrm{\Sigma }:=\left\{x\in \mathcal{C}:\beta x\in {\xi }_{\tau }\left(M\right)\mathit{for}\mathit{some}\beta >1\right\}.$$

If Σ is a bounded subset of $\mathcal{C}$, then ${\xi }_{\tau }$ admits at least one fixed point, that is, there exists $x^{*}\in \mathcal{C}$ such that $x^{*}\in {\xi }_{\tau }\left(x^{*}\right)$.

Theorem 2.

Under the assumptions $\left(A1\right)$ to $\left(A7\right)$, it follows that, for any $U(·)\in {\mathcal{L}}_2(\mathcal{J},\mathcal{U})$, System (1) possesses a mild solution in the space $\mathcal{Y}$, recognizing that $\gamma {\mathfrak{J}}_{\mathrm{\Theta }}^2{\tilde{\mathfrak{J}}}_{\mathrm{\Theta }}^2{\left({\mathfrak{L}}^{*}\theta \right)}^2{𝚤}^{*}<1$.

Proof. 

For $\iota >0$, ∀$x(·)\in \mathcal{Y},{\tilde{x}}_{\rho }\in {\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X)$, U is thereby designated as follows:

$$U\left(t\right)={\mho }^{*}{Q}_{\upsilon }^{*}(\rho -s)\varrho (\iota ,{\mathrm{\Delta }}_0^{\rho })\mathbb{A}\left(x(·)\right),$$

where

$$\begin{array}{ccc}\hfill \mathbb{A}\left(x\right(·\left)\right)& =& \left[\mathbb{E}{\tilde{x}}_{\rho }+{\int }_0^{\rho }\tilde{\eta }\left(s\right)dW\left(s\right)-m\left(x\right)-{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(\rho \right)\mathrm{\Theta }\left\{x_0-m\left(x\right)-y(0,x\left(0\right))\right\}\right]-y(\rho ,x\left(\rho \right))\hfill \\ & & -{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{(\rho -s)}^{\upsilon -1}F\left(s\right)ds-{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{(\rho -s)}^{\upsilon -1}\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & -{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{(\rho -s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & -{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(\rho -s)F\left(s\right)ds-{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(\rho -s)\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & -{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(\rho -s){\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & -\sum _{\kappa =1}^{\gamma }{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }(\rho -t_{\kappa })\mathrm{\Theta }{\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right).\hfill \end{array}$$

${\xi }_{\tau }:\mathcal{Y}\to 2^{\mathcal{Y}}$ is thereby designated as follows:

$$\begin{array}{ccc}\hfill {\xi }_{\tau }\left(x\right)& =& \{\mathrm{\Xi }\in \mathcal{Y}:\mathrm{\Xi }\left(t\right)={\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(t\right)\mathrm{\Theta }\left\{x_0-m\left(x\right)-y(0,x\left(0\right))\right\}+y(t,x\left(t\right))\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\left\{\mho U\left(s\right)+F\left(s\right)\right\}ds\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\left\{\mho U\left(s\right)+F\left(s\right)\right\}ds\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s){\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +\sum _{\kappa =1}^{\gamma }{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }(t-t_{\kappa })\mathrm{\Theta }{\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right).\}\hfill \end{array}$$

We segregated our evidence into five various steps, as shown beneath.

Step 1.

For every $x\in \mathcal{Y}$, the operator ${\xi }_{\tau }\left(x\right)$ is nonempty, convex, and has weakly compact values. By applying Lemma 3, it follows that ${\xi }_{\tau }\left(x\right)\subset \mathbb{G}\left(x\right)$ is nonempty and its values are weakly compact. Furthermore, since $\mathbb{G}\left(x\right)$ is convex-valued, any convex combination $\sigma {\eta }_1+(1-\sigma ){\eta }_2$, with ${\eta }_1,{\eta }_2\in \mathbb{G}\left(x\right)$ and $\sigma \in (0,1)$, also belongs to $\mathbb{G}\left(x\right)$. Therefore, ${\xi }_{\tau }\left(x\right)$ is convex.

Step 2.

The operator ${\xi }_{\tau }\left(x\right)$ maps $\mathcal{Y}$ into itself.

According to the set ${\mathrm{\Sigma }}_{\varsigma }$’s description in $\mathcal{Y}$, we aim to demonstrate the existence of a constant $ϵ>0$ such that ∀$\mathrm{\Xi }\in {\xi }_{\tau }\left(x\right)$, where $x\in {\mathrm{\Sigma }}_{\varsigma }$. The subsequent inequality is genuine: ${\mathbb{E}\parallel \mathrm{\Xi }\left(t\right)\parallel }^2\le ϵ.$ Assuming $\mathrm{\Xi }\in {\xi }_{\tau }\left(x\right)$, ∃ is a function $F\in \mathbb{G}\left(x\right)$ such that, we have

$$\begin{array}{ccc}\hfill \mathrm{\Xi }\left(t\right)& =& {\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(t\right)\mathrm{\Theta }\{x_0-m\left(x\right)-y(0,x\left(0\right))\}+y(t,x\left(t\right))\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\{\mho U(s)+F(s)\}ds\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\left\{\mho U\left(s\right)+F\left(s\right)\right\}ds\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s){\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +\sum _{\kappa =1}^{\gamma }{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }(t-t_{\kappa })\mathrm{\Theta }{\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right)t\in \mathcal{J}.\hfill \end{array}$$

(5)

Then,

$$\begin{array}{ccc}& & \mathbb{E}{∥\mathrm{\Xi }\left(t\right)∥}^2\le 11\mathbb{E}{∥{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(t\right)\mathrm{\Theta }\{x_0-m\left(x\right)-y(0,x\left(0\right))\}∥}^2+11\mathbb{E}{∥y(t,x(t\left)\right)∥}^2\hfill \\ & & +11\mathbb{E}{∥{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\mho U\left(s\right)ds∥}^2+11\mathbb{E}{∥{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\xi \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}F\left(s\right)ds∥}^2\hfill \\ & & +11\mathbb{E}{∥{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\zeta (s,x\left(s\right))dW\left(s\right)∥}^2\hfill \\ & & +11\mathbb{E}{∥{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})∥}^2\hfill \\ & & +11\mathbb{E}{∥{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\mho U\left(s\right)ds∥}^2+11\mathbb{E}{∥{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)F\left(s\right)ds∥}^2\hfill \\ & & +11\mathbb{E}{∥{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\zeta (s,x\left(s\right))dW\left(s\right)∥}^2\hfill \\ & & +11\mathbb{E}{∥{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s){\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})∥}^2\hfill \\ & & +11\mathbb{E}{∥\sum _{\kappa =1}^{\gamma }{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }(t-t_{\kappa })\mathrm{\Theta }{\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right)∥}^2.\hfill \end{array}$$

