Langevin neutral impulsive fractional stochastic system along fractional Brownian motion-A controllability analysis

Abstract

The purpose of this work is to investigate the controllability of Langevin-type stochastic neutral impulsive integro-differential equations governed by the Caputo fractional derivative and driven by fractional Brownian motion, which arise naturally in systems exhibiting memory, impulsive effects, and stochastic disturbances. Using resolvent operators and fixed-point techniques, necessary and sufficient controllability conditions are established for the associated linear system, while the controllability of the nonlinear system is demonstrated via the Banach contraction principle. The theoretical results confirm that appropriate control functions can steer the system to a desired state within a finite time interval. Finally, illustrative numerical examples are provided to demonstrate the applicability and effectiveness of the obtained results, highlighting their relevance to practical stochastic control problems.

1. Introduction

Fractional-order derivatives play a pivotal role in accurately representing intricate dynamics found in natural and engineered systems, where traditional integer-order derivatives often fall short. Fractional calculus has been extensively employed across multiple domains, offering refined analytical tools and enhanced modeling capabilities (see 1 and 2). On the other hand, stochastic differential equations (SDEs) bridge probability theory, stochastic analysis, and differential equations, effectively modeling systems influenced by random fluctuations or external noise (see 3–5). This work particularly investigates systems influenced by Fractional Brownian Motion (fBm) (see 6–8), a Gaussian process exhibiting self-similarity and long-range dependence, governed by the Hurst parameter H ∈ (0, 1). Initially introduced by Kolmogorov in 1940, fBm extends classical Brownian motion and has been the subject of extensive research in stochastic fractional differential equations (see 6, 9–12).

Fractional calculus provides a framework for constructing models that consider not only the current state but also historical data, making it highly relevant in physical phenomena with memory effects. The Caputo fractional derivative is adopted in this study since it permits the use of classical initial conditions and offers a physically meaningful and analytically convenient framework for the controllability analysis of fractional stochastic systems. Many real-world systems are subjected to sudden changes or impulses, necessitating the use of impulsive differential equations to describe their dynamics (see 13–16). Controllability, a fundamental concept in control theory, ensures the ability to steer such systems to a desired state, and it remains a focal point in control system design (see 5, 17–19).

In recent years, significant attention has been devoted to fractional stochastic Langevin neutral impulsive differential equations (see 20 and 21). Ahmed Salem et al. ²² (2021) explored the existence of solutions for a coupled Langevin fractional system of Caputo type with Riemann-Stieltjes boundary conditions. Harikrishnan et al. ²³ (2018) presented an analytical study on impulsive Langevin equations involving the Hilfer fractional derivative. Additionally, Rizwan Rizwan et al. ²⁴ (2022) analyzed nonlinear impulsive Langevin equations under Hilfer fractional derivatives from a qualitative perspective. However, to the best of our knowledge, there is no comprehensive work addressing the controllability and solution of Langevin neutral impulsive stochastic systems governed by the Caputo fractional derivative in the presence of fBm.^25,26

The notable contribution of this chapter as follows:

• The fixed point method is implemented to establish appropriate circumstances for the controllability of Langevin neutral impulsive stochastic system over the Caputo derivative and fBm.

• The conclusions of this study expand on various prior discoveries in the literature, with a focus on the fBm framework within fractional stochastic systems.

• The system terms are considered as a bounded linear operator rather than an An indefinite generators of a moderately everlasting semi-group, recognizing that Caputo derivatives lack the semigroup or commutative properties intrinsic to integer-order derivatives.

• To demonstrate the outcomes for stochastic systems, bounded linear operators prove to be more effective.

• The Gramian operator is employed to examine the ability of linear systems to be controlled.

• For nonlinear systems, controllability is examined by utilizing the Banach expansion.

• Numeric instances are provided to demonstrate the outcomes of the nonlinear and linear scenarios.

The study is structured thereby: Section 2 investigates the essential terminologies and obstacles. Section 3 addresses the actual presence of substances to the linear system. Section 4 establishes the essential requirements required for linear system control. Section 5 describes the key requirements required for the ability to manage nonlinear systems using the Banach fixed point method. Section 6 provides a pair computational instances to demonstrate the applicability and usefulness of the derived ability to control conclusions. The chapter finishes with observations and remarks in Section 7.

2. Preface

2.1. Fractional Brownian motion

A fBm along $\mathcal{H}\in (1/\mathrm{2,1})$ is characterized by the covariance operator:

$\mathbb{E}[{\mathcal{B}}_{\mathcal{H}}(s){\mathcal{B}}_{\mathcal{H}}(v)]=\frac{1}{2}\left(|s{|}^{2\mathcal{H}}+|v{|}^{2\mathcal{H}}-|s-v{|}^{2\mathcal{H}}\right),\forall s,v\ge 0.$

When $\mathcal{H}=1/2$ , ${\mathcal{B}}_{\mathcal{H}}$ coincides with standard Bm. The fBm has the integral form as:

${\mathcal{B}}_{\mathcal{H}}(v)={\int }_0^v{\mathcal{R}}^{\mathcal{H}}(v-s)d\mathrm{\Theta }(s),v\ge 0,$

Where {Θ(s)} represents a Bm, and the kernel ${\mathcal{R}}^{\mathcal{H}}(v,s)$ , as:

${\mathcal{R}}^{\mathcal{H}}(v,s)={\theta }_{\mathcal{H}}{s}^{\frac{1}{2}-\mathcal{H}}{\int }_s^v{(u-s)}^{\mathcal{H}-\frac{3}{2}}{u}^{\mathcal{H}-\frac{1}{2}}du,\text{for\hspace{0.17em}}v>s,$

Along ${\theta }_{\mathcal{H}}=\sqrt{\frac{\mathcal{H}(2\mathcal{H}-1)}{\gamma (2-2\mathcal{H},\mathcal{H}-\frac{1}{2})}}$ . Here, γ(⋅, ⋅) represents the Beta function and ${\mathcal{R}}^{\mathcal{H}}(v,s)=0$ when v ≤ s.

2.2. Caputo fractional derivative (CFD)

Definition 2.1

2 Let [p, q] denote a bounded interval on $\mathbb{R}$ . The fractional Riemann-Liouville integral of order η > 0, where m − 1 < η ≤ m, $m\in \mathbb{N}$ , as:

$I_{p+}^{\eta }h(x)=\frac{1}{\mathrm{\Gamma }(\eta )}{\int }_p^x{(x-u)}^{\eta -1}h(u)du$

Definition 2.2

2 The CFD of order η > 0, where m − 1 < η < m, as

$({}^{C}D_{0+}^{\eta }f)(p)=\frac{1}{\mathrm{\Gamma }(m-\eta )}{\int }_0^p{(p-u)}^{m-\eta -1}{f}^{(m)}(u)du$

Where the function f(p) possesses derivatives that are absolutely continuous up to order (m − 1). For 0 < η < 1,

$({}^{C}D_{0+}^{\eta }f)(p)=\frac{1}{\mathrm{\Gamma }(1-\eta )}{\int }_0^p\frac{f^{\prime }(u)}{{(p-u)}^{\eta }}du.$

Note: The Caputo derivative of fractional combines an integration of order (n − ξ) along standard n^th-order derivative.

${}^C\mathcal{D}^{\xi }p(t)={\mathcal{I}}^{n-\xi }{\mathcal{D}}^{n}p(t)$

Specifically, for 0 < ξ < 1

$\left({\mathcal{I}}_{0+}^{\xi }{}^C\mathcal{D}_{0+}^{\xi }\right)p(t)=p(t)-f(0).$

Definition 2.3

1 The Mittag-Leffler operator of a bounded linear operator $\mathcal{B}$ is represented as

${\mathcal{E}}_{\gamma ,\gamma }(\mathcal{B})=\sum _{n=0}^{\infty }\frac{{\mathcal{B}}^n}{\mathrm{\Gamma }(n\gamma +\gamma )}$

where γ > 0. In particular, when γ = 1, the expression simplifies to

${\mathcal{E}}_{\gamma ,1}(\mathcal{B})={\mathcal{E}}_{\gamma }(\mathcal{B})=\sum _{n=0}^{\infty }\frac{{\mathcal{B}}^n}{\mathrm{\Gamma }(n\gamma +1)}.$

Lemma 2.4

5 Let $\mathcal{B}$ be a Banach space, and let L be a bounded linear operator on $\mathcal{B}\ni$ : ‖L‖ < 1. Then, the operator (I − L)⁻¹ ∃, is bounded, and satisfies

${(\mathbf{I}-L)}^{-1}=\sum _{n=0}^{\infty }{L}^n,$

Where I represents the identity operator.

