Stability Analysis of Finite Time for a Class of Nonlinear Time-Delay Fractional-Order Systems

Abstract

In this study, we delve into the examination of Finite Time Stability (FTS) within a specific class of Fractional-Order Systems (FOS) with time delays. By applying a fixed-point theorem, we establish novel sufficient conditions to ensure FTS for time-delayed FOS within  $1<\sigma <2$. Moreover, we investigate the existence and uniqueness of global solutions for this particular system. To demonstrate the credibility of our results, we substantiate our findings through the presentation of two illustrative examples.

1. Introduction

The principles of fractional-order differentiation and integration are adept at elucidating the dynamics of a substantial class of infinite-dimensional dynamical systems [1]. Their utility spans diverse domains, including electricity, thermal systems, chemistry, signal processing, and control theory. These concepts find applications in various scenarios, including, but not limited to, asymptotic stability [2], finite-time stability [3,4], observer design [5], stabilization [6], and fault estimation [7]. The versatility of non-integer-order calculus renders it an invaluable tool for comprehending and analyzing the intricate behaviors exhibited by complex systems in scientific and engineering disciplines.

In the literature, the exploration of FTS for time-delay Fractional-Order Systems (FOS) has been pursued through various approaches and concepts. One notable example is the utilization of Lyapunov functions [4,8] and Gronwall inequalities [3,9,10,11] to analyze and understand the dynamics involved. Lyapunov functions were employed by the authors of [8] to develop a method for analyzing FTS within FOS that incorporates time-varying delay and nonlinear perturbation. This approach is rooted in the utilization of the Laplace transform and the “inf-sup” method. These authors formulated new delay-dependent criteria for analyzing the FTS of FOS with interval time-varying delays employing the suggested methodology.

Moreover, Thanh et al. introduced a groundbreaking perspective on FTS for time-delay FOS by proposing an innovative explanation based on Lyapunov functions in their work [4]. Expanding on the analysis of time-delay FOS, Naifar et al. conducted a comprehensive investigation in [3], where they employed the Caputo derivative and the generalized Gronwall inequality (CFD) to provide an analysis of the FTS associated with these systems. Furthermore, researchers in [10] delved into the realm of time-delay FOS and proposed a robust FTS approach, contributing valuable insights to the understanding of their dynamic behavior.

In a distinct study, detailed in [9], the authors showcased a demonstrative application of FTS principles to FOS neural networks with proportional delay. This demonstration not only provides theoretical advancements but also practical implications for real-world applications involving neural networks with fractional dynamics. Additionally, in [11], the authors established comprehensive criteria aimed at ensuring the FTS for a specific class of nonlinear fractional-order systems characterized by discrete-time delays. This work contributes valuable tools and guidelines for the analysis and verification of FTS in nonlinear fractional-order systems, further enriching the theoretical foundation in this field.

The format of this paper is as follows. In Section 2, we summarize our basic notions and definitions. In Section 3, we delve into the analysis of FTS within a specific class of time-delay FOS by leveraging the fixed-point technique. Furthermore, we establish the existence and uniqueness of global solutions for this particular class of systems. Drawing inspiration from the findings in [12,13], we harness a fixed-point theorem to systematically investigate the existence and uniqueness of global solutions, as well as the characteristics of FTS for time-delay FOS within the parameter range  $1<\sigma <2$. In Section 4, we illustrate our results with two examples.

2. Preliminaries

In this part, important notations and a few definitions and lemmas will be introduced since they will be used to ensure the existence and FTS of the solution for (3) in the subsequent sections.

Definition 1

([14]). Given  $\sigma >0$ and  $t_0\ge 0$, the Riemann–Liouville fractional integral for a function ζ is defined as

$$I_{t_0}^{\sigma }\zeta \left(t\right)=\frac{1}{\Gamma \left(\sigma \right)}{\int }_{t_0}^t{(t-\lambda )}^{\sigma -1}\zeta \left(\lambda \right)d\lambda \mathit{where}t>t_0$$

(1)

where  ${\displaystyle \Gamma \left(\sigma \right)={\int }_0^{\infty }{u}^{\sigma -1}{e}^{-u}du}$ is the gamma function, provided the right integral converges.