Consequently,

$$\mathbb{E}{∥\mathrm{\Xi }\left(t\right)∥}^2\le 11\left\{1+{\left[{\displaystyle \frac{{\mathfrak{L}}_1^{*}{\varsigma }_{\mho }\theta \psi (1-\upsilon ){\tilde{\mathfrak{J}}}_{\mathrm{\Theta }}}{\iota \mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}\right]}^2{\displaystyle \frac{{\rho }^{2\upsilon -1}}{2\upsilon -1}}+{\left({\displaystyle \frac{\upsilon \theta {\varsigma }_{\mho }{\tilde{\mathfrak{J}}}_{\mathrm{\Theta }}{\mathfrak{L}}_1^{*}}{\iota \mu \left(\upsilon \right)}}\right)}^2{\displaystyle \frac{{\rho }^{2\upsilon -1}}{2\upsilon -1}}\right\}\times \mathcal{K},$$

where

$$\begin{array}{ccc}& & \mathcal{K}=\{{\mathfrak{J}}_{\mathrm{\Theta }}^2{\tilde{\mathfrak{J}}}_{\mathrm{\Theta }}^2{\left({\mathfrak{L}}^{*}\theta \right)}^2\left[\left(\mathbb{E}{∥x_0∥}^2+H_1\varsigma +H_2\right)(1+k_y)+\gamma \sum _{\kappa =1}^{\gamma }{d}_{\kappa }+k_y\right]+k_y(1+\varsigma )\hfill \\ & & +\left[{\left({\displaystyle \frac{\theta \psi (1-\upsilon ){\tilde{\mathfrak{J}}}_{\mathrm{\Theta }}}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}\right)}^2+{\left({\displaystyle \frac{\upsilon \theta {\tilde{\mathfrak{J}}}_{\mathrm{\Theta }}{\mathfrak{L}}_1^{*}}{\mu \left(\upsilon \right)}}\right)}^2\right]{\displaystyle \frac{{\rho }^{2\upsilon -1}}{2\upsilon -1}}\left[{\displaystyle ∥{\Im }_F∥+k_F\rho \varsigma +\left(k_H+Tr{\rm Y}{k}_{\zeta }\right)(1+\varsigma )}\right]\}\hfill \\ & & +\left\{{\left[{\displaystyle \frac{\theta \psi (1-\upsilon ){\tilde{\mathfrak{J}}}_{\mathrm{\Theta }}}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}\right]}^2+{\left({\displaystyle \frac{\upsilon \theta {\tilde{\mathfrak{J}}}_{\mathrm{\Theta }}{\mathfrak{L}}_1^{*}}{\mu \left(\upsilon \right)}}\right)}^2\right\}{\displaystyle \frac{{\rho }^{2\upsilon -1}}{2\upsilon -1}}\left[2\mathbb{E}{∥{\tilde{x}}_{\rho }∥}^2+2{\int }_0^{\rho }\mathbb{E}{∥\tilde{\eta }\left(s\right)∥}^{2}ds+H_1\varsigma +H_2\right].\hfill \end{array}$$

Step 3.

For every $x\in \mathcal{Y}$, the set ${\xi }_{\tau }\left(x\right)$ is closed.

Let ${\left\{{\mathrm{\Xi }}_n\left(t\right)\right\}}_{n\ge 0}\subset {\xi }_{\tau }\left(x\right)$ such that as $n\to \infty$, ${\mathrm{\Xi }}_n\to \mathrm{\Xi }\in \mathcal{Y}$. Then, ∃$F_n\in \mathbb{G}\left(x\right)$ such that ∀$t\in (t_{\kappa },t_{\kappa +1}]$:

$$\begin{array}{ccc}\hfill {\mathrm{\Xi }}_n\left(t\right)& =& {\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(t\right)\mathrm{\Theta }\{x_0-m\left(x\right)-y(0,x\left(0\right))\}+y(t,x\left(t\right))\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}{F}_n\left(s\right)ds\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\{\mho {\mho }^{*}{Q}_{\upsilon }^{*}(\rho -s)\varrho (\iota ,{\mathrm{\Delta }}_0^{\rho }){\mathbb{A}}_n\left(x(·)\right)\}ds\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)F_n\left(s\right)ds\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\{\mho {\mho }^{*}{Q}_{\upsilon }^{*}(\rho -s)\varrho (\iota ,{\mathrm{\Delta }}_0^{\rho }){\mathbb{A}}_n\left(x(·)\right)\}ds\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s){\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +\sum _{\kappa =1}^{\gamma }{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }(t-t_{\kappa })\mathrm{\Theta }{\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\mathbb{A}}_n\left(x(·)\right)& =& \left[\mathbb{E}{\tilde{x}}_{\rho }+{\int }_0^{\rho }\tilde{\eta }\left(s\right)dW\left(s\right)-m\left(x\right)-{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(\rho \right)\mathrm{\Theta }\{x_0-m\left(x\right)-y(0,x\left(0\right))\right]\}-y(\rho ,x\left(\rho \right))\hfill \\ & & -{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{(\rho -s)}^{\upsilon -1}{F}_n\left(s\right)ds-{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{(\rho -s)}^{\upsilon -1}\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & -{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{(\rho -s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & -{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(\rho -s)F_n\left(s\right)ds-{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(\rho -s)\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & -{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(\rho -s){\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & -\sum _{\kappa =1}^{\gamma }{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }(\rho -t_{\kappa })\mathrm{\Theta }{\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right).\hfill \end{array}$$

To the extent that $\mathbb{G}\left(x\right)$ is weakly compact, it follows that $\left\{F_n\right\}$ admits a weakly convergent subsequence with limit $F\in \mathbb{G}\left(x\right)$. Consequently, ${\mathrm{\Xi }}_n\left(t\right)\to \mathrm{\Xi }\left(t\right)$ as $n\to \infty$.