Lemma 2.5

17 : Banach Fixed Point Theorem

Assume (M, ρ) ≠ 0 is a complete metric space. Let 0 ≤ κ < 1 and suppose that G: M → M satisfies the condition:

$\rho (Gx,Gy)\le \kappa \rho (x,y),\forall x,y\in M.$

Then, G possesses a unique fixed point x*. Additionally, for any initial point x₀ ∈ M, the iterative sequence ${(G^j{x}_0)}_{j=1}^{\infty }$ converges to x*.

3. Linear systems and solution formulation

In this section, we use $\mathcal{X}$ and $\mathcal{V}$ to represent separable Hilbert spaces.

We analyze the Langevin neutral impulsive stochastic integro-differential equation governed along fBm, expressed as:

$\begin{array}{ccc}\hfill {}^C\mathcal{D}^{\gamma }({}^C\mathcal{D}^{\eta }-\mathcal{A})\xi (p)+\varphi (p,\xi (p))& =\hfill & \mathcal{B}(p)+{\int }_0^p\psi (p,s,\xi (s))ds+\mathcal{C}v(p)\hfill \\ \hfill & \hfill & +\mathrm{\Theta }(p)d{W}_H(p),\hfill \\ \hfill & \hfill & \text{for\hspace{0.17em}}p\in [0,\mathbb{P}]\backslash \{p_1,p_2,\dots ,p_m\},\hfill \\ \hfill \mathrm{\Delta }\xi (p)& =\hfill & \xi (p_k^{+})-\xi (p_k^{-})={\mathcal{I}}_k(\xi (p_k)),k=1,\dots ,m,\hfill \\ \hfill {}^C\mathcal{D}^{\eta }\xi (p){|}_{p=0}={\xi }_0,\\ \hfill \xi (0)& =\hfill & {\mathrm{\Upsilon }}_0.\hfill \end{array}$

(1)

Here:

• ${}^C\mathcal{D}^{\eta }$ and ${}^C\mathcal{D}^{\gamma }$ denote the Caputo derivatives with orders 0 < η, γ < 1,

• $\xi (\cdot )\in \mathcal{X}$ ,

• $\varphi :{\mathbb{R}}_{+}\times \mathcal{X}\to \mathcal{X}$ is a continuous mapping,

• $\mathcal{A}:\mathcal{X}\to \mathcal{X}$ is a bounded linear operator,

• v(⋅) is the control,

• $\mathcal{C}:\mathcal{V}\to \mathcal{X}$ is a bounded linear operator,

• $\mathcal{B}:{\mathbb{R}}_{+}\to \mathcal{X}$ is a continuous function,

• $\psi :{\mathbb{R}}_{+}\times \mathcal{X}\times \mathcal{X}\to \mathcal{X}$ is continuous,

• W _H (t) is a fBm,

• $\mathrm{\Theta }:{\mathbb{R}}_{+}\to {\mathcal{L}}_2^0$ ,

• ξ₀ and ϒ₀ are the initial states,

• ${\mathcal{I}}_k:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is continuous ∀ k = 1, 2, …, m, with $\xi (p_k^{+})={\lim }_{ϵ\to 0^{+}}\xi (p_k+ϵ)$ and $\xi (p_k^{-})={\lim }_{ϵ\to 0^{-}}\xi (p_k+ϵ)$ , representing the right and left limits of ξ(p) at p = p _k .

Note: ${}^{C}D^{\gamma }({}^{C}D^{\eta }-P)\xi (p)$ Langevin equation used for: In physics, the Langevin equation is a stochastic system that describes how a system changes when it is subjected to a combination of deterministic and fluctuating (or random) forces.

Key Assumption: $({\mathbb{H}}_1)$ The operator $\mathcal{A}$ interacts with the fractional incremental operator ${\mathcal{I}}^{\eta }$ on $\mathcal{X}$ , satisfying $\Vert \mathcal{A}{\Vert }^2\le (2\eta -1)\mathrm{\Gamma }{(\eta )}^2/{\mathbb{P}}^{2\eta }$ . Similarly, $\mathcal{A}$ commutes with ${\mathcal{I}}^{\eta +\gamma }$ , ensuring $\Vert \mathcal{A}{\Vert }^2\le (2\eta +2\gamma -1)\mathrm{\Gamma }{(\eta +\gamma )}^2/{\mathbb{P}}^{2(\eta +\gamma )}$ .

Solution Representation: For 0 < η < 1, the solution of (1) is expressed as:

$\begin{array}{ccc}\hfill \xi (p)& =\hfill & E_{\eta }(A{p}^{\eta })[{\xi }_0+{\mathrm{\Upsilon }}_0+\varphi (0,{\mathrm{\Upsilon }}_0)]+E_{\eta }(A{p}^{\eta })[{\mathrm{\Upsilon }}_0+\varphi (0,{\mathrm{\Upsilon }}_0)]\hfill \\ \hfill & \hfill & -\varphi (p,\xi (p))+{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\hfill \\ \hfill & \hfill & \times \left[A\varphi (s,\xi (s))+\mathcal{B}(s)+\mathcal{C}v(s)+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\mathrm{\Theta }(s)d{W}_H(s),\hfill \\ \hfill & \text{for\hspace{0.17em}}\hfill & p\in (0,p1]\hfill \\ \hfill \xi (p)& =\hfill & E_{\eta }(A{p}^{\eta })[{\xi }_0+{\mathrm{\Upsilon }}_0+\varphi (0,{\mathrm{\Upsilon }}_0)]+E_{\eta }(A{p}^{\eta })[{\mathrm{\Upsilon }}_0+\varphi (0,{\mathrm{\Upsilon }}_0)]\hfill \\ \hfill & \hfill & -\varphi (p,\xi (p))+{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\hfill \\ \hfill & \hfill & \times \left[A\varphi (s,\xi (s))+\mathcal{B}(s)+\mathcal{C}v(s)+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & +{\mathcal{I}}_1(\xi (p_1^{-}))+{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\hfill \end{array}$

$\begin{array}{ccc}\hfill & \times \hfill & \mathrm{\Theta }(s)d{W}_H(s),\text{for\hspace{0.17em}}p\in (p_1,p_2]\hfill \\ \hfill & .\hfill & \hfill \\ \hfill & .\hfill & \hfill \\ \hfill & .\hfill & \hfill \\ \hfill \xi (p)& =\hfill & E_{\eta }(A{p}^{\eta })[{\xi }_0+{\mathrm{\Upsilon }}_0+\varphi (0,{\mathrm{\Upsilon }}_0)]+E_{\eta }(A{p}^{\eta })[{\mathrm{\Upsilon }}_0+\varphi (0,{\mathrm{\Upsilon }}_0)]\hfill \\ \hfill & \hfill & -\varphi (p,\xi (p))+\sum _{i=1}^k{\mathcal{I}}_i(\xi (p_i^{-}))+{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\hfill \\ \hfill & \hfill & \times \left[A\varphi (s,\xi (s))+\mathcal{B}(s)+\mathcal{C}v(s)+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\mathrm{\Theta }(s)d{W}_H(s),\hfill \\ \hfill & \hfill & \text{for\hspace{0.17em}}p\in (p_k,p_{k+1}],k=\mathrm{1,2},\dots ,m\hfill \end{array}$

4. Linear System’s controllability

Definition 4.1

Define the operator $\mathrm{\Xi }:\mathcal{V}\to \mathcal{X}$ as

$\mathrm{\Xi }v={\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(\mathbb{P}-s)}^{\eta })\mathcal{C}v(s)ds$

Clearly, the adjoint operator Ξ* of Ξ, ${\mathrm{\Xi }}^{*}:\mathcal{X}\to \mathcal{V}$ as

$({\mathrm{\Xi }}^{*}z)(p)={(\mathbb{P}-p)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(P^{*}{(\mathbb{P}-p)}^{\eta }){\mathcal{C}}^{*}E\{p/{\mathcal{F}}_p\}.$

Definition 4.2

The controllability Grammian operator $\mathrm{\Theta }:\mathcal{X}\to \mathcal{X}$

$\begin{array}{ccc}\hfill Q(\xi )& =\hfill & {\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{2(\eta +\gamma )-2}{E}_{\eta ,\eta +\gamma }(A{(\mathbb{P}-s)}^{\eta })E_{\eta ,\eta +\gamma }(A^{*}{(\mathbb{P}-s)}^{\eta })\hfill \\ \hfill & \hfill & \mathcal{C}{\mathcal{C}}^{*}E\{p/{\mathcal{F}}_s\}ds.\hfill \end{array}$

Here * represents adjoint operator.

