Finite-Time Stability for a Class of Fractional Itô–Doob Stochastic Time Delayed Systems

Abstract

This paper addresses the finite-time stability of a class of fractional Itô–Doob stochastic systems with time delays. Novel stability criteria are established using a combination of Gronwall-type, Hölder’s, and Burkholder–Davis–Gundy (BDG) inequalities, thereby generalizing classical integer-order stability theory to the fractional domain. Furthermore, the analysis uniquely integrates stochastic perturbations and time delays, providing a comprehensive framework for systems exhibiting both memory and randomness. The effectiveness of the proposed approach is demonstrated through a numerical example of a three-dimensional stochastic delayed system with fractional dynamics.

1. Introduction

Fractional differential Equations (FDEs) have emerged as powerful tools for modeling systems with memory and hereditary properties, leading to significant applications across diverse fields. Their ability to incorporate long-term memory effects and describe multi-scale phenomena captured by operators such as the Riemann–Liouville or Caputo derivatives often provides a better fit to empirical data compared to traditional integer-order models. In bioengineering, FDEs precisely describe the non-Newtonian flow of blood and the viscoelastic mechanics of tissues. In control theory, fractional-order controllers deliver more robust performance for systems with inherent latency. Similarly, in electrochemistry, they are indispensable for modeling the impedance of batteries and supercapacitors, where ion diffusion exhibits fractal-like behavior [1,2,3,4].

In recent years, the study of fractional-order systems with time delays has emerged as a vibrant and critical research frontier. The convergence of these two elements, the non-local memory of fractional calculus and the feedback latency of time delays, presents profound mathematical challenges and practical opportunities. Consequently, contemporary research explores a wide range of problems, extending beyond foundational stability analysis to encompass advanced control synthesis, observer design, synchronization in complex networks, and the numerical simulation of these systems. This expanded focus is largely motivated by the superior modeling capabilities of fractional-order delay models in applications such as viscoelastic materials, biological systems, and network control infrastructures [5,6,7,8].

Building upon prior research [9,10,11,12,13], we investigate the finite-time stability of a class of fractional stochastic time-delayed systems. Our approach builds on established results in the literature and integrates recent advancements in finite-time stability. The main contributions of this paper are the following:

•

Establishing finite-time stability results based on Gronwall-type inequalities.

•

The work in [14] addresses finite-time stability for Riemann–Liouville fractional stochastic systems, providing important results for this class. In contrast, our study focuses on fractional Itô–Doob stochastic systems, which require different analytical tools due to the distinct characteristics of the fractional operator structure. Furthermore, our approach incorporates time delays and stochastic perturbations within a fractional framework, thereby extending the applicability of stability results to a broader class of complex dynamical systems.

•

We integrate stochastic effects and time delays in a novel manner to capture complex dynamical behaviors.

The remainder of this paper is organized as follows: Section 2 presents the main results and the mathematical framework; Section 3 illustrates these findings with examples; and Section 4 concludes the paper.

2. Main Results

Let us consider the fractional stochastic system with time delay given by the following equation:

$$\left\{\begin{array}{c}dZ\left(t\right)=\left[{\mathbf{a}}_{1}Z\left(t\right)+{\mathbf{a}}_{2}Z(t-\tau )\right]dt+\left[{\mathbf{b}}_{1}Z\left(t\right)+{\mathbf{b}}_{2}Z(t-\tau )\right]d{t}^{\alpha ,\psi }\\ +\left[{\mathbf{c}}_{1}Z\left(t\right)+{\mathbf{c}}_{2}Z(t-\tau )\right]d\mathcal{B}\left(t\right),\\ Z\left(t\right)=w,t\in [a-\tau ,a],\end{array}\right.$$

(1)

where$t\in [a,\mathcal{T}],a\in {\mathbb{R}}_{+},$ $0<\alpha <1$, the matrices ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{b}}_1,{\mathbf{b}}_2,{\mathbf{c}}_1,{\mathbf{c}}_2\in {\mathbb{R}}^{n\times n}$, and $\mathcal{B}\left(t\right)$ are standard Brownian motion defined on a complete probability space $\mathcal{R}=\{\mathcal{X},\mathfrak{F},{\left({\mathfrak{F}}_{\varkappa }\right)}_{\varkappa \ge 0},\mathfrak{B}\}$. $\psi \in C^1\left(\left(a,\mathcal{T}\right),\mathbb{R}\right),$ $k\ge {\psi }^{\prime }\left(t\right)\ge h>0,$ $(k,h\in {\mathbb{R}}_{+}^{*}).$

The initial condition $w\in C\left([a-\tau ,a],{\mathbb{R}}^n\right)$, ${\mathfrak{F}}_0$-measurable and endowed with the norm ${∥X∥}_{\infty }={sup}_{a-\tau \le l\le a}{∥X\left(l\right)∥}_2.$

Definition 1 

([1]). Let $\alpha \in \left(0,1\right)$ and let f be a continuous function. Then, the integral of f with respect to $d{x}^{\alpha ,\psi }$ is given by

$$\underset{a}{\overset{t}{{\displaystyle \int }}}f\left(x\right)d{x}^{\alpha ,\psi }=\alpha \underset{a}{\overset{t}{{\displaystyle \int }}}{\left(\psi \left(t\right)-\psi \left(x\right)\right)}^{\alpha -1}f\left(x\right){\psi }^{\prime }\left(x\right)dx.$$

(2)

Definition 2 

([15]). Equation (1) is said to be finite-time stochastically stable with parameters $\{q,\varrho ,v,$T}, $\varrho <v$, (denoted (FTSS)-$\{q,\varrho ,v,$T}), if:

$$\mathbb{E}{∥w∥}_{\infty }^q<\varrho ,$$

imply

$$\mathbb{E}{∥Z\left(t\right)∥}^q<v,\forall t\in [a,\mathcal{T}].$$

Consider $q,p>0$: ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$. We define $m_1,m_2,m_3,m_4,m_5$ by

$$\begin{array}{ccc}\hfill m_1& =& {\left(\frac{\mathrm{\Gamma }\left(p\left(\alpha -1\right)+1\right)}{p^{p\left(\alpha -1\right)+1}}\right)}^{\frac{1}{p}},\hfill \\ \hfill m_2& =& \frac{{∥{\mathbf{a}}_1∥}_2^q}{p^{\frac{q}{P}}{h}^q}+{\alpha }^q{∥{\mathbf{b}}_1∥}_2^q{m}_1^q,\hfill \\ \hfill m_3& =& \frac{{∥{\mathbf{a}}_2∥}_2^q}{p^{\frac{q}{P}}{h}^q}+{\alpha }^q{∥{\mathbf{b}}_2∥}_2^q{m}_1^q,\hfill \\ \hfill m_4& =& 7^{q-1}\left(m_2+\frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right),\hfill \\ \hfill m_5& =& 7^{q-1}\left(m_3+\frac{{∥{\mathbf{c}}_2∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right),\hfill \end{array}$$

where $\mathrm{\Gamma }\left(.\right)$ is the gamma function.