Step 4.

${\xi }_{\tau }\left(x\right)$ is upper semi-continuous and condensing.

Express ${\xi }_{\tau }$ as, ${\xi }_{\tau }={\xi }_{{\tau }_1}+{\xi }_{{\tau }_2}$, where ${\xi }_{{\tau }_1}$ and ${\xi }_{{\tau }_2}$ are defined as follows:

$$\begin{array}{ccc}\hfill {\xi }_{{\tau }_1}& =& \sum _{\kappa =1}^{\gamma }{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }(t-t_{\kappa })\mathrm{\Theta }{\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right),\hfill \\ \hfill {\xi }_{{\tau }_2}& =& {\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(t\right)\mathrm{\Theta }\{x_0-m\left(x\right)-y(0,x\left(0\right))\}+y(t,x\left(t\right))\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\{\mho U(s)+F(s)\}ds\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\{\mho U(s)+F(s)\}ds\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s){\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P}),t\in \mathcal{J}\setminus \{t_1,\dots ,t_{\gamma }\}.\hfill \end{array}$$

It can be shown that the operator ${\xi }_{{\tau }_1}$ exhibits a contractive nature, whereas ${\xi }_{{\tau }_2}$ qualifies as a completely continuous mapping for all elements $x,\overline{x}\in X$ and $t\in (t_{\kappa },t_{\kappa +1}]$. Thus, we estimate the norm of the impulsive operator ${\xi }_{{\tau }_1}$ as follows:

$$\begin{array}{ccc}\hfill \mathbb{E}{∥{\xi }_{{\tau }_1}\left(x\right)\left(t\right)-{\xi }_{{\tau }_1}\left(\overline{x}\right)\left(t\right)∥}^2& =& \mathbb{E}{∥\sum _{\kappa =1}^{\gamma }{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }(t-t_{\kappa })\mathrm{\Theta }\left[{\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right)-{\mathcal{I}}_{\kappa }\left(\overline{x}\left(t_{\kappa }^{-}\right)\right)\right]∥}^2\hfill \\ & \le & \gamma \sum _{\kappa =1}^{\gamma }\mathbb{E}{∥{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }(t-t_{\kappa })\mathrm{\Theta }∥}^2\mathbb{E}{∥{\mathcal{I}}_{\kappa }\left(x\right)-{\mathcal{I}}_{\kappa }\left(\overline{x}\right)∥}^2\hfill \\ & \le & \gamma {\left({\tilde{\mathfrak{J}}}_{\mathrm{\Theta }}\theta {\mathfrak{L}}^{*}{\mathfrak{J}}_{\mathrm{\Theta }}\right)}^2·{𝚤}^{*}\mathbb{E}{\parallel x-\overline{x}\parallel }^2.\hfill \end{array}$$

Therefore, the mapping ${\xi }_{{\tau }_1}$ is contractive under the following sufficient condition:

$$\gamma {\mathfrak{J}}_{\mathrm{\Theta }}^2{\tilde{\mathfrak{J}}}_{\mathrm{\Theta }}^2{\left({\mathfrak{L}}^{*}\theta \right)}^2{𝚤}^{*}<1.$$

Lemma 4.

Let ${\left\{Q_{\upsilon }\left(t\right)\right\}}_{t\ge 0}$ be an analytic semigroup generated by a sectorial operator on a Banach space $\mathcal{C}$. Then, for every $t>0$, the operator $Q_{\upsilon }\left(t\right)$ is compact.

Proof. 

Since the generator is sectorial and the semigroup is analytic, it is strongly continuous and maps bounded subsets of $\mathcal{C}$ into relatively compact subsets for $t>0$ (see [39]). Hence, $Q_{\upsilon }\left(t\right)$ is compact for each $t>0$. □

Remark 5.

Based on Lemma 4, and noting that the operator $Q_{\upsilon }\left(t\right)$ is compact for each $t>0$, then the operators ${\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(·)$ appearing in the stochastic and deterministic integral terms of ${\xi }_{{\tau }_2}$ inherit this compactness. The integrands in ${\xi }_{{\tau }_2}$ (such as $\zeta (·,·)$, $H(·,·,·)$, and $F(·)$) are assumed to be measurable and satisfy Lipschitz conditions, and the integrals are of the Bochner and Itô types. These preserve compactness when composed with compact operators (see [26]). Therefore, each term in ${\xi }_{{\tau }_2}$ maps bounded subsets of X into relatively compact subsets. We conclude that ${\xi }_{{\tau }_2}$ is a compact operator on X for all $t\in \mathcal{J}\setminus \{t_1,\dots ,t_{\gamma }\}$, satisfying the requirements of Theorem 1.

• We proceeded to verify that ${\xi }_{{\tau }_2}$ is upper semi-continuous and completely continuous, and the argument is structured through three separate assertions.

Claim 1.

${\xi }_{{\tau }_2}$ sends bounded subsets of $\mathcal{Y}$ into uniformly bounded ones. The conclusion follows directly by applying the method outlined in Step 2.

Claim 2.

$\left\{{\xi }_{{\tau }_2}\left(x\right):x\in {\mathrm{\Sigma }}_{\varsigma }\right\}$ is equicontinuous.