Definition 2

([14]). Given  $1<\sigma <2$ and  $t_0\ge 0$, the Caputo fractional derivative (CFD) of order σ for a continuous function  $\zeta :[t_0,+\infty )⟶\mathbb{R}$ is defined by

$${}^C{D}_{t_0}^{\sigma }\zeta \left(t\right)=\frac{1}{\Gamma (2-\sigma )}\frac{d^2}{d{t}^2}{\int }_{t_0}^t{(t-\lambda )}^{-\sigma +1}\left(\zeta \left(\lambda \right)-\zeta \left(t_0\right)-{\zeta }^{\prime }\left(t_0\right)(\lambda -t_0)\right)d\lambda ,t>t_0.$$

(2)

Lemma 1.

For  $\sigma >0$,  $\nu >0$ and  $H\left(t\right)=e^{\nu t}$, we have

$$I_0^{\sigma }H\left(t\right)\le \frac{H\left(t\right)}{{\nu }^{\sigma }}.$$

Remark 1.

Note that in [11], the authors derived FTS results for time-delay FOS with  $1<\sigma <2$ using Lemma 1. Furthermore, it is worth highlighting that the authors of [15,16] delved into the exploration of the existence and uniqueness of global solutions, along with the assessment of exponential boundedness, for time-delay FOS in the case where  $0<\sigma <1$. These investigations were conducted with the aid of Lemma 1.

Definition 3

([17]). Let us consider a nonempty set F. A mapping  $\alpha :F\times F\to [0,\infty ]$ is called a generalized metric if:

S1 

$\alpha ({\kappa }_1,{\kappa }_2)=0$ iff  ${\kappa }_1={\kappa }_2$;

S2 

$\alpha ({\kappa }_1,{\kappa }_2)=\alpha ({\kappa }_2,{\kappa }_1)$ for all  ${\kappa }_1,{\kappa }_2\in F$;

S3 

$\alpha ({\kappa }_1,{\kappa }_3)\le \alpha ({\kappa }_1,{\kappa }_2)+\alpha ({\kappa }_3,{\kappa }_2)$ for all  ${\kappa }_1,{\kappa }_2,{\kappa }_3\in F$.

Theorem 1

([17]). Let us consider a generalized complete metric space  $(F,\alpha )$. Assume that  $\Phi :F\to F$ is a contractive operator with  $k<1$. If there is an integer  $m_0\ge 0$ such that  $\alpha ({\Phi }^{m_0+1}{u}_0,{\Phi }^{m_0}{u}_0)<\infty$ for some  $u_0\in F$, then:

(a) 

${\Phi }^n{u}_0$ converges to a fixed point  $u_1$ of Φ;

(b) 

$u_1$ is the unique fixed point of Φ in  $F^{*}:=\{u_2\in F:\alpha ({\Phi }^{m_0}{u}_0,u_2)<\infty \}$;

(c) 

If  $u_2\in F^{*}$, then  $\alpha (u_1,u_2)\le \frac{1}{1-d}\alpha (\Phi u_2,u_2)$.

Consider the following time-delay FOS:

$${}^C{D}_{t_0}^{\sigma }x\left(t\right)=B_{0}x\left(t\right)+B_{1}x(t-\iota \left(t\right))+B_2\nu \left(t\right)+g(t,x\left(t\right),x(t-\iota \left(t\right)),\nu \left(t\right)),t\ge t_0,$$

(3)

with the initial condition  $x\left(y\right)=\zeta \left(y\right)$ for  $t_0-\iota \le y\le t_0$ and  $x^{\prime }\left(t_0\right)=x_1$, where  $1<\sigma <2$,  $\iota \left(t\right)$ is continuous,  $0\le \iota \left(t\right)\le \iota$,  $\nu \left(t\right)\in {\mathbb{R}}^q$,  $\zeta \in C^1\left([t_0-\iota ,t_0],{\mathbb{R}}^m\right)$,  $B_0\in {\mathbb{R}}^{m\times m}$,  $B_1\in {\mathbb{R}}^{m\times m}$, and  $B_2\in {\mathbb{R}}^{m\times q}$.