Theorem 1. 

Let $q,p,h,k,\varrho ,v>0$ and $\frac{1}{p}+\frac{1}{q}=1$. Equation (1) is (FTSS)-$\{q,\varrho ,v,$T}, if

$$\left(7^{q-1}+{\displaystyle \frac{m_5}{q}}{e}^{-q\psi \left(a\right)}\right)e^{\left(m_4+{\displaystyle \frac{k}{h}}{m}_5+q\right)\psi \left(\mathcal{T}\right)}\le {\displaystyle \frac{v}{\varrho }},$$

(3)

where $m_4,m_5$ are given.

Proof. 

Equation (1) has a solution Z defined by

$$\begin{array}{cc}\hfill Z\left(t\right)& =Z\left(a\right)+{\int }_a^t\left({\mathbf{a}}_{1}Z\left(l\right)+{\mathbf{a}}_{2}Z(l-\tau )\right)dl\hfill \\ \hfill & +{\int }_a^t\left({\mathbf{b}}_{1}Z\left(l\right)+{\mathbf{b}}_{2}Z(l-\tau )\right)d{l}^{\alpha ,\psi }\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & +{\int }_a^t\left({\mathbf{c}}_{1}Z\left(l\right)+{\mathbf{c}}_{2}Z(l-\tau )\right)d\mathcal{B}\left(l\right),\hfill \end{array}$$

(5)

then,

$$\begin{array}{cc}\hfill Z\left(t\right)& =Z\left(a\right)+{\int }_a^t\left({\mathbf{a}}_{1}Z\left(l\right)+{\mathbf{a}}_{2}Z(l-\tau )\right)dl\hfill \\ \hfill & +\alpha \underset{a}{\overset{t}{{\displaystyle \int }}}{\left(\psi \left(t\right)-\psi \left(l\right)\right)}^{\alpha -1}{\mathbf{b}}_{1}Z\left(l\right){\psi }^{\prime }\left(l\right)dl\hfill \\ \hfill & +\alpha {\int }_a^t{\left(\psi \left(t\right)-\psi \left(l\right)\right)}^{\alpha -1}{\mathbf{b}}_{2}Z(l-\tau ){\psi }^{\prime }\left(l\right)dl\hfill \\ \hfill & +{\int }_a^t\left({\mathbf{c}}_{1}Z\left(l\right)+{\mathbf{c}}_{2}Z(l-\tau )\right)d\mathcal{B}\left(l\right).\hfill \end{array}$$

(6)

It follows from the Hölder’s inequality that

$$\begin{array}{cc}\hfill {\int }_a^t{\mathbf{a}}_{1}Z\left(l\right)dl& \le \frac{{∥{\mathbf{a}}_1∥}_2}{h}{\int }_a^t∥Z\left(l\right)∥{\psi }^{\prime }\left(l\right)dl\hfill \\ \hfill & =\frac{{∥{\mathbf{a}}_1∥}_2}{h}{\int }_a^t{\left({\psi }^{\prime }\left(l\right)\right)}^{\frac{1}{P}}{\left({\psi }^{\prime }\left(l\right)\right)}^{\frac{1}{q}}{e}^{\psi \left(l\right)}{e}^{-\psi \left(l\right)}∥Z\left(l\right)∥dl\hfill \\ \hfill & \le \frac{{∥{\mathbf{a}}_1∥}_2}{h}{\left(\underset{a}{\overset{t}{\int }}{\left({\psi }^{\prime }\left(l\right)\right)}^{p\frac{1}{P}}{e}^{p\psi \left(l\right)}dl\right)}^{\frac{1}{P}}{\left(\underset{a}{\overset{t}{\int }}{\left({\psi }^{\prime }\left(l\right)\right)}^{q\frac{1}{q}}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^{q}dl\right)}^{\frac{1}{q}}\hfill \\ \hfill & \le \frac{{∥{\mathbf{a}}_1∥}_2}{h}{\left(\underset{a}{\overset{t}{\int }}{\psi }^{\prime }\left(l\right)e^{p\psi \left(l\right)}dl\right)}^{\frac{1}{P}}{\left(\underset{a}{\overset{t}{\int }}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}\hfill \\ \hfill & \le \frac{{∥{\mathbf{a}}_1∥}_2}{h}{\left({\left.\frac{1}{p}{e}^{p\psi \left(l\right)}\right|}_a^t\right)}^{\frac{1}{P}}{\left(\underset{a}{\overset{t}{\int }}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}\hfill \\ \hfill & \le \frac{{∥{\mathbf{a}}_1∥}_2}{h}\frac{1}{p^{\frac{1}{P}}}{\left(e^{p\psi \left(t\right)}-e^{p\psi \left(a\right)}\right)}^{\frac{1}{P}}{\left(\underset{a}{\overset{t}{\int }}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}\hfill \\ \hfill & \le \frac{{∥{\mathbf{a}}_1∥}_2}{h}\frac{1}{p^{\frac{1}{P}}}{e}^{p\psi \left(t\right)\frac{1}{P}}{\left(\underset{a}{\overset{t}{\int }}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}\hfill \\ \hfill & \le \frac{{∥{\mathbf{a}}_1∥}_2}{p^{\frac{1}{P}}h}{e}^{\psi \left(t\right)}{\left(\underset{a}{\overset{t}{\int }}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}.\hfill \end{array}$$

(7)

In this way, we have proven that

$$\begin{array}{ccc}\hfill {\int }_a^t{\mathbf{a}}_{1}Z\left(l\right)dl& \le & \frac{{∥{\mathbf{a}}_1∥}_2}{p^{\frac{1}{P}}h}{e}^{\psi \left(t\right)}{\left(\underset{a}{\overset{t}{\int }}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill {\int }_a^t{\mathbf{a}}_{2}Z(l-\tau )dl& \le & \frac{{∥{\mathbf{a}}_2∥}_2}{p^{\frac{1}{P}}h}{e}^{\psi \left(t\right)}{\left(\underset{a}{\overset{t}{\int }}{e}^{-q\psi \left(l\right)}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}.\hfill \end{array}$$