Lemma 4.3

The operator Q(ξ) = ΞΞ* is well defined and bounded for any η, γ ∈ (1/2, 1].

Proof

$\begin{array}{ccc}\hfill E\Vert \mathrm{\Xi }w{\Vert }^2& =\hfill & E\Vert {\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(\mathbb{P}-s)}^{\eta })\mathcal{C}v(s)ds{\Vert }^2\hfill \\ \hfill & =\hfill & E\Vert {\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta +\gamma -1}\sum _{n=0}^{\infty }\frac{A^k{(\mathbb{P}-s)}^{n\eta }}{\mathrm{\Gamma }(n\eta +\eta +\gamma )}\mathcal{C}v(s)ds{\Vert }^2\hfill \\ \hfill & =\hfill & \sum _{n=0}^{\infty }E\Vert {\int }_0^{\mathbb{P}}\frac{P^n{(\mathbb{P}-s)}^{n\eta +\eta +\gamma -1}}{\mathrm{\Gamma }(n\eta +\eta +\gamma )}\mathcal{C}v(s)ds{\Vert }^2\hfill \\ \hfill & \le \hfill & \sum _{n=0}^{\infty }{\int }_0^{\mathbb{P}}\frac{{(\mathbb{P}-s)}^{n\eta +\eta +\gamma -1}}{\mathrm{\Gamma }(n\eta +\eta +\gamma )}\frac{{(\mathrm{\Gamma }(\eta +\gamma ))}^{2n}(2n\eta +2n\gamma -1)}{{\mathbb{P}}^{2n(\eta +\gamma )}}\hfill \\ \hfill & \hfill & \underset{p\in J}{\sup }E\Vert \mathcal{C}{\Vert }^2\underset{p\in J}{\sup }E\Vert v(s){\Vert }^{2}ds\hfill \\ \hfill & \le \hfill & \Vert v{\Vert }^2{L}_1{\mathbb{P}}^{2\eta +2\gamma -1}\sum _{n=0}^{\infty }\frac{{(\mathrm{\Gamma }(\eta +\gamma ))}^{2n}}{(2n\eta +2n\gamma +2\eta +2\gamma -1){(2\eta +2\gamma -1)}^n}\hfill \\ \hfill & ≔\hfill & v_1\hfill \end{array}$

where $L_1={\sup }_{p\in J}\Vert \mathcal{C}{\Vert }^2$ , The last series converges for any η, γ ∈ (1/2, 1] Ξ and Ξ* are bounded for any p ∈ J.

Theorem 4.4

If the Grammian operator Q(ξ) is bounded, then the linear system (1) is completely controllable.

Proof

The reasoning against this Theorem is transparent.

For the system (1), the control as

$\begin{array}{ccc}\hfill v(p)& =\hfill & LE\left\{Q^{-1}\left({\xi }_1-E_{\eta }(A{p}^{\eta })[{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]\right.\right.\hfill \\ \hfill & \hfill & -E_{\eta }(A{\mathbb{P}}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]+f(p,\xi (p))\hfill \\ \hfill & \hfill & -{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(\mathbb{P}-s)}^{\eta })\hfill \\ \hfill & \hfill & \times \left[Af(s,\xi (s))+\mathcal{B}(s)+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & -{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(\mathbb{P}-s)}^{\eta })\mathrm{\Theta }(s)d{W}_H(s)\hfill \\ \hfill & \hfill & \left.\left.\times E\left\{p/{\mathcal{F}}_s\right\}\right)\right\},forp\in (0,p_1]\hfill \end{array}$

(2)

$\begin{array}{ccc}\hfill v(p)& =\hfill & LE\left\{Q^{-1}\left({\xi }_1-E_{\eta }(A{p}^{\eta })[{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]\right.\right.\hfill \\ \hfill & \hfill & -E_{\eta }(A{\mathbb{P}}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]+f(p,\xi (p))\hfill \\ \hfill & \hfill & -\sum _{i=1}^k{I}_i(\xi (p_i^{-}))-{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta +\gamma -1}\hfill \\ \hfill & \hfill & \times E_{\eta ,\eta +\gamma }(A{(\mathbb{P}-s)}^{\eta })[Af(s,\xi (s))+\mathcal{B}(s)\hfill \\ \hfill & \hfill & \left.+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & -{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(\mathbb{P}-s)}^{\eta })\hfill \\ \hfill & \hfill & \left.\left.\times \mathrm{\Theta }(s)d{W}_H(s)E\left\{p/{\mathcal{F}}_s\right\}\right)\right\},\hfill \\ \hfill & \hfill & forp\in (p_k,p_{k+1}],k=\mathrm{1,2,3},\dots ,m\hfill \end{array}$

(3)

Where $L={(\mathbb{P}-p)}^{\eta +\gamma -1}{T}^{*}{E}_{\eta ,\eta +\gamma }(A^{*}{(\mathbb{P}-p)}^{\eta })$ .

By substitute (3) in the following solution,

$\begin{array}{ccc}\hfill \xi (p)& =\hfill & E_{\eta }(A{p}^{\eta })[{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]\hfill \\ \hfill & \hfill & +E_{\eta }(A{p}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]\hfill \\ \hfill & \hfill & -f(p,\xi (p))+\sum _{i=1}^k{I}_i(\xi (p_i^{-}))\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\hfill \\ \hfill & \hfill & \times \left[Af(s,\xi (s))+\mathcal{B}(s)+\mathcal{C}v(s)\right.\hfill \\ \hfill & \hfill & \left.+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\hfill \\ \hfill & \hfill & \times \mathrm{\Theta }(s)d{W}_H(s),\hfill \\ \hfill & \hfill & forp\in (p_k,p_{k+1}],k=\mathrm{1,2},\dots ,m\hfill \end{array}$

And evaluating ξ(p) equation at $p=\mathbb{P}$ ,

We obtain ξ(p) = ξ₁.

Since ξ₁ is an arbitrary point in $\mathcal{C}$ ,

Hence v(p) directs the system to every points in $\mathcal{C}$ .

5. Controllability of non-linear system

In this section, we extend the controllability results of the linear fractional system to the corresponding nonlinear model. The controllability of the associated linear system serves as a fundamental prerequisite for the nonlinear analysis, as it guarantees the invertibility of the controllability operator and allows the explicit construction of an admissible control function. Building on this linear framework, the nonlinear system is treated as a perturbation of the controllable linear system. Under appropriate Lipschitz continuity and boundedness assumptions, the resulting nonlinear operator is shown to be a contraction on a suitable function space. Consequently, the Banach Fixed Point Theorem is applied to establish the controllability of the nonlinear fractional system.

Evaluate the non-linear equation as

$\begin{array}{ccc}\hfill {}^{C}D^{\gamma }({}^{C}D^{\eta }-A)\xi (p)+f(p,\xi (p))& =\hfill & \mathcal{B}(p,\xi (p))+{\int }_0^t\psi (p,s,\xi (s))ds+\mathcal{C}v(p)\hfill \\ \hfill & \hfill & +\mathrm{\Theta }(p,\xi (p))d{W}_H(t)\hfill \\ \hfill & \hfill & forp\in [0,\mathbb{P}]-\{p_1,p_2,\dots ,t_m\}\hfill \\ \hfill \delta \xi (p)& =\hfill & \xi (p_k^{+})-\xi (p_k^{-})=I_k(Z(p_k)),k=1,\dots ,m\hfill \\ \hfill {}^{C}D^{\eta }\xi (p)|p=0& =\hfill & {\xi }_0\hfill \\ \hfill \xi (0)& =\hfill & {\mathrm{\Upsilon }}_0\hfill \end{array}$

(4)

• • $\mathcal{B}:{\mathbb{R}}_{+}\times \mathcal{X}\to \mathcal{X}$ is continuous,

• • $\mathrm{\Theta }:{\mathbb{R}}_{+}\times \mathcal{X}\to L_2^0$ .

Let us assume that:

$({\mathcal{C}}_{\mathit{11}})$ : The function $\mathcal{X}\mapsto f$ is continuous on $\mathcal{X}$ for a.e. $p\in {\mathbb{R}}_{+}$ .