In the following, we suppose that the functionals g and  $\nu$ satisfy the next conditions:

(i) 

g is continuous,  $g(t,0,0,0)=0$, and for all  $(t,{\xi }_1,{\xi }_2,{\xi }_3,{\tilde{\xi }}_1,{\tilde{\xi }}_2,{\tilde{\xi }}_3)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^m\times {\mathbb{R}}^m\times {\mathbb{R}}^q\times {\mathbb{R}}^m\times {\mathbb{R}}^m\times {\mathbb{R}}^q$ such that

$$∥g(t,{\xi }_1,{\xi }_2,{\xi }_3)-g(t,{\tilde{\xi }}_1,{\tilde{\xi }}_2,{\tilde{\xi }}_3)∥\le \mu \left(t\right)\left(∥{\xi }_1-{\tilde{\xi }}_1∥+∥{\xi }_2-{\tilde{\xi }}_2∥+∥{\xi }_3-{\tilde{\xi }}_3∥\right),$$

(4)

where  $\mu$ is a continuous function and  $∥.∥$ is the Euclidean norm.

(ii) 

For all  $t\ge t_0$,  $\nu$ is a continuous function, and there exist  $\varrho >0$ such that

$${\nu }^T\left(t\right)\nu \left(t\right)\le {\varrho }^2.$$

(5)

Remark 2.

The integral equation associated with the time-delay System (3) is given by (see [11])

$$\begin{array}{ccc}\hfill x\left(t\right)& =& \zeta \left(t_0\right)+(t-t_0)x_1+\frac{1}{\Gamma \left(\sigma \right)}{\int }_{t_0}^t{(t-s)}^{\sigma -1}\left(B_{0}x\left(s\right)+B_{1}x(s-\iota \left(s\right))\right.\hfill \\ & & \left.+B_2\nu \left(s\right)+g(s,x\left(s\right),x(s-\iota \left(s\right)),\nu \left(s\right))\right)ds,t\in [t_0,T],\hfill \\ \hfill x\left(t\right)& =& \zeta \left(t\right),t\in [t_0-\iota ,t_0].\hfill \end{array}$$

(6)

Definition 4.

The time-delay System (3) is FTS w.r.t.  $\{{\delta }_1,{\delta }_2,\varrho ,T\}$,  ${\delta }_1<{\delta }_2$ if

$$\parallel \zeta \parallel \le {\delta }_{1}and\parallel x_1\parallel \le {\delta }_1$$

imply

$$\parallel x\left(t\right)\parallel \le {\delta }_2,\mathit{for}{t}_0\le t\le T,$$

for every ν satisfying (5).

3. Main Results

In this part, our goal is to show the existence, uniqueness, and FTS of the associated solutions of System (3). Let us consider

$$b_p=\underset{l\in [t_0,T]}{max}\left(\mu \left(l\right)+\parallel B_p\parallel \right),\mathrm{for}p\in \{0,1,2\}.$$

(7)

The following theorem shows both the existence and uniqueness of the solutions and establishes the FTS of System (3).

Theorem 2.

The time-delay System (3) is FTS w.r.t.  $\{{\delta }_1,{\delta }_2,\varrho ,T\}$,  ${\delta }_1<{\delta }_2$ if there is  $\theta >0$ with

$$D({\delta }_1,\varrho )=\left(1-t_0+c_1{e}^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(T-t_0)}+T\right){\delta }_1+c_2{e}^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(T-t_0)}\varrho \le {\delta }_2,$$

(8)

where

$$c_1=\frac{M\left(b_0+b_1+\theta \right)(b_0+b_1)\left(1+T-t_0\right)}{\theta \Gamma (\sigma +1)},c_2=\frac{M\left(b_0+b_1+\theta \right)b_2}{\theta \Gamma (\sigma +1)}$$

and

$$M=\underset{\tau \in [t_0,T]}{sup}\left(\frac{{(\tau -t_0)}^{\sigma }}{e^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(\tau -t_0)}}\right).$$

Proof. 

Let  $\zeta \in C\left([t_0-\iota ,t_0],{\mathbb{R}}^m\right)$ and  $x_1\in {\mathbb{R}}^m$ such that  $∥\zeta ∥\le {\delta }_1$ and  $∥x_1∥\le {\delta }_1$. Let  $F:=C\left([t_0-\iota ,T],{\mathbb{R}}^m\right)$ and consider the metric  $\alpha$ on F by

$$\alpha (y_1,y_2)=inf\left\{l\in [0,\infty ]:\frac{∥y_1\left(t\right)-y_2\left(t\right)∥}{\psi \left(t\right)}\le l,\forall t\in [t_0-\iota ,T]\right\},$$

where  $\psi$ is given by  $\psi \left(y\right)=1$ for  $y\in [t_0-\iota ,t_0]$ and  $\psi \left(y\right)=e^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(y-t_0)}$ for  $t_0\le y\le T$. Let us consider the operator  $\mathcal{G}:F\to F$, defined as follows:

$$\begin{array}{ccc}\hfill \left(\mathcal{G}z\right)\left(\tau \right)& =& \zeta \left(t_0\right)+(\tau -t_0)x_1+\frac{1}{\Gamma \left(\sigma \right)}{\int }_{t_0}^{\tau }{(\tau -s)}^{\sigma -1}\left(B_{0}z\left(s\right)+B_{1}z(s-\iota \left(s\right))\right.\hfill \\ & & \left.+B_2\nu \left(s\right)+g(s,z\left(s\right),z(s-\iota \left(s\right)),\nu \left(s\right))\right)ds,\mathrm{for}{t}_0\le \tau \le T\hfill \\ \hfill \left(\mathcal{G}z\right)\left(\tau \right)& =& \zeta \left(\tau \right),\mathrm{for}{t}_0-\iota \le \tau \le t_0.\hfill \end{array}$$

(9)

Note that  $(F,\alpha )$ is a generalized complete metric space,  $\mathcal{G}$ is well-defined,  $\alpha \left(\mathcal{G}{z}_0,z_0\right)<\infty$, and  $\left\{z_1\in F:\alpha (z_0,z_1)<\infty \right\}=F$ for all  $z_0\in F$.

Let  $z_1,$ $z_2\in F$. It is easy to show that  $\left(\mathcal{G}{z}_1\right)\left(\tau \right)-\left(\mathcal{G}{z}_2\right)\left(\tau \right)=0$ for  $t_0-\iota \le \tau \le t_0$. Moreover, for  $t_0\le \tau \le T$, we have

$$\begin{array}{ccc}& & ∥\left(\mathcal{G}{z}_1\right)\left(\tau \right)-\left(\mathcal{G}{z}_2\right)\left(\tau \right)∥\hfill \\ & & \le {\int }_{t_0}^{\tau }\frac{{(\tau -l)}^{\sigma -1}}{\Gamma \left(\sigma \right)}\left(\left(\mu \left(l\right)+∥B_0∥\right)∥z_1\left(l\right)-z_2\left(l\right)∥\right.\hfill \\ & & \left.+\left(\mu \left(l\right)+∥B_1∥\right)∥z_1(l-\iota \left(l\right))-z_2(l-\iota \left(l\right))∥\right)dl\hfill \\ & & \le b_0{\int }_{t_0}^{\tau }{(\tau -l)}^{\sigma -1}\frac{∥z_1\left(l\right)-z_2\left(l\right)∥}{\Gamma \left(\sigma \right)}dl\hfill \\ & & +b_1{\int }_{t_0}^{\tau }{(\tau -l)}^{\sigma -1}\frac{∥z_1(l-\iota \left(l\right))-z_2(l-\iota \left(l\right))∥}{\Gamma \left(\sigma \right)}dl\hfill \\ & & \le \frac{b_0}{\Gamma \left(\sigma \right)}{\int }_{t_0}^{\tau }{(\tau -l)}^{\sigma -1}\frac{∥z_1\left(l\right)-z_2\left(l\right)∥}{\psi \left(l\right)}\psi \left(l\right)dl\hfill \\ & & +\frac{b_1}{\Gamma \left(\sigma \right)}{\int }_{t_0}^{\tau }{(\tau -l)}^{\sigma -1}\frac{∥z_1(l-\iota \left(l\right))-z_2(l-\iota \left(l\right))∥}{\psi (l-\iota (l\left)\right)}\psi (l-\iota \left(l\right))dl\hfill \\ & & \le \frac{b_0}{\Gamma \left(\sigma \right)}\alpha (z_1,z_2){\int }_{t_0}^{\tau }{(\tau -l)}^{\sigma -1}\psi \left(l\right)dl\hfill \\ & & +\frac{b_1}{\Gamma \left(\sigma \right)}\alpha (z_1,z_2){\int }_{t_0}^{\tau }{(\tau -l)}^{\sigma -1}\psi (l-\iota \left(l\right))dl\hfill \\ & & \le \frac{b_0}{\Gamma \left(\sigma \right)}\alpha (z_1,z_2){\int }_{t_0}^{\tau }{e}^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(s-t_0)}{(\tau -s)}^{\sigma -1}ds\hfill \\ & & +\frac{b_1}{\Gamma \left(\sigma \right)}\alpha (z_1,z_2){\int }_{t_0}^{\tau }{e}^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(s-t_0)}{(\tau -s)}^{\sigma -1}ds\hfill \\ & & \le \frac{b_0+b_1}{\Gamma \left(\sigma \right)}\alpha (z_1,z_2){\int }_0^{\tau -t_0}{e}^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}s}{(\tau -t_0-s)}^{\sigma -1}ds.\hfill \end{array}$$