$\forall x\in {\mathrm{\Sigma }}_{\varsigma },\mathrm{\Xi }\in {\xi }_{\tau }\left(x\right)$, ∃$F\in \mathbb{G}\left(x\right)$ such that ∀$t\in \mathcal{J}$, (5) holds. For $0<t_1<t_2<\rho$, we acquire

$$\begin{array}{ccc}& & \mathbb{E}{∥\mathrm{\Xi }\left(t_2\right)-\mathrm{\Xi }\left(t_1\right)∥}^2\le 8\mathbb{E}{∥{\mathrm{\Theta }}^{-1}\Re \{V_{\upsilon }\left(t_2\right)-V_{\upsilon }\left(t_1\right)\}\mathrm{\Theta }\{x_0-m\left(x\right)-y(0,x\left(0\right))\}∥}^2\hfill \\ & & +8\mathbb{E}{∥y(t_2,x\left(t_2\right))-y(t_1,x\left(t_1\right))∥}^2\hfill \\ & & +8\mathbb{E}\Vert {\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}[{\int }_{t_1}^{t_2}{\mathrm{\Theta }}^{-1}{(t_2-s)}^{\upsilon -1}F\left(s\right)ds\hfill \\ & & +{\int }_0^{t_1}{\mathrm{\Theta }}^{-1}[{(t_2-s)}^{\upsilon -1}-{(t_1-s)}^{\upsilon -1}]{F\left(s\right)ds]\Vert }^2\hfill \\ & & +8\mathbb{E}\Vert {\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}\{{\int }_{t_1}^{t_2}{\mathrm{\Theta }}^{-1}{(t_2-s)}^{\upsilon -1}\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\int }_0^{t_1}{\mathrm{\Theta }}^{-1}\left[{(t_2-s)}^{\upsilon -1}-{(t_1-s)}^{\upsilon -1}\right]{\zeta (s,x\left(s\right))dW\left(s\right)\}\Vert }^2\hfill \\ & & +8\mathbb{E}\Vert {\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}\{{\int }_{t_1}^{t_2}{\mathrm{\Theta }}^{-1}{(t_2-s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +{\int }_0^{t_1}{\mathrm{\Theta }}^{-1}\left[{(t_2-s)}^{\upsilon -1}-{(t_1-s)}^{\upsilon -1}\right]{\int }_{\mathrm{\Pi }}{H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\}\Vert }^2\hfill \\ & & +8\mathbb{E}\Vert {\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}\{{\int }_{t_1}^{t_2}{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t_2-s)F\left(s\right)ds\hfill \\ & & +{\int }_0^{t_1}{\mathrm{\Theta }}^{-1}\left[Q_{\upsilon }(t_2-s)-Q_{\upsilon }(t_1-s)\right]{F\left(s\right)ds\}\Vert }^2\hfill \\ & & +8\mathbb{E}\Vert {\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}\{{\int }_{t_1}^{t_2}{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t_2-s)\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\int }_0^{t_1}{\mathrm{\Theta }}^{-1}\left[Q_{\upsilon }(t_2-s)-Q_{\upsilon }(t_1-s)\right]{\zeta (s,x\left(s\right))dW\left(s\right)\}\Vert }^2\hfill \\ & & +8\mathbb{E}\Vert {\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}\{{\int }_{t_1}^{t_2}{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t_2-s){\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +{\int }_0^{t_1}{\mathrm{\Theta }}^{-1}\left[Q_{\upsilon }(t_2-s)-Q_{\upsilon }(t_1-s)\right]{\int }_{\mathrm{\Pi }}{H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\}\Vert }^2\hfill \\ & & \to 0as{t}_2\to t_1.\hfill \end{array}$$

Owing to the compactness of $Q_{\upsilon }$ and the continuity of $\mathrm{\Xi }$, we acquire the result we desire.

Claim 3.

${\xi }_{{\tau }_2}\left(x\right)$ is a completely continuous operator.

If $\forall t\in \mathcal{J},t>0$, then we argue that $\mathcal{V}\left(t\right)=\left\{\mathrm{\Xi }\left(t\right):\mathrm{\Xi }\in {\xi }_{{\tau }_2}\left({\mathrm{\Sigma }}_{\varsigma }\right)\right\}$ is relatively compact. In fact, $\mathcal{V}\left(0\right)$ is relatively compact in ${\mathrm{\Sigma }}_{\varsigma }$. Pretend that $0<t\le \rho$, $0<\epsilon <t$, for $x\in {\mathrm{\Sigma }}_{\varsigma }$; thus, we specify

$$\begin{array}{ccc}\hfill {\mathrm{\Xi }}^{\epsilon }\left(t\right)& =& {\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(t\right)\mathrm{\Theta }\{x_0-m\left(x\right)-y(0,x\left(0\right))\}+y(t,x\left(t\right))\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{t-\epsilon }{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\{\mho U(s)+F(s)\}ds\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{t-\epsilon }{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{t-\epsilon }{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{t-\epsilon }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\{\mho U(s)+F(s)\}ds\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{t-\epsilon }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\zeta (s,x\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{t-\epsilon }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s){\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P}),\hfill \end{array}$$

where $F\in \mathbb{G}\left(x\right)$. Due to the compactness of $Q_{\upsilon }\left(t\right)$, consequently, $\mathcal{V}\left(t\right)$ is relatively compact in X, ∀$\epsilon \in (0,\rho )$. Furthermore, we possess

$$\begin{array}{ccc}\hfill \mathbb{E}{∥\mathrm{\Xi }\left(t\right)-{\mathrm{\Xi }}^{\epsilon }\left(t\right)∥}^2& \le & 8\mathbb{E}{∥{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_{t-\epsilon }^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\mho U\left(s\right)ds∥}^2\hfill \\ & & +8\mathbb{E}{∥{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_{t-\epsilon }^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}F\left(s\right)ds∥}^2\hfill \\ & & +8\mathbb{E}{∥{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_{t-\epsilon }^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\zeta (s,x\left(s\right))dW\left(s\right)∥}^2\hfill \\ & & +8\mathbb{E}{∥{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_{t-\epsilon }^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})∥}^2\hfill \\ & & +8\mathbb{E}{∥{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_{t-\epsilon }^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\mho U\left(s\right)ds∥}^2\hfill \\ & & +8\mathbb{E}{∥{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_{t-\epsilon }^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(\mathfrak{T}-s)F\left(s\right)ds∥}^2\hfill \\ & & +8\mathbb{E}{∥{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\xi \right)}}{\int }_{t-\epsilon }^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\zeta (s,x\left(s\right))dW\left(s\right)∥}^2\hfill \\ & & +8\mathbb{E}{∥{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_{t-\epsilon }^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s){\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})∥}^2.\hfill \end{array}$$

Thus,

$$\mathbb{E}{∥\mathrm{\Xi }\left(t\right)-{\mathrm{\Xi }}^{\epsilon }\left(t\right)∥}^2\to 0,when\epsilon \to 0^{+}.$$

Then, the set $\mathcal{V}\left(t\right)$ is relatively compact. However, the operator ${\xi }_{\tau }$ is completely continuous

Step 5.

