A Sufficient Condition for the Practical Stability of Riemann-Liouville Fractional Nonlinear Systems with Time Delays

Abstract

This study addresses the practical stability analysis of Riemann-Liouville fractional-order nonlinear systems with time delays. We first establish a rigorous formulation of initial conditions that aligns with the properties of Riemann-Liouville fractional derivatives. Subsequently, a generalized definition of practical stability is introduced, specifically tailored to accommodate the hybrid dynamics of fractional calculus and time-delay phenomena. By constructing appropriate Lyapunov-Krasovskii functionals and employing an enhanced Razumikhin-type technique, we derive sufficient conditions ensuring practical stability in the $L_p$-norm sense. The theoretical findings are validated through illustrative example for fractional order nonlinear systems with time delays.

1. Introduction

The study of the stability of fractional-order systems, particularly those with time delays, is a significant research topic due to its applications in various fields such as control systems [1], signal processing [2,3], and biological systems [4] and so on, more applications can be found in refs. [5,6,7,8,9]. The Riemann-Liouville fractional nonlinear time delay systems are a class of systems where the order of the derivative is a fraction rather than an integer, and they often exhibit complex dynamic behaviors due to the presence of time delays. One of the key challenges in analyzing these systems is establishing sufficient(or sufficient and necessary) conditions for their stability. Practical stability, in particular, refers to the system’s ability to remain within a certain bounded region over time, despite the presence of disturbances or uncertainties. This type of stability is crucial for ensuring that the system performs reliably in real-world applications [10,11,12,13,14].

In the context of fractional-order systems with time delays, several approaches have been proposed to address the stability problem. For instance, the work on the Mittag-Leffler stability of impulsive nonlinear fractional-order systems with time delays provides a framework for analyzing the stability of such systems by establishing novel sufficient conditions using mathematical techniques like fixed point theory and Lyapunov methods [1]. The investigation of input-to-state stability and integral input-to-state stability for switched nonlinear neutral systems with multiple time-varying delays and asynchronous switching offers insights into handling the complexities introduced by switching phenomena and neutral terms [2]. This research highlights the importance of considering the effects of switches and time-varying delays on the stability properties of the system. The practical stabilization techniques of the time delayed fractional order systems with constraints is investigated by parametric controllers. It has been explored for fractional-order systems with time delays. By utilizing methods like partial eigenvalue assignment, researchers aim to design controllers that ensure the system’s stability while meeting specific performance criteria [10]. These approaches demonstrate the potential for achieving practical stability through careful controller design and parameter optimization. Practical stability or the fractional order systems with state independent delay is studied in [11]. Variational Lyapunov method is discussed for the stability of the fractional order systems is investigated in [15]. Practical stability of the Riemann-Liouville fractional order systems with delay is studied in [16,17] by using different Razumikhin functional methods. Ref. [18] addresses output feedback practical stabilization of continuous-time nonlinear systems by choosing control actions from a finite set and [19,20,21,22] for uncertain linear systems. Refs. [21,23,24,25,26,27,28,29,30] discuss the practical stabilization for fractional systems with time delay by parametric controllers and other mathematical techniques, ref. [27] address the practical stability for positive linear systems, refs. [31] for fuzzy systems and [32] for Markovian jump systems with time delays, [33] for continuous descriptor linear systems.

In summary, the study of sufficient conditions for the practical stability of Riemann-Liouville fractional nonlinear systems with time delays is enriched by various methodologies that address the unique challenges posed by fractional derivatives and time delays. The integration of theoretical frameworks and practical stabilization techniques continues to advance our understanding and ability to control these complex systems effectively.

From the above observations, fractional order systems have gained considerable attention due to their ability to more accurately model real-world phenomena compared to integer-order models [1]. Among these, Riemann-Liouville fractional order systems stand out for their unique characteristics and applications in various fields such as physics, engineering, and control theory. However, the practical stability analysis of these systems, especially when time delays are involved, remains a challenging task. This paper aims to address the practical stability of Riemann-Liouville fractional order nonlinear systems with time delays. We define the initial conditions for the Riemann-Liouville fractional derivative in an appropriate manner and introduce the concept of practical stability for such systems. Using Lyapunov functions and an improved Razumikhin method, we derive a sufficient condition for practical stability. Illustrative examples are provided to validate the rightness of our results.

The principal contributions of this work are twofold:

1.

We introduce a rigorous definition of practical stability for Riemann-Liouville fractional-order nonlinear systems with time-varying delays, explicitly accounting for the non-local memory effects inherent in fractional derivatives. This extends classical stability concepts to accommodate the hybrid dynamics of fractional calculus and phenomena with time delays.