(10)

We obtain from Lemma 1

$${\int }_0^{\tau -t_0}{e}^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}s}{(\tau -t_0-s)}^{\sigma -1}ds\le \frac{\Gamma \left(\sigma \right)}{b_0+b_1+\theta }{e}^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(\tau -t_0)}.$$

Therefore,

$$∥\left(\mathcal{G}{z}_1\right)\left(\tau \right)-\left(\mathcal{G}{z}_2\right)\left(\tau \right)∥\le \frac{b_0+b_1}{b_0+b_1+\theta }\alpha (z_1,z_2)e^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(\tau -t_0)}.$$

Thus,

$$\alpha (\mathcal{G}{z}_1,\mathcal{G}{z}_2)\le \frac{b_0+b_1}{b_0+b_1+\theta }\alpha (z_1,z_2).$$

Hence,  $\mathcal{G}$ is contractive.

Let  $x_0$ be a function given by

$$\begin{array}{ccc}\hfill x_0\left(\tau \right)& =& \zeta \left(t_0\right)+(\tau -t_0)x_1,\mathrm{for}\tau \in [t_0,T],\hfill \\ \hfill x_0\left(\tau \right)& =& \zeta \left(\tau \right),\mathrm{for}{t}_0-\iota \le \tau \le t_0.\hfill \end{array}$$

It is clear that  $\left(\mathcal{G}{x}_0\right)\left(\tau \right)-x_0\left(\tau \right)=0$ for  $\tau \in [t_0-\iota ,t_0]$. However, for  $t_0\le s\le T$, we have

$$\begin{array}{ccc}\hfill ∥\left(\mathcal{G}{x}_0\right)\left(s\right)-x_0\left(s\right)∥& =& \parallel \frac{1}{\Gamma \left(\sigma \right)}{\int }_{t_0}^s{(s-\tau )}^{\sigma -1}\left(B_0{x}_0\left(\tau \right)+B_1{x}_0(\tau -\iota \left(\tau \right))\right.\hfill \\ & & +\left.B_2\nu \left(\tau \right)+g\left(\tau ,x_0\left(\tau \right),x_0(\tau -\iota \left(\tau \right)),\nu \left(\tau \right)\right)\right)d\tau \parallel \hfill \\ & \le & \frac{1}{\Gamma \left(\sigma \right)}{\int }_{t_0}^s{(s-l)}^{\sigma -1}\left(\left(b_0+b_1\right)\left(∥\zeta ∥+(T-t_0)\parallel x_1\parallel \right)+b_2\varrho \right)dl\hfill \\ & \le & \frac{\left(\left(b_0+b_1\right)\left(\parallel \zeta \parallel +(T-t_0)\parallel x_1\parallel \right)+b_2\varrho \right)}{\Gamma (\sigma +1)}{(s-t_0)}^{\sigma }.\hfill \end{array}$$

(11)

Consequently,

$$\frac{∥\left(\mathcal{G}{x}_0\right)\left(s\right)-x_0\left(s\right)∥}{\psi \left(s\right)}\le \frac{\left(\left(b_0+b_1\right)\left(\parallel \zeta \parallel +(T-t_0)\parallel x_1\parallel \right)+b_2\varrho \right)}{\Gamma (\sigma +1)}M.$$

Therefore,

$$\alpha (\mathcal{G}{x}_0,x_0)\le \frac{\left(\left(b_0+b_1\right)\left(\parallel \zeta \parallel +(T-t_0)\parallel x_1\parallel \right)+b_2\varrho \right)}{\Gamma (\sigma +1)}M.$$

It follows from Theorem 1 that there is a unique solution x of (3) with initial conditions  $\zeta$,  $x_1$ such that

$$\begin{array}{ccc}\hfill \alpha (x,x_0)& \le & \frac{(b_0+b_1+\theta )}{\theta }\frac{\left[\left(b_0+b_1\right)\left(\parallel \zeta \parallel +(T-t_0)\parallel x_1\parallel \right)+b_2\varrho \right]}{\Gamma (\sigma +1)}M\hfill \\ & \le & c_1{\delta }_1+c_2\varrho .\hfill \end{array}$$