${\xi }_{{\tau }_2}$ has a closed graph.

Let $x^{\left(n\right)}\to x^{*}(n\to \infty )$ in $\mathcal{Y}$ and ${\mathrm{\Xi }}^{\left(n\right)}\to {\mathrm{\Xi }}^{*}(n\to \infty )$, ${\mathrm{\Xi }}^{\left(m\right)}\in {\xi }_{{\tau }_2}\left(x^{\left(n\right)}\right)$ in $\mathcal{Y}$. We will then argue that ${\mathrm{\Xi }}^{*}\in {\xi }_{{\tau }_2}\left(x^{*}\right)$. Since ${\mathrm{\Xi }}^{\left(n\right)}\in {\xi }_{{\tau }_2}\left(x^{\left(n\right)}\right)$ signifies that $\mathrm{\exists } F^n\in \mathbb{G}\left(x^n\right)$ such that ∀$t\in (0,\rho )$, we have

$$\begin{array}{ccc}\hfill {\mathrm{\Xi }}^{\left(n\right)}\left(t\right)& =& {\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(t\right)\mathrm{\Theta }\{x_0-n\left(x^n\right)-y(0,x\left(0\right))\}+y(t,x^n\left(t\right))\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\{\mho U(s)+F^n(s)\}ds\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\zeta (s,x^n\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x^n\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\{\mho U(s)+F^n(s)\}ds\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\zeta (s,x^n\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s){\int }_{\mathrm{\Pi }}H(s,x^n\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P}).\hfill \end{array}$$

Utilizing $\left(A2\right)–\left(A7\right)$, one can readily establish that

$${\left\{y(·,x^{\left(n\right)}),F^{\left(n\right)},\zeta (·,x^{\left(n\right)}),H(·,x^{\left(n\right)},·)\right\}}_{n\ge 1}$$

is bounded. Moreover,

$$\left\{y(·,x^{\left(n\right)}),F^{\left(n\right)},\zeta (·,x^{\left(n\right)}),H(·,x^{\left(n\right)},·)\right\}\to \left\{y(·,x^{*}),F^{*},\zeta (·,x^{*}),H(·,x^{*},·)\right\}$$

is weak in $X\times {\mathcal{L}}_{\mathcal{F}}^2(\mathcal{J},X)\times {\mathcal{L}}_{{\rm Y}}^2(Y,X)\times X$. Owing to the compactness of $Q_{\upsilon }$, consequently, we have

$$\begin{array}{ccc}\hfill {\mathrm{\Xi }}^{\left(n\right)}\left(t\right)& \to & {\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(t\right)\mathrm{\Theta }\{x_0-m\left(x^{*}\right)-y(0,x\left(0\right))\}+y(t,x^{*}\left(t\right))\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\{\mho U(s)+F^{*}(s)\}ds\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}\zeta (s,x^{*}\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{(t-s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x^{*}\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\{\mho U(s)+F^{*}(s)\}ds\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s)\zeta (s,x^{*}\left(s\right))dW\left(s\right)\hfill \\ & & +{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^t{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(t-s){\int }_{\mathrm{\Pi }}H(s,x^{*}\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P}).\hfill \end{array}$$

(6)

By concentrating on the convergence ${\mathrm{\Xi }}^{\left(n\right)}\to {\mathrm{\Xi }}^{*}$ in $\mathcal{Y}$, and applying Lemma 3, we conclude that $F^{*}\in \mathbb{G}\left(x^{*}\right)$, which implies ${\mathrm{\Xi }}^{*}\in {\xi }_{{\tau }_2}\left(x^{*}\right)$. Hence, ${\xi }_{{\tau }_2}$ possesses a closed graph and is a completely continuous multivalued operator with compact values. It follows that ${\xi }_{{\tau }_2}$ is upper semi-continuous by Theorem 1, and the operator ${\xi }_{\tau }$ admits a fixed point in ${\mathrm{\Sigma }}_{\varsigma }$, corresponding to a mild solution of Problem (1). □

Theorem 3.

Provided that Assumptions $\left(A1\right)–\left(A7\right)$ are fulfilled, the system described by (1) is approximately controllable over $\mathcal{J}$.

Proof. 

Let $x^{ϵ}$ be a fixed point of ${\xi }_{\tau }$ in ${\mathrm{\Sigma }}_{\varsigma }$; thus, it can be easily seen that

$$\begin{array}{ccc}\hfill x^{ϵ}\left(\rho \right)& =& {\tilde{x}}_{\rho }-ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })[\left[\mathbb{E}{\tilde{x}}_{\rho }+{\int }_0^{\rho }\tilde{\eta }\left(s\right)dW\left(s\right)-m\left(x^{ϵ}\right)-{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(\rho \right)\mathrm{\Theta }\{x_0-m\left(x^{ϵ}\right)-y(0,x\left(0\right))\}\right]\hfill \\ & & -y(\rho ,x^{ϵ}\left(\rho \right))\hfill \\ & & -{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{(\rho -s)}^{\upsilon -1}{F}^{ϵ}\left(s\right)ds-{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{(\rho -s)}^{\upsilon -1}\zeta (s,x^{ϵ}\left(s\right))dW\left(s\right)\hfill \\ & & -{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{(\rho -s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x^{ϵ}\left(\mathfrak{S}\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & -{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(\rho -s)F^{ϵ}\left(s\right)ds-{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(\rho -s)\zeta (s,x^{ϵ}\left(s\right))dW\left(s\right)\hfill \\ & & -{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}{\int }_0^{\rho }{\mathrm{\Theta }}^{-1}{Q}_{\upsilon }(\rho -s){\int }_{\mathrm{\Pi }}H(s,x^{ϵ}\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})\hfill \\ & & -\sum _{\kappa =1}^{\gamma }{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }(\rho -t_{\kappa })\mathrm{\Theta }{\mathcal{I}}_{\kappa }\left(x^{ϵ}\left(t_{\kappa }^{-}\right)\right)].\hfill \end{array}$$