∃ γ _f > 0 ∋:, ‖f(p, y) − f(p, ξ)‖² < γ _f ‖y − ξ‖², for all $y,\xi \in \mathcal{X}$ .

$({\mathcal{C}}_{\mathit{12}})$ : The function $\mathcal{X}\mapsto Q(p,\xi (p))$ is continuous on $\mathcal{X}$ for a.e. $p\in {\mathbb{R}}_{+}$ .

∃ γ _q > 0 ∋: ‖Q(p, y) − Q(p, ξ)‖² < γ _q ‖y − ξ‖², for all $y,\xi \in \mathcal{X}$ .

$({\mathcal{C}}_{\mathit{13}})$ : The function $\mathcal{X}\mapsto \psi (p,s,\xi (p))$ is continuous on $\mathcal{X}$ for a.e. $p\in {\mathbb{R}}_{+}$ .

∃ γ _r > 0 ∋: ‖ψ(p, s, y) − ψ(p, s, ξ)‖² < γ _r ‖y − ξ‖², for all $y,\xi \in \mathcal{X}$ .

$({\mathcal{C}}_{\mathit{14}})$ : The function $\mathcal{X}\mapsto \mathrm{\Theta }(p,\xi (p))$ is continuous on $\mathcal{X}$ for a.e. $p\in {\mathbb{R}}_{+}$ .

∃ γ_Θ > 0 ∋: ‖Θ(p, y) − Θ(p, ξ)‖² < γ_Θ‖y − ξ‖², for all $y,\xi \in \mathcal{X}$ .

$({\mathcal{C}}_{\mathit{15}})$ : $\Vert y-\xi {\Vert }_{L_1}^2={\int }_0^{\mathbb{P}}\Vert y(p)-\xi (p){\Vert }^2\le K\Vert y-\xi {\Vert }^2$ .

$({\mathcal{C}}_{\mathit{16}})$ : The operator Q = ΞΞ* is well defined and bounded ∃ a constant v₁ > 0 ∋: E‖Ξw‖² ≤ v₁.

A function $\xi \in \mathcal{X}$ is said to be a mild solution of equation (4) if it satisfies the equation.

$\xi (p)=\left\{\begin{array}{c}{E}_{\eta }(A{p}^{\eta })[{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]+E_{\eta }(A{p}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]\hfill \\ -f(p,\xi (p))+{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\left[Af(s,\xi (s))\right.\hfill \\ \left.+\mathcal{B}(s,\xi (s))+\mathcal{C}v(s)+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds+{\int }_0^p{(p-s)}^{\eta +\gamma -1}\hfill \\ \times E_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\mathrm{\Theta }(s,\xi (s))d{W}_H(s),forp\in (0,p_1]\hfill \\ E_{\eta }(A{p}^{\eta })[{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]+E_{\eta }(A{p}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]\hfill \\ -f(p,\xi (p))+\sum _{i=1}^k{I}_i(\xi (p_i^{-}))+{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\hfill \\ \times \left[Af(s,\xi (s))+\mathcal{B}(s,\xi (s))+\mathcal{C}v(s)+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds+{\int }_0^p{(p-s)}^{\eta +\gamma -1}\hfill \\ \times E_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\mathrm{\Theta }(s,\xi (s))d{W}_H(s),forp\in (p_k,p_{k+1}],k=\mathrm{1,2},\dots ,m\hfill \end{array}\right.$

(5)

Theorem 5.1

Under assumptions $({\mathcal{C}}_{11})$ – $({\mathcal{C}}_{16})$ , if the linear fractional dynamical system (1) is controllable, then the nonlinear fractional dynamical system (4) is controllable, provided that

${\mathrm{\Theta }}_{{\gamma }_q,{\gamma }_{\mathrm{\Theta }},{\gamma }_r,\mathbb{P},\eta ,K}\Vert A{\Vert }_{L(Z)}^2+L_1{p}_1<1.$

• Proof: Step 1: Definition of the operator. Define the operator

$F:C_{1-\eta }({\mathbb{R}}_{+},\mathcal{X})\to C_{1-\eta }({\mathbb{R}}_{+},\mathcal{X})$

by

$\begin{array}{ll}\hfill (F\xi )(p)& =E_{\eta }(A{p}^{\eta })[{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]+E_{\eta }(A{p}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]-f(p,\xi (p))\hfill \\ \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })[Af(s,\xi (s))+\mathcal{B}(s,\xi (s))+\mathcal{C}v(s)\hfill \\ \hfill & +{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau ]ds\hfill \\ \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\mathrm{\Theta }(s,\xi (s))d{W}_H(s),\hfill \end{array}$

for p ∈ (p _k , p_k+1], k = 1, 2, …, m.

• Step 2: Construction of the admissible set. Let

$B_1=\left\{\xi \in C_{1-\eta }({\mathbb{R}}_{+},\mathcal{X}):\Vert \xi {\Vert }_{\eta }^2\le \epsilon \right\},$

where

$\epsilon \ge 2\left[\Vert {\mathrm{\Upsilon }}_0{\Vert }^2+\frac{M+M_1+M_2+N+K\Vert A{\Vert }_{L(Z)\psi (\mathbb{P})}^2}{\mathrm{\Gamma }(\eta +1)}\right].$

• Step 3: Choice of control. Since the linear system (1) is controllable, the operator Q is invertible. Define the control v(⋅) by ∀ p ∈ (0, p₁],

$\begin{array}{ccc}\hfill v(p)& =\hfill & {(\mathbb{P}-p)}^{\eta -1}{\mathcal{C}}^{*}{E}_{\eta ,\eta }(A^{*}{(\mathbb{P}-p)}^{\eta })E\{Q^{-1}({\xi }_1-E_{\eta }(A{\mathbb{P}}^{\eta })\hfill \\ \hfill & \hfill & \times [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]-E_{\eta }(A{\mathbb{P}}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]\hfill \\ \hfill & \hfill & +f(p,\xi (p))-{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}{E}_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\hfill \\ \hfill & \hfill & \times \left[Af(s,\xi (s))+\mathcal{B}(s,\xi (s))-{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & \left.\left.+{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}{E}_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\mathrm{\Theta }(s,\xi (s))d{W}_H(s)E\left\{p/{\mathcal{F}}_s\right\}\right)\right\}\hfill \end{array}$

For p ∈ (p _k , p_k+1], k = 1, 2, 3, …, m,

$\begin{array}{ccc}\hfill v(p)& =\hfill & {(\mathbb{P}-p)}^{\eta -1}{T}^{*}{E}_{\eta ,\eta }(A^{*}{(\mathbb{P}-p)}^{\eta })E\left\{Q^{-1}\left({\xi }_1-E_{\eta }(A{\mathbb{P}}^{\eta })\right.\right.\hfill \\ \hfill & \hfill & \times [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]-E_{\eta }(A{\mathbb{P}}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]\hfill \\ \hfill & \hfill & -\sum _{i=1}^k{I}_i(\xi (p_i^{-}))+f(p,\xi (p))-{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}\hfill \\ \hfill & \hfill & \times E_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\left[Af(s,\xi (s))+\mathcal{B}(s,\xi (s))\right.\hfill \\ \hfill & \hfill & \left.-{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds+{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}\hfill \\ \hfill & \hfill & \times E_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\mathrm{\Theta }(s,\xi (s))d{W}_H(s)\hfill \\ \hfill & \hfill & +{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}{E}_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\left[Af(s,\xi (s))\right.\hfill \\ \hfill & \hfill & \left.+\mathcal{B}(s,\xi (s))+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & +{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}{E}_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\hfill \\ \hfill & \hfill & \left.\left.\times \mathrm{\Theta }(s,\xi (s))d{W}_H(s)E\left\{p/{\mathcal{F}}_s\right\}\right)\right\}\hfill \end{array}$

(6)

and similarly for p ∈ (p _k , p_k+1], k = 1, 2, …, m, as given in the statement.