Hence,

$$∥x_0\left(t\right)-x\left(t\right)∥\le \left(c_1{\delta }_1+c_2\varrho \right)e^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(t-t_0)},\mathrm{for}\mathrm{all}{t}_0\le t\le T.$$

Therefore, for every  $t\in [t_0,T]$

$$\begin{array}{ccc}\hfill ∥x\left(t\right)∥& \le & ∥x_0\left(t\right)∥+∥x\left(t\right)-x_0\left(t\right)∥\hfill \\ & \le & \left(1+T-t_0+c_1{e}^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(t-t_0)}\right){\delta }_1+c_2{e}^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(t-t_0)}\varrho .\hfill \end{array}$$

Thus, if (8) holds, then

$$∥x\left(t\right)∥\le {\delta }_2,\mathrm{for}\mathrm{all}{t}_0\le t\le T,$$

and this completes the proof. □

Remark 3.

For  $t\ge t_0$, we have

$$\frac{{(t-t_0)}^{\sigma }}{e^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(t-t_0)}}\le \frac{{\sigma }^{\sigma }{e}^{-\sigma }}{b_0+b_1+\theta }.$$

Then,

$$c_1\le \frac{{\sigma }^{\sigma }{e}^{-\sigma }}{\Gamma (\sigma +1)\theta }(b_0+b_1)(1+T-t_0)\mathit{and}{c}_2\le \frac{{\sigma }^{\sigma }{e}^{-\sigma }}{\Gamma (\sigma +1)\theta }{b}_2.$$

Thus, the Condition (8) can be relaxed by

$$\begin{array}{ccc}\hfill C({\delta }_1,\varrho )& =& \left(1+T-t_0+\frac{{\sigma }^{\sigma }{e}^{-\sigma }}{\Gamma (\sigma +1)\theta }(b_0+b_1)(1+T-t_0)e^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(T-t_0)}\right){\delta }_1\hfill \\ & & +\frac{{\sigma }^{\sigma }{e}^{-\sigma }}{\Gamma (\sigma +1)\theta }{b}_2{e}^{{(b_0+b_1+\theta )}^{\frac{1}{\sigma }}(T-t_0)}\varrho \le {\delta }_2.\hfill \end{array}$$

(12)

4. Examples

In this section, we illustrate two examples to demonstrate the applicability of our major results of the theory stated in Section 3. Moreover, we conduct a simulation of both examples based on [18,19,20].

Example 1.

Let  $\sigma =1.6$,  $\iota \left(s\right)=0.1,$ and  $x_1=(0,0)$. Let us consider the following time-delay FOS of order σ:

$${}^C{D}_{t_0}^{\sigma }x\left(t\right)=B_{0}x\left(t\right)+B_{1}x(t-\iota \left(t\right))+B_2\nu \left(t\right)+g(t,x\left(t\right),x(t-\iota \left(t\right)),\nu \left(t\right)),t\ge 0,$$

(13)

withthe initial condition

$$\zeta \left(t\right)={\left(6\times {10}^{-2},0\right)}^T,\mathit{for}t\in [-{10}^{-1},0],$$

where the function g and the perturbation ν are given as follows:

$$g(s,x\left(s\right),x(s-\iota \left(s\right)),\nu \left(s\right))=0.01{\left(sin\left(x_2(s-\iota \left(s\right))\right),sin\left(x_1\left(s\right)\right)\right)}^T,\nu \left(t\right)={(0.1,0.1)}^T.$$

and

$$B_0=\left(\begin{array}{cc}0& -0.1\\ 0.2& 0\end{array}\right),B_1=\left(\begin{array}{cc}0.3& 0\\ 0.2& 0\end{array}\right),B_2=\left(\begin{array}{cc}0.2& 0\\ 0.1& 0\end{array}\right).$$

Hence, from (7), it is straightforward to verify that  $b_0=0.21$,  $b_1=0.3706$, and  $b_2=0.2336$. For  $\theta =1$,  $\varrho =1$,  ${\delta }_1=0.2$,  ${\delta }_2=50$, and  $T=4.61$, we obtain  $C({\delta }_1,\varrho )\simeq 49.9<{\delta }_2$. It follows from Theorem 2 that the time-delay FOS (13) is FTS w.r.t.  $\left(0.2,50,1,4.61\right)$.