Therefore, there is, in fact, a subsequence $\{F^{ϵ}\left(t\right),y(t,x^{ϵ}\left(t\right)),\zeta (t,x^{ϵ}\left(t\right)),$$H(t,x^{ϵ}\left(t\right),\mathbb{P}\left(t\right))\}$ that converges weakly to $\left\{F\right(t),y(t,x\left(t\right)),\zeta (t,x\left(t\right)),H(t,x\left(t\right),\mathbb{P}\left(t\right)\left)\right\}$ in $X\times {\mathcal{L}}_{{\rm Y}}^2(Y,X)$.

$$\begin{array}{ccc}& & \mathbb{E}{∥x^{ϵ}\left(\rho \right)-{\tilde{x}}_{\rho }∥}^2\le 17\mathbb{E}{∥ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })y(\rho ,x^{ϵ}\left(\rho \right))∥}^2+17Tr{\rm Y}{\int }_0^{\rho }\mathbb{E}{∥ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })\tilde{\eta }∥}_{{\mathcal{L}}_{{\rm Y}}^2}^{2}ds\hfill \\ & & +17\mathbb{E}{∥ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })\left[\mathbb{E}{\tilde{x}}_{\rho }-{\mathrm{\Theta }}^{-1}\Re V_{\upsilon }\left(\rho \right)\mathrm{\Theta }{x}_0-m\left(x\right)-y(0,x\left(0\right))\right]∥}^2\hfill \\ & & +17{\left[{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}\right]}^2\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho }){(\rho -s)}^{\upsilon -1}\left\{F^{ϵ}\left(s\right)-F\left(s\right)\right\}∥ds\right\}}^2\hfill \\ & & +17{\left[{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}\right]}^2\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho }){(\rho -s)}^{\upsilon -1}F\left(s\right)∥ds\right\}}^2\hfill \\ & & +17Tr{\rm Y}\left[{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}\right]\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\zeta (ϵ,{\mathrm{\Delta }}_0^{\rho }){(\rho -s)}^{\upsilon -1}\left\{\zeta (s,x^{ϵ}\left(s\right))-\zeta (s,x\left(s\right))\right\}∥ds\right\}}^2\hfill \\ & & +17Tr{\rm Y}\left[{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}\right]\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho }){(\rho -s)}^{\upsilon -1}\zeta (s,x\left(s\right))∥ds\right\}}^2\hfill \\ & & +17{\left[{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}\right]}^2\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho }){(\rho -s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}\left\{H(s,x^{ϵ}\left(s\right),\mathbb{P}\left(s\right))-H(s,x\left(s\right),\mathbb{P}\left(s\right))\right\}\aleph (ds,d\mathbb{P})∥\right\}}^2\hfill \\ & & +17{\left[{\displaystyle \frac{\wp \Re (1-\upsilon )}{\mu \left(\upsilon \right)\mathrm{\Gamma }\left(\upsilon \right)}}\right]}^2\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho }){(\rho -s)}^{\upsilon -1}{\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})∥\right\}}^2\hfill \\ & & +17{\left[{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}\right]}^2\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })Q_{\upsilon }(\rho -s)\left\{F^{ϵ}\left(s\right)-F\left(s\right)\right\}∥ds\right\}}^2\hfill \\ & & +17{\left[{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}\right]}^2\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })Q_{\upsilon }(\rho -s)F\left(s\right)∥ds\right\}}^2\hfill \\ & & +17Tr{\rm Y}\left[{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}\right]\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })Q_{\upsilon }(\rho -s)\left\{\zeta (s,x^{ϵ}\left(s\right))-\zeta (s,x\left(s\right))\right\}∥ds\right\}}^2\hfill \\ & & +17Tr{\rm Y}\left[{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}\right]\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })Q_{\upsilon }(\rho -s)\zeta (s,x\left(s\right))∥ds\right\}}^2\hfill \\ & & +17{\left[{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}\right]}^2\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })Q_{\upsilon }(\rho -s){\int }_{\mathrm{\Pi }}\left\{H(s,x^{ϵ}\left(s\right),\mathbb{P}\left(s\right))-H(s,x\left(s\right),\mathbb{P}\left(s\right))\right\}\aleph (ds,d\mathbb{P})∥\right\}}^2\hfill \\ & & +17{\left[{\displaystyle \frac{\upsilon {\Re }^2}{\mu \left(\upsilon \right)}}\right]}^2\mathbb{E}{\left\{{\int }_0^{\rho }∥{\mathrm{\Theta }}^{-1}ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })Q_{\upsilon }(\rho -s){\int }_{\mathrm{\Pi }}H(s,x\left(s\right),\mathbb{P}\left(s\right))\aleph (ds,d\mathbb{P})∥\right\}}^2\hfill \\ & & +17{\left[∥{\mathrm{\Theta }}^{-1}\Re \mathrm{\Theta }ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })∥\right]}^2\mathbb{E}{\left\{∥\sum _{\kappa =1}^{\gamma }{V}_{\upsilon }(\rho -t_{ϵ})\left\{{\mathcal{I}}_{\kappa }\left(x^{ϵ}\left(t_{\kappa }^{-}\right)\right)-{\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right)\right\}∥\right\}}^2\hfill \\ & & +17{\left[∥{\mathrm{\Theta }}^{-1}\Re \mathrm{\Theta }ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })∥\right]}^2\mathbb{E}{\left\{∥\sum _{\kappa =1}^{\gamma }{V}_{\upsilon }(\rho -t_{ϵ}){\mathcal{I}}_{\kappa }\left(x\left(t_{\kappa }^{-}\right)\right)∥\right\}}^2.\hfill \end{array}$$

Since $ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })\to 0$ is strong as $ϵ\to 0^{+}$, and as $\parallel ϵ\varrho (ϵ,{\mathrm{\Delta }}_0^{\rho })\parallel \le 1$, the Lebesgue dominated convergence theorem together with the compactness of $Q_{\upsilon }\left(t\right)$ implies that

$$\mathbb{E}{∥x^{ϵ}\left(\rho \right)-{\tilde{x}}_{\rho }∥}^2\to 0\mathrm{as}ϵ\to 0^{+}.$$