• Step 4: Invariance of B ₁ . la Let ξ ∈ B₁. Using the assumptions $({\mathcal{C}}_{11})$ – $({\mathcal{C}}_{16})$ , properties of Mittag–Leffler functions, and stochastic integral estimates,

we obtain

$\underset{p\in J}{\sup }E\Vert (F\xi )(p){\Vert }^2\le r,$

where r > 0 is independent of ξ. Hence,

$F(B_1)\subset B_1.$

• Step 5: Contraction property of F . For $\xi ,y\in C_{1-\eta }({\mathbb{R}}_{+},\mathcal{X})$ and p ∈ (p _k , p_k+1], k = 1, 2, …, m, we have

$\Vert (F\xi )(p)-(Fy)(p){\Vert }^2\le \left[{\mathrm{\Theta }}_{{\gamma }_q,{\gamma }_{\mathrm{\Theta }},{\gamma }_r,\mathbb{P},\eta ,K}\Vert A{\Vert }_{L(Z)}^2+L_1{p}_1\right]\Vert \xi -y{\Vert }_{\eta }^2.$

• Step 6: Conclusion. Since

${\mathrm{\Theta }}_{{\gamma }_q,{\gamma }_{\mathrm{\Theta }},{\gamma }_r,\mathbb{P},\eta ,K}\Vert A{\Vert }_{L(Z)}^2+L_1{p}_1<1,$

the operator F is a contraction on B₁. By the Banach contraction principle, F admits a unique fixed point, which corresponds to a mild solution steering the system from ξ₀ to ξ₁. Therefore, the nonlinear system (4) is controllable.

□ The operator $F:C_{1-\eta }({\mathbb{R}}_{+},\mathcal{X})\mapsto C_{1-\eta }({\mathbb{R}}_{+},\mathcal{X})$ , ∀ p ∈ (0, s₀] as

$(F\xi )(p)=E_{\eta }(A{p}^{\eta })[{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]+E_{\eta }(A{p}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]-f(p,\xi (p))$

$\begin{array}{ccc}\hfill & +\hfill & {\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\left[Af(s,\xi (s))+\mathcal{B}(s,\xi (s))\right.\hfill \\ \hfill & \hfill & \left.+\mathcal{C}v(s)+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds+{\int }_0^p{(p-s)}^{\eta +\gamma -1}\hfill \\ \hfill & \hfill & \times E_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\mathrm{\Theta }(s,\xi (s))d{W}_H(s)\hfill \end{array}$

for each for p ∈ (p _k , p_k+1], k = 1, 2, …, m, we obtain:

$\begin{array}{ccc}\hfill (F\xi )(p)& =\hfill & E_{\eta }(A{p}^{\eta })[{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]+E_{\eta }(A{p}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]-f(p,\xi (p))\hfill \\ \hfill & \hfill & +\sum _{i=1}^k{I}_i(\xi (p_i^{-}))+{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\left[Af(s,\xi (s))\right.\hfill \\ \hfill & \hfill & \left.+\mathcal{B}(s,\xi (s))+\mathcal{C}v(s)+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}{E}_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta })\mathrm{\Theta }(s,\xi (s))d{W}_H(s)\hfill \end{array}$

Let $B_1=\{z\in C_{1-\eta }({\mathbb{R}}_{+},\mathcal{X}),\Vert \xi {\Vert }_{\eta }^2\le ϵ\}$ ,

Where $ϵ\ge 2\left[\Vert {\mathrm{\Upsilon }}_0{\Vert }^2+(M+M_1+M_2+N+K\Vert A{\Vert }_{L(Z)\psi (\mathbb{P})}^2)/\mathrm{\Gamma }(\eta +1)\right]$ .

Since the linear system (1) corresponding to the nonlinear system (4) is controllable. We have, Q is invertible and the control as, ∀ p ∈ (0, p₁],

$\begin{array}{ccc}\hfill v(p)& =\hfill & {(\mathbb{P}-p)}^{\eta -1}{\mathcal{C}}^{*}{E}_{\eta ,\eta }(A^{*}{(\mathbb{P}-p)}^{\eta })E\left\{Q^{-1}\left({\xi }_1-E_{\eta }(A{\mathbb{P}}^{\eta })\right.\right.\hfill \\ \hfill & \hfill & \times [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]-E_{\eta }(A{\mathbb{P}}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]\hfill \\ \hfill & \hfill & +f(p,\xi (p))-{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}{E}_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\hfill \\ \hfill & \hfill & \times \left[Af(s,\xi (s))+\mathcal{B}(s,\xi (s))-{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & \left.\left.+{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}{E}_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\mathrm{\Theta }(s,\xi (s))d{W}_H(s)E\left\{p/{\mathcal{F}}_s\right\}\right)\right\}\hfill \end{array}$

For p ∈ (p _k , p_k+1], k = 1, 2, 3, …, m,

$\begin{array}{ccc}\hfill v(p)& =\hfill & {(\mathbb{P}-p)}^{\eta -1}{T}^{*}{E}_{\eta ,\eta }(A^{*}{(\mathbb{P}-p)}^{\eta })E\left\{Q^{-1}\left({\xi }_1-E_{\eta }(A{\mathbb{P}}^{\eta })\right.\right.\hfill \\ \hfill & \hfill & \times [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]-E_{\eta }(A{\mathbb{P}}^{\eta })[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]\hfill \\ \hfill & \hfill & -\sum _{i=1}^k{I}_i(\xi (p_i^{-}))+f(p,\xi (p))-{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}\hfill \\ \hfill & \hfill & \times E_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\left[Af(s,\xi (s))+\mathcal{B}(s,\xi (s))\right.\hfill \\ \hfill & \hfill & \left.-{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds+{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}\hfill \\ \hfill & \hfill & \times E_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\mathrm{\Theta }(s,\xi (s))d{W}_H(s)\hfill \end{array}$

$\begin{array}{ccc}\hfill & +\hfill & {\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}{E}_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\left[Af(s,\xi (s))+\mathcal{B}(s,\xi (s))\right.\hfill \\ \hfill & \hfill & \left.+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds+{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\eta -1}\hfill \\ \hfill & \hfill & \times E_{\eta ,\eta }(A{(\mathbb{P}-s)}^{\eta })\mathrm{\Theta }(s,\xi (s))d{W}_H(s)E\left.\left.\{p/{\mathcal{F}}_s\}\right)\right\}\hfill \end{array}$

(7)

Then, we first need to show that F(B₁) ⊂ B₁.

Let ξ ∈ B₁ and set $M={\sup }_{p\in [0,{\mathbb{R}}_{+}]}\Vert f(p,0){\Vert }^2$ ,

$M_1={\sup }_{p\in [0,{\mathbb{R}}_{+}]}\Vert \mathcal{B}(t,0){\Vert }^2$ ,

$M_2={\sup }_{p\in [0,{\mathbb{R}}_{+}]}\Vert \psi (p,s,0){\Vert }^2$ ,

$K_1={\int }_0^{{\mathbb{R}}_{+}}|\xi (p)|dp$ , $K_2={\int }_0^{{\mathbb{R}}_{+}}|v(p)|dp$ ,

$N={\sup }_{p\in [0,{\mathbb{R}}_{+}]}\Vert \mathrm{\Theta }(p,0){\Vert }^2$ ,

$G_1={\sup }_{0\le p\le \mathbb{P}}\Vert E_{\eta }(A{t}^{\eta }){\Vert }^2$ ,

$G_2={\sup }_{0\le p\le \mathbb{P}}\Vert E_{\eta ,\eta +\gamma }(A{t}^{\eta }){\Vert }^2$ , for each p ∈ (0, p₁]