(9)

Now, we proceed to estimate the third and fourth terms on the right-hand side of the equality:

$$\begin{array}{cc}\hfill & \underset{a}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(l\right)\right)}^{\alpha -1}∥Z\left(l\right)∥{\psi }^{\prime }\left(l\right)dl\hfill \\ \hfill & =\underset{a}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(l\right)\right)}^{\alpha -1}{\left({\psi }^{\prime }\left(l\right)\right)}^{\frac{1}{P}}{\left({\psi }^{\prime }\left(l\right)\right)}^{\frac{1}{q}}{e}^{\psi \left(l\right)}{e}^{-\psi \left(l\right)}∥Z\left(l\right)∥dl\hfill \\ \hfill & \le {\left(\underset{a}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(l\right)\right)}^{p\left(\alpha -1\right)}{\left({\psi }^{\prime }\left(l\right)\right)}^{p\frac{1}{P}}{e}^{p\psi \left(l\right)}dl\right)}^{\frac{1}{P}}{\left(\underset{a}{\overset{t}{\int }}{\left({\psi }^{\prime }\left(l\right)\right)}^{q\frac{1}{q}}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^{q}dl\right)}^{\frac{1}{q}}\hfill \\ \hfill & \le {\left(\underset{a}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(l\right)\right)}^{p\left(\alpha -1\right)}{e}^{p\psi \left(l\right)}{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{p}}{\left({\int }_a^t{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \le {\left(\underset{a}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(l\right)\right)}^{p\left(\alpha -1\right)}{e}^{p\psi \left(l\right)}{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{p}}{\left({\int }_a^t{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}},\hfill \end{array}$$

(11)

we put $u=\psi \left(t\right)-\psi \left(l\right)$; then, $du=-{\psi }^{\prime }\left(l\right)dl,$

$$\begin{array}{ccc}\hfill {\int }_a^t{\left(\psi \left(t\right)-\psi \left(l\right)\right)}^{p\left(\alpha -1\right)}{e}^{p\psi \left(l\right)}{\psi }^{\prime }\left(l\right)dl& =& {\int }_0^{\psi \left(t\right)-\psi \left(a\right)}{u}^{p\left(\alpha -1\right)}{e}^{p\left(\psi \left(t\right)-u\right)}du\hfill \\ & =& e^{p\psi \left(t\right)}{\int }_0^{\psi \left(t\right)-\psi \left(a\right)}{u}^{p\left(\alpha -1\right)}{e}^{-pu}du\hfill \\ & =& \frac{e^{p\psi \left(t\right)}}{p^{p\left(\alpha -1\right)}}{\int }_0^{\psi \left(t\right)-\psi \left(a\right)}{\left(pu\right)}^{\left[p\left(\alpha -1\right)+1\right]-1}{e}^{-pu}du\hfill \\ & \le & \frac{e^{p\psi \left(t\right)}}{p^{p\left(\alpha -1\right)+1}}\mathrm{\Gamma }\left(p\left(\alpha -1\right)+1\right),\hfill \end{array}$$

(12)

then,

$$\begin{array}{cc}\hfill & \underset{a}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(l\right)\right)}^{\alpha -1}∥Z\left(l\right)∥{\psi }^{\prime }\left(l\right)dl\hfill \\ \hfill & \le {\left(\frac{e^{p\psi \left(t\right)}}{p^{p\left(\alpha -1\right)+1}}\mathrm{\Gamma }\left(p\left(\alpha -1\right)+1\right)\right)}^{\frac{1}{p}}{\left({\int }_a^t{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}\hfill \\ \hfill & \le e^{\psi \left(t\right)}{\left(\frac{\mathrm{\Gamma }\left(p\left(\alpha -1\right)+1\right)}{p^{p\left(\alpha -1\right)+1}}\right)}^{\frac{1}{p}}{\left({\int }_a^t{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}},\hfill \end{array}$$

(13)

we put

$$m_1={\left(\frac{\mathrm{\Gamma }\left(p\left(\alpha -1\right)+1\right)}{p^{p\left(\alpha -1\right)+1}}\right)}^{\frac{1}{p}},$$

then,

$$\underset{a}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(l\right)\right)}^{\alpha -1}∥Z\left(l\right)∥{\psi }^{\prime }\left(l\right)dl\le e^{\psi \left(t\right)}{m}_1{\left({\int }_a^t{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}.$$

(14)

So,

$${\int }_a^t{\left(\psi \left(t\right)-\psi \left(l\right)\right)}^{\alpha -1}∥Z(l-\tau )∥{\psi }^{\prime }\left(l\right)dl\le e^{\psi \left(t\right)}{m}_1{\left(\underset{a}{\overset{t}{\int }}{e}^{-q\psi \left(l\right)}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}},$$

(15)

then,

$$\begin{array}{ccc}\hfill ∥Z\left(t\right)∥& \le & {∥w∥}_{\infty }+\frac{{∥{\mathbf{a}}_1∥}_2}{p^{\frac{1}{P}}h}{e}^{\psi \left(t\right)}{\left(\underset{a}{\overset{t}{\int }}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}\hfill \\ & & +\frac{{∥{\mathbf{a}}_2∥}_2}{p^{\frac{1}{P}}h}{e}^{\psi \left(t\right)}{\left(\underset{a}{\overset{t}{\int }}{e}^{-q\psi \left(l\right)}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}\hfill \\ & & +\alpha {∥{\mathbf{b}}_1∥}_2{e}^{\psi \left(t\right)}{m}_1{\left({\int }_a^t{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}\hfill \\ & & +\alpha {∥{\mathbf{b}}_2∥}_2{e}^{\psi \left(t\right)}{m}_1{\left(\underset{a}{\overset{t}{\int }}{e}^{-q\psi \left(l\right)}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl\right)}^{\frac{1}{q}}\hfill \\ & & +∥{\int }_a^t{\mathbf{c}}_{1}Z\left(l\right)d\mathcal{B}\left(l\right)∥+∥{\int }_a^t{\mathbf{c}}_{2}Z(l-\tau )d\mathcal{B}\left(l\right)∥,\hfill \end{array}$$