By using the above parameters and with the time step  $2\times {10}^{-6}$, we show in Figure 1 the trajectory simulations of the associated solution of the time-delay FOS (13). Moreover, in Figure 2, we depict the trajectory of  $∥x\left(t\right)∥$. According to Definition 4 of the FTS, it is evident from Figure 2 that  $∥x\left(t\right)∥$ does not exceed  ${\delta }_2$ for each value of  $t\in [0,T]$. Consequently, Figure 2 shows the FTS of System (13) with respect to  $\left(0.2,50,1,4.61\right)$. Then, the simulation results verify the effectiveness of the theoretical results.

Example 2.

Let  $\sigma =1.4$,  $\iota \left(s\right)=0.1,$ and  $x_1=(0,0,0)$. Let us consider the following time-elay FOS of order σ:

$${}^C{D}_{t_0}^{\sigma }x\left(t\right)=B_{0}x\left(t\right)+B_{1}x(t-\iota \left(t\right))+B_2\nu \left(t\right)+g(t,x\left(t\right),x(t-\iota \left(t\right)),\nu \left(t\right)),t\ge 0,$$

(14)

with the initial condition

$$\zeta \left(\tau \right)={\left(3\times {10}^{-2},0,4\times {10}^{-2}\right)}^T,\mathit{for}\tau \in [-{10}^{-1},0],$$

where the function g and the perturbation ν are given as follows:

$$g(s,x\left(s\right),x(s-\iota \left(s\right)),\nu \left(s\right))=0.01{\left(sin\left(x_2(s-\iota \left(s\right))\right),sin\left(x_3\left(s\right)\right),sin\left(x_1(s-\iota \left(s\right))\right)\right)}^T,$$

$$\nu \left(t\right)={(0.1,0,0.1)}^T$$

and

$$B_0=\left(\begin{array}{ccc}0.2& -0.1& 0.15\\ -0.2& 0.15& 0.05\\ 0.02& 0.1& -0.25\end{array}\right),B_1=\left(\begin{array}{ccc}-0.1& -0.2& 0.3\\ 0.15& 0.2& 0.15\\ -0.15& 0.2& -0.25\end{array}\right),$$

$$B_2=\left(\begin{array}{ccc}-0.05& 0.06& 0.1\\ -0.07& -0.08& 0.09\\ 0.11& 0.13& 0.15\end{array}\right).$$

Therefore, from (7), it is easy to prove that  $b_0=0.3766$,  $b_1=0.4928$, and  $b_2=0.2521$.

For  $\theta =1$,  $\varrho =1$,  ${\delta }_1=0.3$,  ${\delta }_2=100$, and  $T=3.448$, we obtain  $C({\delta }_1,\varrho )\simeq 99<{\delta }_2$. It follows from Theorem 2 that the time-delay FOS (14) is FTS w.r.t.  $\left(0.3,100,1,3.448\right)$.

By utilizing the specified parameters and employing a time step of  $2\times {10}^{-6}$, the trajectory simulations for the solution associated with the time-delay FOS (14) are presented in Figure 3. Additionally, Figure 4 illustrates the trajectory of  $∥x\left(t\right)∥$. In accordance with Definition 4 of the FTS, it is evident from Figure 4 that  $∥x\left(t\right)∥$ does not exceed  ${\delta }_2$ for each value of  $t\in [0,T]$. Consequently, Figure 4 displays the FTS of System (14) with respect to  $\left(0.3,100,1,3.448\right)$. The simulation results affirm the effectiveness of the theoretical findings.

5. Conclusions

In this work, we delved into the exploration of a novel and robust FTS for time-delay FOS. Our findings highlight the feasibility of constructing a novel essential constraint crucial for ensuring the robustness of FTS in these systems, employing an innovative fixed-point theory method. To underscore the practical applicability and effectiveness of our approach, we presented two distinct examples that vividly demonstrate the precision and reliability of the analytical framework established in this study. This work contributes not only to the theoretical advancements in FTS but also offers valuable insights for practical implementations in the analysis and control of time-delay FOS. In future work, we plan to expand the application of our robust FTS methodology to diverse complex systems, refine the proposed constraint under varying conditions, and validate its practical utility through experiments or case studies.