$\begin{array}{ccc}\hfill E\Vert (F\xi )(p){\Vert }^2& \le \hfill & \Vert E_{\eta }(A{p}^{\eta }){\Vert }^{2}E\Vert [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2+\Vert E_{\eta }(A{p}^{\eta }){\Vert }^2\hfill \\ \hfill & \hfill & \times E\Vert [{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2-\Vert f(p,\xi (p)){\Vert }^2\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}\Vert E_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta }){\Vert }^2\left[\Vert A{\Vert }^2\right.\hfill \\ \hfill & \hfill & \times \Vert f(s,\xi (s)){\Vert }^2+\Vert \mathcal{B}(s,\xi (s)){\Vert }^2+\Vert \mathcal{C}{\Vert }^2\Vert v(s){\Vert }^2\hfill \\ \hfill & \hfill & \left.+{\int }_0^s\Vert \psi (s,\tau ,\xi (\tau )){\Vert }^{2}d\tau \right]ds+{\int }_0^p{(p-s)}^{\eta +\gamma -1}\hfill \\ \hfill & \hfill & \times \Vert E_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta }){\Vert }^2\Vert \mathrm{\Theta }(s,\xi (s)){\Vert }^{2}d{W}_H(s)\hfill \\ \hfill & \le \hfill & \Vert E_{\eta }(A{p}^{\eta }){\Vert }^{2}E\Vert [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2+\Vert E_{\eta }(A{p}^{\eta }){\Vert }^2\hfill \\ \hfill & \hfill & \times E\Vert [{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2-\Vert f(p,\xi (p)){\Vert }^2+{\int }_0^p{(p-s)}^{\eta +\gamma -1}\hfill \\ \hfill & \hfill & \times \Vert E_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta }){\Vert }^2\left[\Vert A{\Vert }^2\Vert f(s,\xi (s)){\Vert }^2\right.\hfill \\ \hfill & \hfill & +\Vert \mathcal{B}(s,\xi (s))-\mathcal{B}(s,0){\Vert }^2+\Vert \mathcal{B}(s,0){\Vert }^2+\Vert \mathcal{C}{\Vert }^2\Vert v(s){\Vert }^2\hfill \\ \hfill & \hfill & \left.+{\int }_0^s\Vert \psi (s,\tau ,z(\tau ))-\psi (s,\mathrm{0,0}){\Vert }^2+\Vert \psi (s,\mathrm{0,0}){\Vert }^{2}d\tau \right]ds\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}\Vert E_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta }){\Vert }^2\Vert \mathrm{\Theta }(s,\xi (s))\hfill \\ \hfill & \hfill & -\mathrm{\Theta }(s,0){\Vert }^2+\Vert \mathrm{\Theta }(s,0){\Vert }^{2}d{W}_H(s)\hfill \end{array}$

$\begin{array}{ccc}\hfill & \le \hfill & G_{1}E\Vert [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2+G_{1}E\Vert [{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2-M\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}{G}_2\left[\Vert A{\Vert }^{2}M+\Vert \mathcal{B}(s,\xi (s))-Q(s,0){\Vert }^2+M_1\right.\hfill \\ \hfill & \hfill & \left.+\Vert \mathcal{C}{\Vert }^2{K}_2+{\int }_0^s\Vert r(s,\tau ,z(\tau ))-r(s,\mathrm{0,0}){\Vert }^{2}d\tau +M_2\right]ds\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}{G}_2\Vert \mathrm{\Theta }(s,\xi (s))-\mathrm{\Theta }(s,0){\Vert }^{2}d{W}_H(s)+N\hfill \\ \hfill & \le \hfill & 4G_{1}E\Vert [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2+G_{1}E\Vert [{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2\hfill \\ \hfill & \hfill & +4G_2{L}_1\frac{\Vert w{\Vert }^2{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}+4G_{2}X\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\hfill \\ \hfill & \hfill & +4G_2{X}^1\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\hfill \\ \hfill & \le \hfill & r\hfill \end{array}$

Where

$\begin{array}{ccc}\hfill X& =\hfill & \left({\gamma }_f+{\gamma }_f{G}_2{K}_1+M_1+{\gamma }_f+{\gamma }_q+{\gamma }_q{G}_2{K}_1+M_1+{\gamma }_q\right.\hfill \\ \hfill & \hfill & \left.+{\gamma }_r+{\gamma }_r{G}_2{K}_1+M_2+{\gamma }_r\right)<\infty \hfill \\ \hfill X^1& =\hfill & ({\gamma }_{\mathrm{\Theta }}+{\gamma }_{\mathrm{\Theta }}{G}_2{K}_1+N+{\gamma }_{\mathrm{\Theta }})<\infty \hfill \end{array}$

Let us consider

$\underset{p\in J}{\sup }E\Vert v(p){\Vert }^2=\left\{\begin{array}{c}\Vert {\mathrm{\Xi }}^{*}{\Vert }^2\Vert Q^{-1}{\Vert }^2\left[E\Vert {\xi }_1{\Vert }^2+E\Vert [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2\right.\hfill \\ +E\Vert [{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2+4G_{2}X\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\hfill \\ +4G_2{X}^1\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1},\hfill \\ \left.forp\in \left(0,s_0\right.\right]\hfill \\ \Vert {\mathrm{\Xi }}^{*}{\Vert }^2\Vert Q^{-1}{\Vert }^2\left[E\Vert {\xi }_1{\Vert }^2+E\Vert [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2\right.\hfill \\ +E\Vert [{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2+\Vert \sum _{i=1}^k{I}_i(\xi (p_i^{-})){\Vert }^2\hfill \\ +4G_{2}X\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}+4G_2{X}^1\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1},\hfill \\ \left.forp\in \left(p_k,s_k\right.\right],k=\mathrm{1,2,3},\dots ,m\hfill \end{array}\right.$

Where sup_p∈JE‖v(p)‖² ≔ p₁ ≤ r. Now take $\xi ,y\in C_{1-\eta }({\mathbb{R}}_{+},\mathcal{X})$ . For each for p ∈ (p _k , p_k+1], k=1,2,…,m, we obtain:

$\begin{array}{ccc}\hfill E\Vert (F\xi )(p){\Vert }^2& =\hfill & \Vert E_{\eta }(A{p}^{\eta }){\Vert }^{2}E\Vert [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2+\Vert E_{\eta }(A{p}^{\eta }){\Vert }^2\hfill \\ \hfill & \hfill & \times E\Vert [{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2-\Vert f(p,\xi (p)){\Vert }^2+\Vert \sum _{i=1}^k{I}_i(\xi (p_i^{-})){\Vert }^2\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}\Vert E_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta }){\Vert }^2\left[\Vert A{\Vert }^2\Vert f(s,\xi (s)){\Vert }^2\right.\hfill \\ \hfill & \hfill & \left.+\Vert \mathcal{B}(s,\xi (s)){\Vert }^2+\Vert \mathcal{C}{\Vert }^2\Vert v(s){\Vert }^2+{\int }_0^s\Vert \psi (s,\tau ,\xi (\tau )){\Vert }^{2}d\tau \right]ds\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}\Vert E_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta }){\Vert }^2\Vert \mathrm{\Theta }(s,\xi (s)){\Vert }^{2}d{W}_H(s)\hfill \\ \hfill & \le \hfill & G_{1}E\Vert [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2+G_{1}E\Vert [{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2\hfill \\ \hfill & \hfill & +\Vert \sum _{i=1}^k{I}_i(\xi (p_i^{-})){\Vert }^2-M+{\int }_0^p{(p-s)}^{\eta +\gamma -1}{G}_2\left[\Vert A{\Vert }^{2}M\right.\hfill \\ \hfill & \hfill & +\Vert \mathcal{B}(s,\xi (s))-\mathcal{B}(s,0){\Vert }^2+M_1+\Vert \mathcal{C}{\Vert }^2{K}_2\hfill \\ \hfill & \hfill & \left.+{\int }_0^s\Vert \psi (s,\tau ,z(\tau ))-\psi (s,\mathrm{0,0}){\Vert }^{2}d\tau +M_2\right]ds+N\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}{G}_2\Vert \mathrm{\Theta }(s,\xi (s))-\mathrm{\Theta }(s,0){\Vert }^{2}d{W}_H(s)\hfill \\ \hfill & \le \hfill & 4G_{1}E\Vert [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2+\Vert \sum _{i=1}^k{I}_i(\xi (p_i^{-})){\Vert }^2\hfill \\ \hfill & \hfill & +G_{1}E\Vert [{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]{\Vert }^2\hfill \\ \hfill & \hfill & +4G_2{L}_1\frac{\Vert u{\Vert }^2{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}+4G_{2}X\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\hfill \end{array}$

$\begin{array}{ccc}\hfill & +\hfill & 4G_2{X}^1\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\hfill \\ \hfill & \le \hfill & \Vert \sum _{i=1}^k{I}_i(\xi (p_i^{-})){\Vert }^2\hfill \\ \hfill & \le \hfill & r\hfill \end{array}$

Which proves that F(B₁) ⊂ B₁.