(16)

consequently,

$$\begin{array}{ccc}\hfill {∥Z\left(t\right)∥}^q& \le & 7^{q-1}\left({∥w∥}_{\infty }^q+{\displaystyle \frac{{∥{\mathbf{a}}_1∥}_2^q}{p^{\frac{q}{P}}{h}^q}}{e}^{q\psi \left(t\right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right.\hfill \\ & & +{\displaystyle \frac{{∥{\mathbf{a}}_2∥}_2^q}{p^{\frac{q}{P}}{h}^q}}{e}^{q\psi \left(t\right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{e}^{-q\psi \left(l\right)}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \\ & & +{\alpha }^q{∥{\mathbf{b}}_1∥}_2^q{e}^{q\psi \left(t\right)}{m}_1^q{\int }_a^t{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \\ & & +{\alpha }^q{∥{\mathbf{b}}_2∥}_2^q{e}^{q\psi \left(t\right)}{m}_1^q\underset{a}{\overset{t}{{\displaystyle \int }}}{e}^{-q\psi \left(l\right)}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \\ & & +\left.{∥{\int }_a^t{\mathbf{c}}_{1}Z\left(l\right)d\mathcal{B}\left(l\right)∥}^q+{∥{\int }_a^t{\mathbf{c}}_{2}Z(l-\tau )d\mathcal{B}\left(l\right)∥}^q\right).\hfill \end{array}$$

(17)

So,

$$\begin{array}{ccc}\hfill {∥Z\left(t\right)∥}^q& \le & 7^{q-1}\left({∥w∥}_{\infty }^q+\left({\displaystyle \frac{{∥{\mathbf{a}}_1∥}_2^q}{p^{\frac{q}{P}}{h}^q}}+{\alpha }^q{∥{\mathbf{b}}_1∥}_2^q{m}_1^q\right)e^{q\psi \left(t\right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right.\hfill \\ & & +\left({\displaystyle \frac{{∥{\mathbf{a}}_2∥}_2^q}{p^{\frac{q}{P}}{h}^q}}+{\alpha }^q{∥{\mathbf{b}}_2∥}_2^q{m}_1^q\right)e^{q\psi \left(t\right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{e}^{-q\psi \left(l\right)}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \end{array}$$

(18)

$$\begin{array}{ccc}& & +\left.{∥{\int }_a^t{\mathbf{c}}_{1}Z\left(l\right)d\mathcal{B}\left(l\right)∥}^q+{∥{\int }_a^t{\mathbf{c}}_{2}Z(l-\tau )d\mathcal{B}\left(l\right)∥}^q\right),\hfill \end{array}$$

(19)

we put

$$m_2={\displaystyle \frac{{∥{\mathbf{a}}_1∥}_2^q}{p^{\frac{q}{P}}{h}^q}}+{\alpha }^q{∥{\mathbf{b}}_1∥}_2^q{m}_1^q,m_3={\displaystyle \frac{{∥{\mathbf{a}}_2∥}_2^q}{p^{\frac{q}{P}}{h}^q}}+{\alpha }^q{∥{\mathbf{b}}_2∥}_2^q{m}_1^q.$$

Then,

$$\begin{array}{ccc}\hfill {∥Z\left(t\right)∥}^q& \le & 7^{q-1}\left({∥w∥}_{\infty }^q+m_2{e}^{q\psi \left(t\right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{e}^{-q\psi \left(l\right)}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right.\hfill \\ & & +m_3{e}^{q\psi \left(t\right)}\underset{a}{\overset{t}{{\displaystyle \int }}}{e}^{-q\psi \left(l\right)}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \\ & & +\left.{∥{\int }_a^t{\mathbf{c}}_{1}Z\left(l\right)d\mathcal{B}\left(l\right)∥}^q+{∥{\int }_a^t{\mathbf{c}}_{2}Z(l-\tau )d\mathcal{B}\left(l\right)∥}^q\right).\hfill \end{array}$$

(20)

Then, using Theorem $7.1$ in [15] (Burkholder–Davis–Gundy (BDG) inequality), we obtain

$$\begin{array}{ccc}\hfill \mathbb{E}{∥{\int }_a^t{\mathbf{c}}_{1}Z\left(l\right)d\mathcal{B}\left(l\right)∥}^q& \le & {∥{\mathbf{c}}_1∥}_2^q{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{(t-a)}^{\frac{q-2}{2}}{\int }_a^t\mathbb{E}{∥Z\left(l\right)∥}^{q}dl\hfill \\ & \le & \frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{(t-a)}^{\frac{q-2}{2}}{\int }_a^t\mathbb{E}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \\ & \le & \frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{(t-a)}^{\frac{q-2}{2}}{e}^{q\psi \left(t\right)}\hfill \\ & & \times {\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl,\hfill \end{array}$$

(21)

then,

$$\begin{array}{ccc}& & \mathbb{E}{∥{\int }_a^t{\mathbf{c}}_{1}Z\left(l\right)d\mathcal{B}\left(l\right)∥}^q\hfill \\ & \le & \frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}{e}^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)d,\hfill \\ & & \mathbb{E}{∥{\int }_a^t{\mathbf{c}}_{2}Z(l-\tau )d\mathcal{B}\left(l\right)∥}^q\hfill \\ & \le & \frac{{∥{\mathbf{c}}_2∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}{e}^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl.\hfill \end{array}$$

(22)

So,

$$\begin{array}{ccc}\hfill \mathbb{E}{∥Z\left(t\right)∥}^q& \le & 7^{q-1}\left(\mathbb{E}{∥w∥}_{\infty }^q+m_2{e}^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\right.\hfill \\ & & +m_3{e}^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \\ & & +\frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}{e}^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \\ & & +\left.\frac{{∥{\mathbf{c}}_2∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}{e}^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl\right).\hfill \end{array}$$

(23)

Then,

$$\begin{array}{cc}\hfill \mathbb{E}{∥Z\left(t\right)∥}^q& \le 7^{q-1}\left(\mathbb{E}{∥w∥}_{\infty }^q+\left(m_2+\frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)\right.\hfill \\ \hfill & \times e^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \\ \hfill & \left.+\left(m_3+\frac{{∥{\mathbf{c}}_2∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)e^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl\right).\hfill \end{array}$$

(24)

Hence,

$$\begin{array}{ccc}\hfill \mathbb{E}{∥Z\left(t\right)∥}^q& \le & 7^{q-1}\mathbb{E}{∥w∥}_{\infty }^q+7^{q-1}\left(m_2+\frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)\hfill \\ & & \times e^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \\ & & +7^{q-1}\left(m_3+\frac{{∥{\mathbf{c}}_2∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)\hfill \\ & & \times e^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl.\hfill \end{array}$$

(25)