2.

By developing an enhanced Razumikhin-type approach combined with Lyapunov-Krasovskii functional analysis, we derive novel sufficient conditions for verifying practical stability in the $L_p$-norm sense. These conditions provide a computationally tractable criterion with reduced conservatism compared to existing fractional-order delay system stability tests.

The remainder of this manuscript is structured as follows: Section 2 establishes the mathematical preliminaries, including fundamental properties of Riemann-Liouville derivatives and a formal definition of practical stability tailored to fractional delay systems. Section 3 presents the core theoretical results, deriving stability conditions through a systematic application of the proposed Razumikhin-functional framework. Section 4 validates the theoretical findings via numerical simulations on a fractional-order neural network model with variable communication delays, demonstrating improved convergence rates over integer-order counterparts. Section 5 concludes with a summary of key insights and discussions on potential extensions to distributed delay systems and fractional-order control applications.

Notations: $R^n$ and $R_{+}$ represent the n dimensional space of real vectors and the positive real numbers respectively. $f\in C([a,b],R^n)$ means the f is a continuous mapping from interval $[a,b]$ onto $R^n$, ${\parallel \varphi (t)\parallel }_0={max}_{t\in [-\tau ,0]}\parallel \varphi (t)\parallel$ means the 0 norm of the function $\varphi (t)$ interval $[-\tau ,0]$, $[h(t)]$ is the integer function of $h(t),{}_0{}^{RL}D_t^{q}x(t)$ is the Riemann-Liouville fractional derivatives of $x(t)$, ${}_{t_0}{}^{GL}D_t^q$ is the Grünwald–Letnikov fractional derivative and ${}_{t_0}{}^{GL}D_t^q$ is the is the fractional Dini derivative, $C_{1-q}([0,T),R)$ is a set of continuous functional from interval $[0,T)\to R$, $M,S_{\beta }$ are sets of functions.

2. Preliminary Knowledge

Consider the Riemann-Liouville(R-L) fractional nonlinear system with time delay and $q\in (0,1)$:

$${}_0{}^{RL}D{}_t{}^{q}x(t)=f(t,x(t)),t>0,$$

(1)

the initial condition is

$$x(t)=\phi (t),t\in [-\tau ,0],$$

$$\underset{t\to t_0^{+}}{lim}\left[t^{1-q}x(t)\right]={\displaystyle \frac{\phi (0)}{\mathsf{\Gamma }(q)}},$$

where $x(t)\in R^n$ is the state of the system (1), ${}_0{}^{RL}D{}_t{}^{q}x(t)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-q)}}{\displaystyle \frac{d}{dt}}{\int }_0^t{(t-s)}^{(-q)}m(s)ds,t>0$ is the Riemann-Liouville fractional derivative, $\mathsf{\Gamma }()$ is the Gamma function, function $f:R_{+}\times R^n\to R^n$, $\tau$ is the time delay, $x_t(\theta )=x(t+\theta )$, $\theta \in [-\tau ,0]$, $\phi (t)\in C([-\tau ,0],R^n)$ is the initial state of the system.

Definition 1. 

[11] Given the definition of the following sets

$$C_{1-q}([0,T),R)=\left\{m(t):[0,T)\to R,t^{1-q}m(t)\in C([0,T),R)\right\},0<T\le \infty ,$$

$${\parallel \phi (t)\parallel }_0=\underset{t\in [-\tau ,0]}{max}\parallel \phi (t)\parallel ,$$

$$M=\left\{\omega \mid \omega \in C(R_{+},R_{+}),\mathit{function}\omega (s)\mathit{is}\mathit{strictly}\mathit{increasing}\mathit{and}\omega (0)=0\right\}.$$

$S_{\beta }=\{x\in R^n:\parallel x\parallel \le \beta \} \mathit{for\; any\; given}$$\beta >0$.

Lemma 1. 

[15] Let $m(t)\in C_{1-q}([0,T),R)$, for any $t_1\in (0,T)$, when $0\le t\le t_1$, $m(t)<0$, if $m(t_1)=0$ holds, then ${}_0{}^{RL}D{}_t{}^{q}m(t)|_{t=t_1}\ge 0.$

Lemma 2. 