Then for each p ∈ [0, p₁], we obtain:

$\begin{array}{ccc}\hfill \Vert (F\xi )(p)-(Fy)(p){\Vert }^2& =\hfill & {\int }_0^p{(p-s)}^{\eta +\gamma -1}\Vert E_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta }){\Vert }^2\left[\Vert A{\Vert }^2\right.\hfill \\ \hfill & \hfill & \times \Vert f(s,\xi (s))-f(s,y(s)){\Vert }^{2}ds+\Vert \mathcal{B}(s,\xi (s))\hfill \\ \hfill & \hfill & -\mathcal{B}(s,y(s)){\Vert }^2+\Vert \mathcal{C}{\Vert }^2\Vert v(s){\Vert }^2\hfill \\ \hfill & \hfill & \left.+{\int }_0^s\Vert \psi (s,\tau ,\xi (\tau ))-\psi (s,\tau ,y(\tau )){\Vert }^{2}d\tau \right]ds\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\eta +\gamma -1}\Vert E_{\eta ,\eta +\gamma }(A{(p-s)}^{\eta }){\Vert }^2\hfill \\ \hfill & \hfill & \times \Vert \mathrm{\Theta }(s,\xi (s))-\mathrm{\Theta }(s,y(s)){\Vert }^{2}d{W}_H(s)\hfill \\ \hfill & \le \hfill & 4G_2{L}_1\frac{p_1{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}+\left[4G_{2}X\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\right.\hfill \\ \hfill & \hfill & \left.+4G_2{X}^1\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\right]\Vert \xi -s{\Vert }_{\eta }^2\hfill \\ \hfill & \le \hfill & [{\mathrm{\Theta }}_{{\gamma }_q,{\gamma }_{\mathrm{\Theta }},{\gamma }_r,\mathbb{P},\eta ,K}{‖A‖}_{L(Z)}^2+L_1{p}_1]{‖\xi -s‖}_{\eta }^2\hfill \end{array}$

where,

$\begin{array}{ccc}\hfill {\mathrm{\Theta }}_{{\gamma }_q,{\gamma }_{\mathrm{\Theta }},{\gamma }_r,\mathbb{P},\eta ,K}\Vert A{\Vert }_{L(Z)}^2+L_1{p}_1& ≔\hfill & 4G_2{L}_1\frac{p_1{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\hfill \\ \hfill & \hfill & +\left[4G_{2}X\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\right.\hfill \\ \hfill & \hfill & \left.+4G_2{X}^1\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\right]\hfill \end{array}$

Now take $\xi ,y\in C_{1-\eta }({\mathbb{R}}_{+},Z).Thenforeachp\in [p_k,p_{k+1}],k=\mathrm{1,2},\dots ,m$ ,

$\begin{array}{ccc}\hfill \Vert (F\xi )(p)-(Fy)(p){\Vert }^2& \le \hfill & \Vert \sum _{i=1}^k{I}_i(\xi (p_i^{-})){\Vert }^2+4G_2{L}_1\frac{\Vert w{\Vert }^2{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\hfill \\ \hfill & \hfill & +\left[4G_{2}X\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\right.\hfill \\ \hfill & \hfill & \left.+4G_2{X}^1\frac{{\mathbb{P}}^{2(\eta +\gamma -1)}}{2(\eta +\gamma -1)-1}\right]\Vert \xi -s{\Vert }_{\eta }^2\hfill \\ \hfill & \le \hfill & \Vert \sum _{i=1}^k{I}_i(\xi (p_i^{-})){\Vert }^2+\left[{\mathrm{\Theta }}_{{\gamma }_q,{\gamma }_{\mathrm{\Theta }},{\gamma }_r,\mathbb{P},\eta ,K}\Vert A{\Vert }_{L(Z)}^2\right.\hfill \\ \hfill & \hfill & \left.+L_1{p}_1\right]\Vert \xi -s{\Vert }_{\eta }^2\hfill \\ \hfill & \le \hfill & [{\mathrm{\Theta }}_{{\gamma }_q,{\gamma }_{\mathrm{\Theta }},{\gamma }_r,\mathbb{P},\eta ,K}\Vert A{\Vert }_{L(Z)}^2+L_1{p}_1]\Vert \xi -s{\Vert }_{\eta }^2\hfill \end{array}$

where ${\mathrm{\Theta }}_{{\gamma }_q,{\gamma }_{\mathrm{\Theta }},{\gamma }_r,\mathbb{P},\eta ,K}\Vert A{\Vert }_{L(Z)}^2+L_1{p}_1$ , a quantity determined solely by the parameters of the problem.

Given that ${\mathrm{\Theta }}_{{\gamma }_q,{\gamma }_{\mathrm{\Theta }},{\gamma }_r,\mathbb{P},\eta ,K}\Vert A{\Vert }_{L(Z)}^2+L_1{p}_1<1$ , the operator F becomes a expansion mapping. Hence the system (4) is controllable. This completes the proof (Figure 1).

Figure 1.

Fractional linear system (8) with η = 1/2, γ = 2/3 with control and without impulsive effect.

6. Examples

Example 6.1

Consider the Langevin neutral impulsive system as

$\begin{array}{ccc}\hfill {}^{C}D^{\gamma }({}^{C}D^{\eta }-A)\xi (p)+f(p,\xi (p))& =\hfill & \mathcal{B}(p)+{\int }_0^6\frac{{(t-s)}^{0.5-1}}{\mathrm{\Gamma }(0.5)}ds+\mathcal{C}v(p)\hfill \\ \hfill & \hfill & +\mathrm{\Theta }(p)d{W}_H(p),\hfill \\ \hfill & \hfill & forp\in [\mathrm{0,6}]-\{1,\dots ,5\}\hfill \\ \hfill \delta \xi (p_i)& =\hfill & \frac{1.5+\xi (p_i^{-})}{4},p_i=i,i=\mathrm{1,2,3}\hfill \\ \hfill {}^{C}D^{\eta }\xi (p)|p=0& =\hfill & {\xi }_0\hfill \\ \hfill z(0)& =\hfill & {\mathrm{\Upsilon }}_0\hfill \end{array}$

(8)

Where η = 1/2; γ = 2/3, $\xi (p)=\left(\begin{array}{c}\hfill {\xi }_1(p)\hfill \\ \hfill {\xi }_2(p)\hfill \end{array}\right)$ ,

• $A=\left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$ ; $\mathcal{C}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}\right)$ ; $\mathrm{\Theta }(t)=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right)$ ; $\mathcal{B}(p)=\left(\begin{array}{c}\hfill e^{-p}\hfill \\ \hfill (e^p+1)\hfill \end{array}\right)$

• ${\xi }_0=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\right)$ and ${\mathrm{\Upsilon }}_0=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right)$

The solution is

$\begin{array}{ccc}\hfill \xi (p)& =\hfill & E_{\frac{1}{2}}\left(A{p}^{\frac{1}{2}}\right)[{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]+E_{\frac{1}{2}}\left(A{p}^{\frac{1}{2}}\right)[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]-f(p,\xi (p))\hfill \\ \hfill & \hfill & +\sum _{i=1}^k{I}_i(\xi (p_i^{-}))+{\int }_0^p{(p-s)}^{\frac{1}{2}+\frac{2}{3}-1}{E}_{\frac{1}{2},\frac{1}{2}+\frac{2}{3}}\left(A{(p-s)}^{\frac{1}{2}}\right)\hfill \\ \hfill & \hfill & \times \left[Af(s,\xi (s))+\mathcal{B}(s)+\mathcal{C}v(s)+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{\frac{1}{2}+\frac{2}{3}-1}{E}_{\frac{1}{2},\frac{1}{2}+\frac{2}{3}}\left(A{(p-s)}^{\frac{1}{2}}\right)\mathrm{\Theta }(s)d{W}_H(s)\hfill \end{array}$

Figures 2 and 3 depict the evolution of the state variable z(t) under fractional dynamics, stochastic perturbations, and impulsive effects. In the absence of impulses, the system evolves smoothly due to the memory property of the Caputo fractional derivative and the influence of fractional Brownian motion.

Figure 2.

Fractional linear system (8) with η = 1/2, γ = 2/3.

Figure 3.

Fractional linear system (8) with η = 1/2, γ = 2/3 with impulsive effect.

The impulsive mechanism introduces abrupt changes in the state at prescribed instants, clearly visible as jumps in the trajectories. These impulses significantly modify the system’s evolution by correcting deviations caused by stochastic disturbances. As a result, the trajectories exhibit improved stabilization after each impulse, demonstrating that impulsive control enhances controllability.

Overall, the numerical results validate the theoretical analysis and highlight the effectiveness of impulsive actions in controlling fractional stochastic systems with memory effects.