Thus,

$$\begin{array}{ccc}\hfill m_4& =& 7^{q-1}\left(m_2+\frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right),\hfill \\ \hfill m_5& =& 7^{q-1}\left(m_3+\frac{{∥{\mathbf{c}}_2∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \mathbb{E}{∥Z\left(t\right)∥}^q& \le & 7^{q-1}\mathbb{E}{∥w∥}_{\infty }^q+m_4{e}^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \\ & & +m_5{e}^{q\psi \left(t\right)}{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl,\hfill \end{array}$$

therefore,

$$\begin{array}{ccc}\hfill e^{-q\psi \left(t\right)}\mathbb{E}{∥Z\left(t\right)∥}^q& \le & 7^{q-1}{e}^{-q\psi \left(t\right)}\mathbb{E}{∥w∥}_{\infty }^q+m_4{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z\left(l\right)∥}^q{\psi }^{\prime }\left(l\right)dl\hfill \\ & & +m_5{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl.\hfill \end{array}$$

For $H\left(t\right)=e^{-q\psi \left(t\right)}\mathbb{E}{∥Z\left(t\right)∥}^q$, then,

$$\begin{array}{ccc}\hfill H\left(t\right)& \le & 7^{q-1}\mathbb{E}{∥w∥}_{\infty }^q+m_4{\int }_a^{t}H\left(l\right){\psi }^{\prime }\left(l\right)dl\hfill \\ & & +m_5{\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl,\hfill \end{array}$$

$l=s+\tau$, then, $dl=ds,$

$$\begin{array}{cc}\hfill {\int }_a^t{e}^{-q\psi \left(l\right)}\mathbb{E}{∥Z(l-\tau )∥}^q{\psi }^{\prime }\left(l\right)dl& ={\int }_{a-\tau }^{t-\tau }{e}^{-q\psi \left(s+\tau \right)}\mathbb{E}{∥Z\left(s\right)∥}^q{\psi }^{\prime }\left(s+\tau \right)ds\hfill \\ \hfill & ={\int }_{a-\tau }^a{e}^{-q\psi \left(s+\tau \right)}\mathbb{E}{∥Z\left(s\right)∥}^q{\psi }^{\prime }\left(s+\tau \right)ds\hfill \\ \hfill & +{\int }_a^{t-\tau }{e}^{-q\psi \left(s+\tau \right)}\mathbb{E}{∥Z\left(s\right)∥}^q{\psi }^{\prime }\left(s+\tau \right)ds\hfill \\ \hfill & \le \mathbb{E}{∥w∥}_{\infty }^q{\int }_{a-\tau }^a{e}^{-q\psi \left(s+\tau \right)}{\psi }^{\prime }\left(s+\tau \right)ds\hfill \\ \hfill & +\frac{k}{h}{\int }_a^{t-\tau }{e}^{-q\psi \left(s\right)}\mathbb{E}{∥Z\left(s\right)∥}^q{\psi }^{\prime }\left(s\right)ds\hfill \\ \hfill & \le \mathbb{E}{∥w∥}_{\infty }^q{\left.\frac{1}{q}{e}^{-q\psi \left(s+\tau \right)}\right|}_a^{a-\tau }+\frac{k}{h}{\int }_a^{t}H\left(s\right){\psi }^{\prime }\left(s\right)ds\hfill \\ \hfill & \le \frac{1}{q}{e}^{-q\psi \left(a\right)}\mathbb{E}{∥w∥}_{\infty }^q+\frac{k}{h}{\int }_a^{t}H\left(s\right){\psi }^{\prime }\left(s\right)ds.\hfill \end{array}$$

(26)

Then,

$$\begin{array}{cc}\hfill H\left(t\right)& \le 7^{q-1}\mathbb{E}{∥w∥}_{\infty }^q+m_4{\int }_a^{t}H\left(l\right){\psi }^{\prime }\left(l\right)dl\hfill \\ \hfill & +m_5{\displaystyle \frac{1}{q}}{e}^{-q\psi \left(a\right)}\mathbb{E}{∥w∥}_{\infty }^q+m_5{\displaystyle \frac{k}{h}}{\int }_a^{t}H\left(s\right){\psi }^{\prime }\left(s\right)ds,\hfill \end{array}$$

we obtain

$$H\left(t\right)\le \left(7^{q-1}+{\displaystyle \frac{m_5}{q}}{e}^{-q\psi \left(a\right)}\right)\mathbb{E}{∥w∥}_{\infty }^q+\left(m_4+{\displaystyle \frac{k{m}_5}{h}}\right){\int }_a^{t}H\left(l\right){\psi }^{\prime }\left(l\right)dl.$$

Using the Gronwall Lemma, we obtain

$$\begin{array}{ccc}\hfill H\left(t\right)& \le & \left(7^{q-1}+m_5\frac{1}{q}{e}^{-q\psi \left(a\right)}\right)\mathbb{E}{∥w∥}_{\infty }^q{e}^{\left(m_4+m_5\frac{k}{h}\right){\int }_a^t{\psi }^{\prime }\left(l\right)dl}\hfill \\ & \le & \left(7^{q-1}+m_5\frac{1}{q}{e}^{-q\psi \left(a\right)}\right)e^{\left(m_4+m_5\frac{k}{h}\right)\psi \left(\mathcal{T}\right)}\mathbb{E}{∥w∥}_{\infty }^q,t\in [a,\mathcal{T}].\hfill \end{array}$$

Consequently,

$$\mathbb{E}{∥Z\left(t\right)∥}^q\le \left(7^{q-1}+\frac{m_5}{q}{e}^{-q\psi \left(a\right)}\right)e^{\left(m_4+\frac{k}{h}{m}_5+q\right)\psi \left(\mathcal{T}\right)}\mathbb{E}{∥w∥}_{\infty }^q,t\in [a,\mathcal{T}].$$

(27)

Therefore, if $\mathbb{E}{∥w∥}_{\infty }^q<\varrho$ and (3) is satisfied then, $\mathbb{E}{∥Z\left(t\right)∥}^q\le v$, for all $t\in [a,\mathcal{T}]$. □

3. Illustrative Examples

In this section, two illustrative examples will be presented to clarify the validity of the theoretical results obtained in the previous sections.

Example 1. 