□

4. Illustrative Example

We assessed a category of impulsive stochastic partial differential inclusions of the Sobolev type that involved the ABC-FD, $\mathcal{P}js$, a control function, and the Clarke sub-differential.

$$\begin{array}{ccc}& & {}^{\mathcal{ABC}}D_{0+}^{3/5}\left(1-{\displaystyle \frac{{\partial }^2}{\partial {\mathfrak{E}}^2}}\right)\{x(t,\mathfrak{E})-y(t,x(t,\mathfrak{E}))\}\in {\displaystyle \frac{{\partial }^2}{\partial {\mathfrak{E}}^2}}\{x(t,\mathfrak{E})-y(t,x(t,\mathfrak{E}))\}+\tau (t,\mathfrak{E})\hfill \\ & & +\left[{\displaystyle \frac{sin\mathfrak{E}+\mid x(t,\mathfrak{E})\mid }{9}}\right]{\displaystyle \frac{dW\left(t\right)}{dt}}+\partial F\left({\displaystyle \frac{t^2-\mid x(t,\mathfrak{E})\mid }{9}}\right)\hfill \\ & & +{\int }_{\mathrm{\Pi }}\{{\displaystyle \frac{cos\mid \mathfrak{E}\mathbb{P}\mid }{5}}+{\displaystyle \frac{(t^2+1)\mid x(t,\mathfrak{E})\mid }{5+\mid x(t,\mathfrak{E})\mid }}\}\aleph (dt,d\mathbb{P}),t\in \mathcal{J}=(0,1],\mathfrak{E}\in [0,\pi ],t\ne t_{\kappa }\hfill \\ & & x(t,0)=x(t,\pi )=0,t\in (0,1],\hfill \\ & & x(t_{\kappa }^{+},\mathfrak{E})-x(t_{\kappa }^{-},\mathfrak{E})={\displaystyle \frac{1}{1+{\mathfrak{E}}^2}}tanh\left(x\left(\mathfrak{E}\right)\right),\forall \mathfrak{E}\in [0,\pi ],\hfill \\ & & x(0,\mathfrak{E})+\sum _{i=1}^n{\delta }_{i}x(t_i,\mathfrak{E})=x_{\mathfrak{E}},\mathfrak{E}\in [0,\pi ],\hfill \end{array}$$

(7)

where ${}^{\mathcal{ABC}}D_{0+}^{3/5}$ is the ABC-FD of order $\upsilon =3/5$, and $\mho U=\tau (\mathfrak{T},\mathfrak{E}),U\in \mathcal{U}$. In addition, $X=Y={\mathcal{L}}^2([0,\pi ],(0,1])$ is the separable Hilbert space. In the example above, the parameter varying in $[0,\pi ]$ represents the spatial variable on which the underlying partial differential operator acts. Thus, the system is posed on a spatial domain $[0,\pi ]$ with appropriate boundary conditions. Here, the impulsive mapping defined by

$${\mathcal{I}}_{\kappa }\left(x\right)\left(\mathfrak{E}\right)={\displaystyle \frac{1}{1+{\mathfrak{E}}^2}}tanh\left(x\left(\mathfrak{E}\right)\right),\mathfrak{E}\in [0,\pi ].$$

(8)

(i)

Complete continuity: The mapping $x\mapsto tanh\left(x\right)$ is Fréchet differentiable and bounded on $\mathbb{R}$. Since ${\displaystyle \frac{1}{1+{\mathfrak{E}}^2}}$ is smooth and decays in $\mathfrak{E}$, the operator ${\mathcal{I}}_{\kappa }$ maps bounded sets in X into equicontinuous and uniformly bounded sets. By the Arzelà–Ascoli theorem and compactness of the embedding into X, this implies that ${\mathcal{I}}_{\kappa }$ is compact (hence completely continuous). Furthermore, for all $x\in X$, we have

$$\begin{array}{cc}\hfill |{\mathcal{I}}_{\kappa }{\left(x\right)\left(\mathfrak{E}\right)|}^2& ={\left|{\displaystyle \frac{1}{1+{\mathfrak{E}}^2}}tanh\left(x\left(\mathfrak{E}\right)\right)\right|}^2\le {\displaystyle \frac{1}{{(1+{\mathfrak{E}}^2)}^2}}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \mathbb{E}\parallel {\mathcal{I}}_{\kappa }{\left(x\right)\parallel }^2& =\mathbb{E}{\int }_0^{\pi }{\left|{\displaystyle \frac{1}{1+{\mathfrak{E}}^2}}tanh\left(x\left(\mathfrak{E}\right)\right)\right|}^{2}d\mathfrak{E}\le {\int }_0^{\pi }{\displaystyle \frac{1}{{(1+{\mathfrak{E}}^2)}^2}}d\mathfrak{E}=:d_{\kappa }.\hfill \end{array}$$

Hence, $\mathbb{E}\parallel {\mathcal{I}}_{\kappa }{\left(x\right)\parallel }^2\le d_{\kappa }$, and the bound is independent of x. This confirms Condition (i) of $\left(A7\right)$.

(ii)

Lipschitz continuity in mean-square norm: Using the Lipschitz property of the hyperbolic tangent function, we estimate the following:

$$\begin{array}{cc}\hfill |{\mathcal{I}}_{\kappa }\left(x\right)\left(\mathfrak{E}\right)-{\mathcal{I}}_{\kappa }\left(\overline{x}\right)\left(\mathfrak{E}\right)|& =\left|{\displaystyle \frac{1}{1+{\mathfrak{E}}^2}}\left[tanh\left(x\left(\mathfrak{E}\right)\right)-tanh\left(\overline{x}\left(\mathfrak{E}\right)\right)\right]\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{1+{\mathfrak{E}}^2}}|x\left(\mathfrak{E}\right)-\overline{x}\left(\mathfrak{E}\right)|.\hfill \end{array}$$

Squaring and integrating is expressed as follows:

$$\begin{array}{cc}\hfill \mathbb{E}\parallel {\mathcal{I}}_{\kappa }\left(x\right)-{\mathcal{I}}_{\kappa }\left(\overline{x}\right){\parallel }^2& \le \mathbb{E}{\int }_0^{\pi }{\left({\displaystyle \frac{1}{1+{\mathfrak{E}}^2}}\right)}^2{|x\left(\mathfrak{E}\right)-\overline{x}\left(\mathfrak{E}\right)|}^{2}d\mathfrak{E}\hfill \\ \hfill & \le \left(\underset{\mathfrak{E}\in [0,\pi ]}{sup}{\displaystyle \frac{1}{{(1+{\mathfrak{E}}^2)}^2}}\right)\mathbb{E}\parallel x-\overline{x}{\parallel }^2={\displaystyle \frac{1}{{(1+{\pi }^2)}^2}}\mathbb{E}{\parallel x-\overline{x}\parallel }^2.\hfill \end{array}$$

Thus, Condition (ii) of $\left(A7\right)$ holds with ${𝚤}^{*}={\displaystyle \frac{1}{{(1+{\pi }^2)}^2}}$.