We have the control of the system (8) is given by

$\begin{array}{ccc}\hfill v(p)& =\hfill & {(\mathbb{P}-p)}^{\frac{1}{2}+\frac{2}{3}-1}{\mathcal{C}}^{*}{E}_{\frac{1}{2},\frac{1}{2}+\frac{2}{3}}\left(A^{*}{(\mathbb{P}-p)}^{\frac{1}{2}}\right)E\left\{Q^{-1}\left({\xi }_1-E_{\frac{1}{2}}\right.\left(A{t}^{\frac{1}{2}}\right)\right.\hfill \\ \hfill & \hfill & \times [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]-E_{\frac{1}{2}}\left(A{\mathbb{P}}^{\frac{1}{2}}\right)[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]+f(p,\xi (p))\hfill \end{array}$

$\begin{array}{ccc}\hfill & -\hfill & \sum _{i=1}^k{I}_i(\xi (p_i^{-}))-{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\frac{1}{2}+\frac{2}{3}-1}{E}_{\frac{1}{2},\frac{1}{2}+\frac{2}{3}}\left(A{(\mathbb{P}-s)}^{\frac{1}{2}}\right)\hfill \\ \hfill & \hfill & \times \left[Af(s,\xi (s))+\mathcal{B}(s)+\mathcal{C}v(s)+{\int }_0^s\psi (s,\tau ,z(\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & \left.\left.-{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{\frac{1}{2}+\frac{2}{3}-1}{E}_{\frac{1}{2},\frac{1}{2}+\frac{2}{3}}\left(A{(\mathbb{P}-s)}^{\frac{1}{2}}\right)\mathrm{\Theta }(s)d{W}_H(s)E\{p/{\mathcal{F}}_s\}\right)\right\}\hfill \end{array}$

The controllability Grammian operator as

$\begin{array}{ccc}\hfill \mathrm{\Theta }& =\hfill & \left(\begin{array}{cc}\hfill 0.1275\hfill & \hfill 0.2364\hfill \\ \hfill 0.2364\hfill & \hfill 0.5259\hfill \end{array}\right)\hfill \\ \hfill & =\hfill & 0.0671-0.0559\hfill \\ \hfill & =\hfill & 0.0112>0\hfill \end{array}$

It is Invertible. Therefore, according to Theorem (4.4), the system described in (8), as presented in the example, is fully controllable on the interval [0, 6] \{1, …, 5}.

Example 6.2

Evaluate the non linear system as

$\begin{array}{ccc}\hfill {}^{C}D^{\gamma }({}^{C}D^{\eta }-A)\xi (p)+f(p,\xi (p))& =\hfill & Q(p,\xi (p))+{\int }_0^{\frac{5}{2}}\frac{{(p-s)}^{0.5-1}}{\mathrm{\Gamma }(0.5)}ds+\mathcal{C}v(p)\hfill \\ \hfill & \hfill & +\mathrm{\Theta }(p,\xi (p))d{W}_H(p),\hfill \\ \hfill & \hfill & p\in \left[0,\frac{5}{2}\right]-\left\{\frac{1}{2},1,\frac{3}{2},2\right\}\hfill \\ \hfill \delta \xi (p_i)& =\hfill & \frac{1.8+\xi (p_i^{-})}{3},p=\frac{i}{2},i=\mathrm{1,2,3,4}\hfill \\ \hfill \xi (0)& =\hfill & 0\hfill \end{array}$

(9)

Where η = 0.8; γ = 0.7; $\xi (p)=\left(\begin{array}{c}\hfill {\xi }_1(p)\hfill \\ \hfill {\xi }_2(p)\hfill \end{array}\right)$ ,

• $A=\left(\begin{array}{cc}\hfill 0.1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.5\hfill \end{array}\right)$ ; $\mathcal{C}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right)$ ; $\mathrm{\Theta }(p,\xi (p))=\left(\begin{array}{c}\hfill (2p^2+1){\xi }_1(p)e^{-p}\hfill \\ \hfill {\xi }_2{e}^{-p}\hfill \end{array}\right)$ ;

• $\mathcal{B}(p,\xi (p))=\left(\begin{array}{c}\hfill e^{-p}{\xi }_1(p)\hfill \\ \hfill (e^p+1){\xi }_2(p)\hfill \end{array}\right){\xi }_0=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\right)$ and ${\mathrm{\Upsilon }}_0=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right)$

The solution is,

$\begin{array}{ccc}\hfill \xi (p)& =\hfill & E_{0.8}(A{t}^{0.8})[{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]+E_{0.8}(A{t}^{0.8})[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]-f(p,\xi (p))\hfill \\ \hfill & \hfill & +\sum _{i=1}^k{I}_i(\xi (p_i^{-}))+{\int }_0^p{(p-s)}^{0.8+0.7-1}{E}_{0.8,0.8+0.7}(A{(p-s)}^{0.8})\hfill \\ \hfill & \hfill & \times \left[Af(s,\xi (s))+\mathcal{B}(s,\xi (s))+\mathcal{C}v(s)+{\int }_0^s\psi (s,\tau ,z(\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & +{\int }_0^p{(p-s)}^{0.8+0.7-1}{E}_{0.8,0.8+0.7}(A{(p-s)}^{0.8})\mathrm{\Theta }(s,\xi (s))d{W}_H(s)\hfill \end{array}$

The control as

$\begin{array}{ccc}\hfill v(p)& =\hfill & {(\mathbb{P}-p)}^{0.8+0.7-1}{T}^{*}{E}_{0.8,0.8+0.7}\left(A^{*}{(\mathbb{P}-p)}^{0.8}E\left\{Q^{-1}\left({\xi }_1-E_{0.8}(A{t}^{0.8})\right.\right.\right.\hfill \\ \hfill & \hfill & \times [{\xi }_0+{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]-E_{0.8}(A{\mathbb{P}}^{0.8})[{\mathrm{\Upsilon }}_0+f(0,{\mathrm{\Upsilon }}_0)]+f(p,\xi (p))\hfill \\ \hfill & \hfill & -\sum _{i=1}^k{I}_i(\xi (p_i^{-}))-{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{0.8+0.7-1}{E}_{0.8,0.8+0.7}(A{(\mathbb{P}-s)}^{0.8})\hfill \\ \hfill & \hfill & \times \left[Af(s,\xi (s))+\mathcal{B}(s,\xi (s))+\mathcal{C}v(s)+{\int }_0^s\psi (s,\tau ,\xi (\tau ))d\tau \right]ds\hfill \\ \hfill & \hfill & -{\int }_0^{\mathbb{P}}{(\mathbb{P}-s)}^{0.8+0.7-1}{E}_{0.8,0.8+0.7}(A{(\mathbb{P}-s)}^{0.8})\hfill \\ \hfill & \hfill & \left.\left.\times \mathrm{\Theta }(s,\xi (s))d{W}_H(s)E\left\{p/{\mathcal{F}}_s\right\}\right)\right\}\hfill \end{array}$

The controllability Grammian operator as

$\begin{array}{ccc}\hfill Q& =\hfill & \left(\begin{array}{cc}\hfill 0.7108\hfill & \hfill -0.4704\hfill \\ \hfill -0.4704\hfill & \hfill 1.2427\hfill \end{array}\right)\hfill \\ \hfill & =\hfill & 0.8833-0.2213\hfill \\ \hfill & =\hfill & 0.6620>0\hfill \end{array}$

It is Invertible. Hence the system (9) is completely able to controlled on [0, 5/2] − {1/2, 1, 3/2, 2}.

6.1. Discussion of results

The numerical simulations demonstrate how fractional memory, stochastic perturbations, and impulsive effects collectively shape the system dynamics. The observed jumps in the state trajectories are primarily caused by the designed impulsive controls, which instantaneously redirect the system toward desired states. Variations in the stochastic intensity and the fractional-order parameters influence the amplitude and rate of convergence, revealing the sensitivity of the system to memory and noise. These results confirm that appropriate tuning of impulses and system parameters can effectively stabilize the nonlinear fractional stochastic system and enhance controllability, providing both theoretical validation and practical insight.

7. Conclusion

This work has presented a comprehensive controllability analysis for Langevin stochastic neutral impulsive integro-differential equations involving the Caputo fractional derivative and fractional Brownian motion. The main contributions of the paper include: (i) the derivation of necessary and sufficient controllability conditions for the corresponding linear fractional stochastic system, (ii) the establishment of controllability for the nonlinear system via the Banach contraction principle, and (iii) the validation of the theoretical results through illustrative examples. The results demonstrate that impulsive effects, memory terms, and fractional stochastic perturbations can be effectively handled within a unified controllability framework. Consequently, the proposed approach provides a useful mathematical foundation for analyzing the dynamical behavior of real-world systems arising in engineering, physics, and applied sciences that exhibit fractional-order dynamics and stochastic influences.