Consider the system

$$\begin{array}{c}dZ\left(t\right)=\left[{\mathbf{a}}_{1}Z\left(t\right)+{\mathbf{a}}_{2}Z(t-0.1)\right]dt+\left[{\mathbf{b}}_{1}Z\left(t\right)+{\mathbf{b}}_{2}Z(t-0.1)\right]d{t}^{\alpha ,\psi }\\ +\left[{\mathbf{c}}_{1}Z\left(t\right)+{\mathbf{c}}_{2}Z(t-0.1)\right]d\mathcal{B}\left(t\right),\\ Z\left(t\right)=w,t\in [a-0.1,a]\end{array}$$

(28)

where $t\in [a,\mathcal{T}],a\in {\mathbb{R}}_{+},$ $\alpha =0.94,$$Z\left(t\right)=\left(\begin{array}{c}{Z}_1\left(t\right)\\ Z_2\left(t\right)\end{array}\right)\in {\mathbb{R}}^2$, ψ$\in C^1\left(a,\mathcal{T}\right),$$k>{\psi }^{\prime }\left(t\right)>h>0,$ $(k,h\in {\mathbb{R}}_{+}^{*})$, the initial condition $w=\left(\begin{array}{c}0.073\\ 0.08\end{array}\right)$, and with the following data:

$$\begin{array}{ccc}\hfill {\mathbf{a}}_1& =& {10}^{-1}\left(\begin{array}{cc}1& 2\\ 3& 1\end{array}\right),{\mathbf{a}}_2={10}^{-2}\left(\begin{array}{cc}5& 3\\ 3& 1\end{array}\right),\hfill \\ \hfill {\mathbf{b}}_1& =& {10}^{-1}\left(\begin{array}{cc}2& 2\\ 0& 1\end{array}\right),{\mathbf{b}}_2={10}^{-2}\left(\begin{array}{cc}4& 2\\ 0& 1\end{array}\right),\hfill \\ \hfill {\mathbf{c}}_1& =& {10}^{-4}\left(\begin{array}{cc}6& 2\\ 3& 1\end{array}\right),{\mathbf{c}}_2={10}^{-4}\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right).\hfill \end{array}$$

Then, we deduce that

${∥{\mathbf{a}}_1∥}_2=0.3618,{∥{\mathbf{a}}_2∥}_2=0.066056,{∥{\mathbf{b}}_1∥}_2=0.29208,{∥{\mathbf{b}}_2∥}_2=0.044954,{∥{\mathbf{c}}_1∥}_2=0.00070711,{∥{\mathbf{c}}_2∥}_2=0.0001618.$

If $\psi \left(t\right)=t,$ $a=0$ and $\mathcal{T}=1,$ then ${\psi }^{\prime }\left(t\right)=1.$ We choose $q=1.4,\varrho =0.45$; then, $p=3.5.$

Hence, we obtain $m_1=0.78936$, $m_2=0.26352$ and $m_3=0.022057$. From inequality (3) and for $v=7.52$, the estimated finite time $\mathcal{T}=1$, $m_4=0.57396$ and $m_5=0.048042$. Then, by Theorem 1, the problem (28) is (FTSS)-$\{q,\varrho ,v,$T}.

$$\begin{array}{ccc}\hfill m_1& =& \left(\frac{\mathrm{\Gamma }\left(p\left(\alpha -1\right)+1\right)}{p^{p\left(\alpha -1\right)+1}}\right)=0.78936.\hfill \\ \hfill m_2& =& \frac{{∥{\mathbf{a}}_1∥}_2^q}{p^{\frac{q}{P}}{h}^q}+{\alpha }^q{∥{\mathbf{b}}_1∥}_2^q{m}_1^q=0.26352.\hfill \\ \hfill m_3& =& \frac{{∥{\mathbf{a}}_2∥}_2^q}{p^{\frac{q}{P}}{h}^q}+{\alpha }^q{∥{\mathbf{b}}_2∥}_2^q{m}_1^q=0.022057.\hfill \\ \hfill m_4& =& 7^{q-1}\left(m_2+\frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)=0.57396.\hfill \\ \hfill m_5& =& 7^{q-1}\left(m_3+\frac{{∥{\mathbf{c}}_2∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)=0.048042.\hfill \end{array}$$

Then,

$$\left(7^{q-1}+{\displaystyle \frac{m_5}{q}}{e}^{-q\psi \left(a\right)}\right)e^{\left(m_4+{\displaystyle \frac{k}{h}}{m}_5+q\right)\psi \left(\mathcal{T}\right)}=16.710\le {\displaystyle \frac{v}{\varrho }}.$$

If $\psi \left(t\right)=ln\left(t\right),$ $a=1$ and $\mathcal{T}=1.1,$ then ${\psi }^{\prime }\left(t\right)={\displaystyle \frac{1}{t}},h={\displaystyle \frac{10}{11}},k=1.$ We choose $q=1.4,\varrho =0.45$; then, $p=3.5$.

Hence, we obtain $m_1=0.78936$, $m_2=0.28436$ and $m_3=0.023983$. From inequality (3) and for $v=1.3$, the estimated finite time $\mathcal{T}=1.1$, $m_4=0.61935$ and $m_5=0.052237$. Then, by Theorem 1, problem (28) is (FTSS)-$\{q,\varrho ,v,$T$\}.$

$$\begin{array}{ccc}\hfill m_1& =& 0.78936.\hfill \\ \hfill m_2& =& \frac{{∥{\mathbf{a}}_1∥}_2^q}{p^{\frac{q}{P}}{h}^q}+{\alpha }^q{∥{\mathbf{b}}_1∥}_2^q{m}_1^q=0.28436.\hfill \\ \hfill m_3& =& \frac{{∥{\mathbf{a}}_2∥}_2^q}{p^{\frac{q}{P}}{h}^q}+{\alpha }^q{∥{\mathbf{b}}_2∥}_2^q{m}_1^q=0.023983.\hfill \\ \hfill m_4& =& 7^{q-1}\left(m_2+\frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)=0.61935.\hfill \\ \hfill m_5& =& 7^{q-1}\left(m_3+\frac{{∥{\mathbf{c}}_2∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)=0.052237.\hfill \end{array}$$

then,

$$\left(7^{q-1}+{\displaystyle \frac{m_5}{q}}{e}^{-q\psi \left(a\right)}\right)e^{\left(m_4+{\displaystyle \frac{k}{h}}{m}_5+q\right)\psi \left(\mathcal{T}\right)}=2.7001\le {\displaystyle \frac{v}{\varrho }}.$$

Example 2. 