[9] If ${}_0{}^{RL}D{}_t{}^{q}m(t)$ exists on interval (0,T), then

$${}_0{}^{RL}D{}_t{}^{q}m(t)={}_{t_0}{}^{GL}D{}_t{}^{q}m(t)={}_{t_0}{}^{GL}D{}_{+}{}^{q}m(t)$$

holds for $t\in (0,T)$, where

${}_{t_0}{}^{GL}D{}_t{}^q={lim}_{h\to 0}{\displaystyle \frac{1}{h^q}}{\sum }_{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}m(t-rh),C_q^r={\displaystyle \frac{q(q-1)\dots (q-r+1)}{r!}},t>0$ is the Grünwald–Letnikov fractional derivative , and

${}_{t_0}{}^{GL}D{}_t{}^q=limsu{p}_{h\to 0^{+}}{\displaystyle \frac{1}{h^q}}{\sum }_{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}m(t-rh),C_q^r={\displaystyle \frac{q(q-1)\dots (q-r+1)}{r!}},t>0$ is the fractional Dini derivative.

The following practical stability definition is given for system (1).

Definition 2. 

Assume there exists a positive pair $(\lambda ,\rho )$ with $0<\lambda <\rho$, if there exists a $T>0$ such that for any initial state $\phi \in ([-\tau ,0],R^n)$, the inequality ${\parallel \phi \parallel }_0<\lambda$ holds, and when $t\ge T$, the inequality $\parallel x(t;\phi )\parallel <\rho$ holds, then system (1) is said to be practically stable with respect to the positive pair $(\lambda ,\rho )$.

3. Main Results

Based on the afore mentioned relevant definitions and lemmas, the determination of practical stability for system (1) is as follows:

Theorem 1. 

Let $V:R_{+}\times R_n\to R_{+}$ be a continuous function that is locally Lipschitz continuous in x uniformly in t. Suppose the following conditions hold:

(i) there exists a $T>0$, such that the following inequality

$$a(\parallel x\parallel )\le V(t,x),$$

(2)

holds for any $t>T$, $x(t)\in R^n$, where $a\in M$.

(ii) there exists a constant $\lambda \ge 0$, for any function $\gamma \in C_{1-q}([0,\infty ),R^n)$ and

$$\underset{t\to 0^{+}}{lim}\left[t^{1-q}\gamma (t)\right]\in S_{\lambda },$$

such that the following

$$\underset{t\to 0^{+}}{lim}{t}^{1-q}V(t,\gamma (t))<b(\lambda ),$$

(3)

holds, where $b\in M$.

(iii) there exists a constant $C\ge 0$, for any initial state condition $\phi (t)\in S_{\lambda }$ and $x(t)=x(t;\phi ),t\in [0,\infty )$ is the corresponding solution to (1), for any $t>0$, and the inequality

$$t^{1-q}V(t,x(t))\ge {(t+s)}^{1-q}V(t+s,x(t+s)),$$

(4)

holds such that

$$D^{+}V(t,x(t))<{\displaystyle \frac{C}{t^q\mathsf{\Gamma }(1-q)}},$$

(5)

where $s\in [-min\left\{\tau ,t\right\},0]$, $D^{+}V(t,x(t))$ is the Dini fractional derivative for the Lyapunov function of system (1):

$$D^{+}V(t,x(t))=\underset{h\to 0}{lim}sup{\displaystyle \frac{1}{h^q}}\left[V(t,x(t))-\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}V(t-rh,x(t)-h^{q}f(t,x_t))\right],$$

(6)

then the Riemann-Liouville fractional nonlinear system (1) with time delay is practically stable with respect to positive pair $(\lambda ,a^{-1}(C+b(\lambda )))$.

Proof. 

The basic idea of proof is: by constructing a suitable functional $V(x,t)$ that satisfies

$$a\left(∥x(t)∥\right)\le V(t,x(t))<C+{\displaystyle \frac{b(\lambda )}{t^{1-q}}},t>T,$$

the left side of the above inequality is the condition (i) of the Theorem 1, we only need prove the right side of the inequality, we use the method of contradiction to prove it, this is the main part of the proof. When $t\ge 1$, $t^{1-q}>1$,

$$∥x(t)∥\le a^{-1}\left(C+b(\lambda )\right),t\in [max\{1,T\},\infty ).$$

it can be obtained that the solution $x(t)$ satisfies the definition of asymptotic stability.

Let $\phi (t)$ be the initial state function of system (1) which satisfies ${\parallel \phi (t)\parallel }_0<\lambda$, and let $x(t)$ represent the corresponding solution that satisfies the initial conditions of system (5.1). According to the condition (ii) of the Theorem 1, the inequality

$$\underset{t\to 0^{+}}{lim}{t}^{1-q}V(t,\gamma (t))<b(\lambda )$$

holds. Based on the definition of the limit, there exists a $\delta >0$ such that $t^{1-q}V(t,x(t))<b(\lambda ),0<t<\delta$, hence there exists a constant $C\ge 0$ such that

$$V(t,x(t))<C+{\displaystyle \frac{b(\lambda )}{t^{1-q}}},t\in (0,\delta ).$$

The following will prove that the inequality

$$V(t,x(t))<C+{\displaystyle \frac{b(\lambda )}{t^{1-q}}},$$

(7)

holds for any $t\in (0,\infty )$ holds.