Hence, the operator ${\mathcal{I}}_{\kappa }$ given in (8) satisfies all parts of Assumption $\left(A7\right)$.

$Z:\mathbf{D}\left(Z\right)\to X$ and $\mathrm{\Theta }:\mathbf{D}\left(\mathrm{\Theta }\right)\to X$ are defined by $Z=\left({\displaystyle \frac{{\partial }^2}{\partial {\mathfrak{E}}^2}}\right)$, $\mathrm{\Theta }=\left(1-{\displaystyle \frac{{\partial }^2}{\partial {\mathfrak{E}}^2}}\right)$ with $\mathbf{D}\left(Z\right)=\mathbf{D}\left(\mathrm{\Theta }\right)=\left\{x\in X:x,{\displaystyle \frac{\partial x}{\partial \mathfrak{E}}},areabsolutelycontinuous,{\displaystyle \frac{{\partial }^{2}x}{\partial {\mathfrak{E}}^2}}\in X,x\left(0\right)=x\left(\pi \right)=0\right\}$. Thus,

$$\begin{array}{ccc}\hfill Zx& =& \sum _{n=1}^{\infty }{n}^2\langle x,x_n\rangle x_n,x\in \mathbf{D}\left(Z\right)\hfill \\ \hfill \mathrm{\Theta }x& =& \sum _{n=1}^{\infty }(1+n^2)\langle x,x_n\rangle x_n,x\in \mathbf{D}\left(\mathrm{\Theta }\right),\hfill \end{array}$$

where $x_n\left(s\right)=\sqrt{2/\pi }sinns,n=1,2,3,\dots$ is an orthogonal set of eigenvectors of Z. However, for $x\in X$, we have

$$\begin{array}{ccc}\hfill {\mathrm{\Theta }}^{-1}x& =& \sum _{n=1}^{\infty }(1/(1+n^2))\langle x,x_n\rangle x_n,\hfill \\ \hfill Z{\mathrm{\Theta }}^{-1}x& =& \sum _{n=1}^{\infty }(n^2/(1+n^2))\langle x,x_n\rangle x_n.\hfill \end{array}$$

Due to the fact that $Z{\mathrm{\Theta }}^{-1}$ is self-adjoint and generates analytic semigroup ${\left\{Q_{\upsilon }\left(t\right)\right\}}_{t\ge 0}$, we specified the following functions:

$$\begin{array}{ccc}\hfill \partial F(t,x(t\left)\right)& =& \partial F\left({\displaystyle \frac{t^2-\mid x(t,\mathfrak{E})\mid }{9}}\right)\hfill \\ \hfill \zeta (t,x(t\left)\right)& =& \left[{\displaystyle \frac{sin\mathfrak{E}+\mid x(t,\mathfrak{E})\mid }{9}}\right]\hfill \\ \hfill H(t,x(t),\mathbb{P}(t\left)\right)& =& \left\{{\displaystyle \frac{cos\mid \mathfrak{E}\mathbb{P}\mid }{5}}+{\displaystyle \frac{(t^2+1)\mid x(t,\mathfrak{E})\mid }{5+\mid x(t,\mathfrak{E})\mid }}\right\}.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill {∥\partial F(t,x(t\left)\right)∥}^2& \le & ∥{\displaystyle \frac{t^2-\mid x(t,\mathfrak{E})\mid }{9}}∥\le {\displaystyle \frac{1}{81}}\left\{1+{∥x∥}^2\right\}\hfill \\ \hfill {∥\zeta (t,x(t\left)\right)∥}^2& \le & {\displaystyle \frac{1}{81}}\left\{1+{∥x∥}^2\right\}\hfill \\ \hfill {\int }_{\mathrm{\Pi }}{∥\hslash (t,x,\mathbb{P})∥}^2\lambda d\mathbb{P}& \le & {\displaystyle \frac{2}{25}}\left\{1+4{∥x∥}^2\right\}.\hfill \end{array}$$

Consequently, System (7) may be adjusted as System (1) under the given conditions, meeting all of the presumptions of Theorems 2 and 3. Accordingly, (7) is approximately controllable on $(0,1]$.

5. Conclusions

In this work, we developed a comprehensive analytical framework for the approximate controllability of a new class of Sobolev-type stochastic impulsive differential inclusions. These systems are governed by the Atangana–Baleanu fractional derivative in the Caputo sense, incorporate stochastic perturbations modeled by Wiener processes, and include Poisson jump discontinuities. Clarke’s generalized sub-differential approach was employed to handle nonsmooth and multivalued nonlinearities, thereby extending classical controllability results to settings that involve memory effects, random fluctuations, and impulsive state transitions. The main theoretical results were established using variational techniques, measurable selection theorems, and multivalued fixed point arguments in Banach spaces. To demonstrate the practical relevance of our results, a carefully constructed example was presented and verified.

Several promising directions remain for future research. One natural extension would be to study systems driven by more general fractional kernels, such as Prabhakar operators or distributed-order formulations, which would capture a wider range of memory effects. Another important direction is to investigate exact controllability and optimal control problems under state-dependent impulsive regimes or systems influenced by more general Lévy noise. The inclusion of delays, non-instantaneous impulses, and time-varying stochastic coefficients would provide further insight into realistic applications. Finally, the design of efficient numerical schemes that respect the structural properties of such hybrid systems remains an open and significant challenge for future work.