Consider the system

$$\begin{array}{c}dZ\left(t\right)=\left[{\mathbf{a}}_{1}Z\left(t\right)+{\mathbf{a}}_{2}Z(t-\tau )\right]dt+\left[{\mathbf{b}}_{1}Z\left(t\right)+{\mathbf{b}}_{2}Z(t-\tau )\right]d{t}^{\alpha ,\psi }\\ +\left[{\mathbf{c}}_{1}Z\left(t\right)+{\mathbf{c}}_{2}Z(t-\tau )\right]d\mathcal{B}\left(t\right),\\ Z\left(t\right)=w,t\in [a-\tau ,a],\end{array}$$

(29)

where $t\in [a,\mathcal{T}],a\in {\mathbb{R}}_{+},$ $\alpha =0.94$, $Z\left(t\right)=(Z_1\left(t\right),Z_2\left(t\right),Z_3\left(t\right))\in {\mathbb{R}}^3,$ ψ$\in C^1\left(a,\mathcal{T}\right),$ $k\ge {\psi }^{\prime }\left(t\right)\ge h>0,$ $(k,h\in {\mathbb{R}}_{+}^{*})$, $\tau =0.02.$

The initial condition $w=\left(\begin{array}{c}0.04\\ 0.02\\ 0.01\end{array}\right),$ and with the following data:

$$\begin{array}{ccc}\hfill {\mathbf{a}}_1& =& {10}^{-2}\left(\begin{array}{ccc}1& 2& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),{\mathbf{a}}_2={10}^{-2}\left(\begin{array}{ccc}0& 0& 1\\ 0& 2& 0\\ 3& 1& 0\end{array}\right),\hfill \\ \hfill {\mathbf{b}}_1& =& {10}^{-2}\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 3& 1& 0\end{array}\right),{\mathbf{b}}_2={10}^{-2}\left(\begin{array}{ccc}0& 0& 2\\ 0& 0& 1\\ 3& 1& 0\end{array}\right),\hfill \\ \hfill {\mathbf{c}}_1& =& {10}^{-2}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 3& 1& 0\end{array}\right),{\mathbf{c}}_2={10}^{-2}\left(\begin{array}{ccc}0& 1& 0\\ 0& 2& 0\\ 1& 0& 1\end{array}\right).\hfill \end{array}$$

Then, we deduce that

${∥{\mathbf{a}}_1∥}_2=0.024142,{∥{\mathbf{a}}_2∥}_2=0.032566,{∥{\mathbf{b}}_1∥}_2=0.033028,{∥{\mathbf{b}}_2∥}_2=0.031623,{∥{\mathbf{c}}_1∥}_2=0.033166,{∥{\mathbf{c}}_2∥}_2=0.022361.$

If $\psi \left(t\right)=t,$$a=0$ and $\mathcal{T}=1,$ then ${\psi }^{\prime }\left(t\right)=1.$ We choose $q=1.4,\varrho =0.02$; then, $p=3.5.$

Hence, we obtain $m_1=0.78936$; $m_2=0.0088571$ and $m_3=0.010245$. From inequality (3) and for $v=0.2$, the estimated finite time $\mathcal{T}=1$, $m_4=0.026876$ and $m_5=0.026681$. Then, by Theorem 1, problem (29) is (FTSS)-$\{q,\varrho ,v,$T$\}.$

$$\begin{array}{ccc}\hfill m_1& =& {\left(\frac{\mathrm{\Gamma }\left(p\left(\alpha -1\right)+1\right)}{p^{p\left(\alpha -1\right)+1}}\right)}^{\frac{1}{p}}=0.78936.\hfill \\ \hfill m_2& =& \frac{{∥{\mathbf{a}}_1∥}_2^q}{p^{\frac{q}{P}}{h}^q}+{\alpha }^q{∥{\mathbf{b}}_1∥}_2^q{m}_1^q=0.0088571.\hfill \\ \hfill m_3& =& \frac{{∥{\mathbf{a}}_2∥}_2^q}{p^{\frac{q}{P}}{h}^q}+{\alpha }^q{∥{\mathbf{b}}_2∥}_2^q{m}_1^q=0.010245.\hfill \\ \hfill m_4& =& 7^{q-1}\left(m_2+\frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)=0.026876.\hfill \\ \hfill m_5& =& 7^{q-1}\left(m_3+\frac{{∥{\mathbf{c}}_2∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)=0.026681.\hfill \end{array}$$

then,

$$\left(7^{q-1}+\frac{m_5}{q}{e}^{-q\psi \left(a\right)}\right)e^{\left(m_4+\frac{k}{h}{m}_5+q\right)\psi \left(\mathcal{T}\right)}=9.3993\le \frac{v}{\varrho }.$$

If $\psi \left(t\right)=lnt,$ $a=1$ and $\mathcal{T}=1.23,$ then ${\psi }^{\prime }\left(t\right)=\frac{1}{t},h=\frac{100}{123},k=1.$ We choose $q=1.4,\varrho =0.02$; then, $p=3.5.$

Hence, we obtain $m_1=0.78936$; $m_2=0.0099658$ and $m_3=0.010245$. From inequality (3) and for $v=0.06$ the estimated finite time $\mathcal{T}=1.23$, $m_4=0.030474$ and $m_5=0.031034$. Then, by Theorem 1, problem (29) is (FTSS)-$\{q,\varrho ,v,$T}.

$$\begin{array}{ccc}\hfill m_1& =& {\left(\frac{\mathrm{\Gamma }\left(p\left(\alpha -1\right)+1\right)}{p^{p\left(\alpha -1\right)+1}}\right)}^{\frac{1}{p}}=0.78936.\hfill \\ \hfill m_2& =& \frac{{∥{\mathbf{a}}_1∥}_2^q}{p^{\frac{q}{P}}{h}^q}+{\alpha }^q{∥{\mathbf{b}}_1∥}_2^q{m}_1^q,m_2=0.0099658.\hfill \\ \hfill m_3& =& \frac{{∥{\mathbf{a}}_2∥}_2^q}{p^{\frac{q}{P}}{h}^q}+{\alpha }^q{∥{\mathbf{b}}_2∥}_2^q{m}_1^q=0.011931.\hfill \\ \hfill m_4& =& 7^{q-1}\left(m_2+\frac{{∥{\mathbf{c}}_1∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)=0.030474.\hfill \\ \hfill m_5& =& 7^{q-1}\left(m_3+\frac{{∥{\mathbf{c}}_2∥}_2^q}{h}{\left(\frac{q(q-1)}{2}\right)}^{\frac{q}{2}}{\mathcal{T}}^{\frac{q-2}{2}}\right)=0.031034.\hfill \end{array}$$

Then,

$$\left(7^{q-1}+\frac{m_5}{q}{e}^{-q\psi \left(a\right)}\right)e^{\left(m_4+\frac{k}{h}{m}_5+q\right)\psi \left(\mathcal{T}\right)}=2.9818\le \frac{v}{\varrho }.$$

Example 3. 