Assume there exists a point $\xi$, $\xi \ge \delta >0$, such that

$$V(\xi ,x(\xi ))=C+{\displaystyle \frac{b(\lambda )}{{\xi }^{1-q}}},V(t,x(t))<C+{\displaystyle \frac{b(\lambda )}{t^{1-q}}},t\in (0,\xi ),$$

(8)

In the reason that $(V(t,x(t))-C-{\displaystyle \frac{b(\lambda )}{t^{1-q}}})\in C_{1-q}([0,\xi ],R)$, and also

$$V(\xi ,x(\xi ))-C-{\displaystyle \frac{b(\lambda )}{{\xi }^{1-q}}}=0,V(t,x(t))-C-{\displaystyle \frac{b(\lambda )}{t^{1-q}}}<0,t\in (0,\xi ),$$

From Lemma 1, we have

$${}_0{}^{RL}D_{\xi }^q(V(t,x(t))-C-{\displaystyle \frac{b(\lambda )}{t^{1-q}}}){|}_{t=\xi }\ge 0,$$

then,

$${}_0{}^{RL}D_{\xi }^q(V(\xi ,x(\xi ))\ge {}_0{}^{RL}D_{\xi }^{q}C+{}_0{}^{RL}D_{\xi }^q{\displaystyle \frac{b(\lambda )}{{\xi }^{1-q}}})={\displaystyle \frac{C}{{\xi }^q\mathsf{\Gamma }(1-q)}}.$$

From Lemma 2, it brings the follows

$${}_0{}^{GL}D_{+}^q{(V(t,x(t))|}_{t=\xi }={}_0{}^{RL}D_{\xi }^q(V(\xi ,x(\xi ))\ge {\displaystyle \frac{C}{{\xi }^q\mathsf{\Gamma }(1-q)}}.$$

(9)

For any $h>0$ and any $t\in [0,\xi ]$, let

$$K(x(t),h)=\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}x(t-rh),$$

From Lemma 2, for the system (1), there exists a $t\in [t_0,\xi ]$, $x(t)$ satisfies

$${}_0{}^{RL}D{}_t{}^{q}x(t)=f(t,x_t)={}_0{}^{GL}D{}_{+}{}^{q}x(t)=\underset{h\to 0^{+}}{lim}sup{\displaystyle \frac{1}{h^q}}(x(t)-K(x(t),h)),$$

then

$$\underset{h\to 0^{+}}{lim}sup{\displaystyle \frac{1}{h^q}}(x(t)-K(x(t),h))=f(t,x_t),$$

therefore, we can obtain

$$K(x(t),h)=x(t)-h^{q}f(t,x_t)-o(h^q),$$

then, for the function $V(t,x(t))$, we have

$$\begin{array}{cc}\hfill V(t,x(t))-& \sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}V(t-rh,x(t-rh))\hfill \\ \hfill =& V(t,x(t))-\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}V(t-rh,x(t)-h^{q}f(t,x_t))\hfill \\ \hfill +& \sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^r\left[V(t-rh,K(x(t),h)+o(h^q))-V(t-rh,x(t-rh))\right].\hfill \end{array}$$

(10)

Because in (10), the function is local Lipschitz continuous in variable x, there must exists a Lipschitz constant $L>0$ such that the following inequality holds:

$$\begin{array}{cc}\hfill & \sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^r\left[V(t-rh,K(x(t),h)+o(h^q))-V(t-rh,x(t-rh))\right]\hfill \\ \hfill & \le L∥\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^r\left(K(x(t),h)+o(h^q)-x(t-rh)\right)∥\hfill \\ \hfill & \le L∥\sum _{r=1}^{⌊{\displaystyle \frac{t-t_0}{h}}⌋}{(-1)}^{r+1}{C}_q^r\sum _{i=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{i+1}{C}_q^{i}x(t-ih)-\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}x(t-rh)∥\hfill \\ \hfill & +L∥\sum _{r=1}^{\left[{\displaystyle \frac{t-t_0}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}o(h^q)∥\hfill \\ \hfill & =L∥\left(\sum _{r=0}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^r\right)\left(\sum _{i=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{i+1}{C}_q^{i}x(t-ih)\right)∥+L∥o(h^q)∥\left|\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^r\right|,\hfill \end{array}$$