Consider the system

$$\left\{\begin{array}{c}dZ\left(t\right)=\left[{\mathbf{a}}_{1}Z\left(t\right)+{\mathbf{a}}_{2}Z(t-\tau )\right]dt+\left[{\mathbf{b}}_{1}Z\left(t\right)+{\mathbf{b}}_{2}Z(t-\tau )\right]d{t}^{\alpha ,\psi }\\ +\left[{\mathbf{c}}_{1}Z\left(t\right)+{\mathbf{c}}_{2}Z(t-\tau )\right]d\mathcal{B}\left(t\right),\\ Z\left(t\right)=w,t\in [a-\tau ,a],\end{array}\right.$$

(30)

where $t\in [1,2],$ $a\in {\mathbb{R}}_{+},$ $\alpha =0.85$, $Z\left(t\right)=\left(\begin{array}{c}{Z}_1\left(t\right)\\ Z_2\left(t\right)\end{array}\right)\in {\mathbb{R}}^2,$ $\psi =lnt,$ $k=1,$ $h={\displaystyle \frac{1}{2}}$, the initial condition is

$$w=\left(\begin{array}{c}t+1\\ t\end{array}\right),\mathit{and}\mathit{with}\mathit{the}\mathit{following}\mathit{data}:$$

$$\begin{array}{ccc}\hfill {\mathbf{a}}_1& =& \left(\begin{array}{cc}t& 2\\ 0& 1\end{array}\right),{\mathbf{a}}_2=\left(\begin{array}{cc}0& t\\ 2& 1\end{array}\right),\hfill \\ \hfill {\mathbf{b}}_1& =& \left(\begin{array}{cc}t+1& 0\\ 0& 1\end{array}\right),{\mathbf{b}}_2=\left(\begin{array}{cc}-t& 2\\ 0& 1\end{array}\right),\hfill \\ \hfill {\mathbf{c}}_1& =& \left(\begin{array}{cc}-t+1& 0\\ 1& 2\end{array}\right),{\mathbf{c}}_2=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right).\hfill \end{array}$$

Then, we deduce that

$${\mathbf{a}}_1^T\times {\mathbf{a}}_1=\left(\begin{array}{cc}t& 0\\ 2& 1\end{array}\right)\times \left(\begin{array}{cc}t& 2\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}{t}^2& 2t\\ 2t& 5\end{array}\right),$$

with the following eigenvalues:

$$\begin{array}{ccc}\hfill \lambda & =& {\displaystyle \frac{1}{2}}\sqrt{t^4+6t^2+25}+{\displaystyle \frac{1}{2}}{t}^2+{\displaystyle \frac{5}{2}},\hfill \\ \hfill \lambda & =& {\displaystyle \frac{1}{2}}{t}^2-{\displaystyle \frac{1}{2}}\sqrt{t^4+6t^2+25}+{\displaystyle \frac{5}{2}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {∥{\mathbf{a}}_1∥}_2& =& \sqrt{{\displaystyle \frac{\sqrt{t^4+6t^2+25}+t^2+5}{2}}},{∥{\mathbf{a}}_2∥}_2=\sqrt{{\displaystyle \frac{\sqrt{t^4-2t^2+25}+t^2+5}{2}}},\hfill \\ \hfill {∥{\mathbf{b}}_1∥}_2& =& t+1,{∥{\mathbf{b}}_2∥}_2=\sqrt{{\displaystyle \frac{\sqrt{t^4+6t^2+25}+t^2+5}{2}}},\hfill \\ \hfill {∥{\mathbf{c}}_1∥}_2& =& \sqrt{{\displaystyle \frac{\sqrt{{\left(t-1\right)}^4+2{\left(t-1\right)}^2+25}+{\left(t-1\right)}^2+5}{2}},}{∥{\mathbf{c}}_2∥}_2=1.618,\hfill \end{array}$$

$${∥w∥}_2=\sqrt{{\left(t+1\right)}^2+t^2},t\in [1,2],$$

$$\begin{array}{ccc}\hfill {∥{\mathbf{a}}_1∥}_2& =& \sqrt{{\displaystyle \frac{\sqrt{2^4+{62}^2+25}+2^2+5}{2}}}=5.972,{∥{\mathbf{a}}_2∥}_2=\sqrt{{\displaystyle \frac{\sqrt{t^4-2t^2+25}+t^2+5}{2}}},\hfill \\ \hfill {∥{\mathbf{b}}_1∥}_2& =& t+1,{∥{\mathbf{b}}_2∥}_2=\sqrt{{\displaystyle \frac{\sqrt{t^4+6t^2+25}+t^2+5}{2}}},\hfill \\ \hfill {∥{\mathbf{c}}_1∥}_2& =& \sqrt{{\displaystyle \frac{\sqrt{{\left(t-1\right)}^4+2{\left(t-1\right)}^2+25}+{\left(t-1\right)}^2+5}{2}},}{∥{\mathbf{c}}_2∥}_2=1.618.\hfill \end{array}$$

4. Conclusions

This paper establishes a set of verifiable sufficient conditions for the finite-time stability of a broad class of fractional Itô–Doob stochastic differential equations with time-varying delays. The analytical framework, based on a combination of Gronwall-type inequalities, Hölder’s inequality, and the Burkholder–Davis–Gundy (BDG) inequality, provides a powerful mechanism for bounding the system’s moments. In doing so, this work successfully extends classical integer-order stability theory into the fractional domain, while simultaneously accounting for the intricate interplay of stochastic noise and hereditary effects. The resulting integrated framework not only offers deeper analytical insights, but also provides a robust theoretical foundation for the stability analysis of complex dynamical systems characterized by memory, randomness, and non-local operators.

This work opens several promising avenues for future research. A natural extension involves applying the developed framework to more intricate system models, including those governed by variable-order fractional operators, which provide higher modeling fidelity for processes with evolving dynamic properties. Furthermore, the analysis could be extended to systems driven by Lévy noise or fractional Brownian motion, leading to a more comprehensive theory for multi-dimensional and non-Gaussian stochastic environments. Beyond model generalization, a parallel research direction involves exploring alternative analytical tools, such as Lyapunov’s second method tailored for fractional stochastic systems or fixed-point theorems, which could yield less conservative stability criteria and handle nonlinearities more effectively. These advancements would significantly broaden the framework’s applicability, paving the way for novel designs in robust control, fault-tolerant systems, and advanced signal processing algorithms for complex, memory-dependent processes.