(11)

Both sides of (10) and (11) are divided by $h^q$ and substitute (10) with (11), then

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{h^q}}\left[V(t,x(t))-\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}V(t-rh,x(t-rh))\right]\hfill \\ \hfill & \le {\displaystyle \frac{1}{h^q}}\left[V(t,x(t))-\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}V(t-rh,x(t)-h^{q}f(t,x_t))\right]\hfill \\ \hfill & +L∥\left(\sum _{r=0}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^r\right)\left({\displaystyle \frac{1}{h^q}}\sum _{i=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{i+1}{C}_q^{i}x(t-ih)\right)∥\hfill \\ \hfill & +L{\displaystyle \frac{∥o(h^q)∥}{h^q}}∥\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^r∥.\hfill \end{array}$$

(12)

Let $h\to 0^{+}$ on both sides of (12), then

$$\begin{array}{cc}\hfill & {}_0{}^{GL}D{}_{+}{}^{q}V(t,x(t))=\underset{h\to 0^{+}}{lim}{\displaystyle \frac{1}{h^q}}\left[V(t,x(t))-\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}V(t-rh,x(t-rh))\right]\hfill \\ \hfill & \le \underset{h\to 0^{+}}{lim}{\displaystyle \frac{1}{h^q}}\left[V(t,x(t))-\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}V(t-rh,x(t)-h^{q}f(t,x_t))\right]\hfill \\ \hfill & +L\underset{h\to 0^{+}}{lim}sup\left|\left(\sum _{r=0}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^r\right)\right|∥\left({\displaystyle \frac{1}{h^q}}\sum _{i=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{i+1}{C}_q^{i}x(t-ih)\right)∥\hfill \\ \hfill & +L\underset{h\to 0^{+}}{lim}{\displaystyle \frac{∥o(h^q)∥}{h^q}}\left|\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^r\right|.\hfill \end{array}$$

(13)

Because when $h\to 0$, ${\displaystyle \frac{\parallel o(h^q)\parallel }{h^q}}\to 0$, also when $\left|\mu \right|\le 1$, ${\sum }_{r=0}^{\infty }{C}_q^r{\mu }^r={(1+\mu )}^q$, so (13) can be changed into

$${}_0{}^{GL}D{}_{+}{}^{q}V(t,x(t))\le \underset{h\to 0^{+}}{lim}{\displaystyle \frac{1}{h^q}}\left[V(t,x(t))-\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}V(t-rh,x(t)-h^{q}f(t,x_t))\right],$$

(14)

Case 1. let $\xi >\tau$, then $min\left\{\xi ,\tau \right\}=\tau$, from(8), we have

$${\xi }^{1-q}V(\xi ,x^{*}(\xi ))-b(\lambda )=C{\xi }^{1-q}>C{t}^{1-q}>t^{1-q}V(t,x(t))-b(\lambda ),t\in (0,\xi ),$$

so

$${\xi }^{1-q}V(\xi ,x^{*}(\xi ))>{(\xi +\theta )}^{1-q}V(\xi +\theta ,x^{*}(\xi +\theta )),\theta \in (-\tau ,0).$$

Case 2. let $\xi \le \tau$, then $min\left\{\xi ,\tau \right\}=\xi$, similar to case 1, it can be obtained that

$${\xi }^{1-q}V\left(\xi ,x^{*}(\xi )\right)>{(\xi +\theta )}^{1-q}V\left(\xi +\theta ,x^{*}(\xi +\theta )\right),\theta \in (-\xi ,0).$$

Based on case 1 and case 2, it can be concluded that

$${\xi }^{1-q}V(\xi ,x(\xi ))>{(\xi +\theta )}^{1-q}V(\xi +\theta ,x(\xi +\theta )),s\in \left(-min\{\tau ,t\},0\right).$$

(15)

According to the condition (iii) of the theorem, it can be concluded that

$$D^{+}V(\xi ,x(\xi ))<{\displaystyle \frac{C}{{\xi }^q\mathsf{\Gamma }(1-q)}},$$

Also from (14), we have

$$\begin{array}{cc}\hfill {\left.{}_0{}^{GL}D_{+}^{q}V(t,x(t))\right|}_{t=\xi } & \le \underset{h\to 0^{+}}{lim}{\displaystyle \frac{1}{h^q}}\left[V(\xi ,x(\xi ))-\sum _{r=1}^{[\xi /h]}{(-1)}^{r+1}{C}_q^{r}V(\xi -rh,x(\xi )-h^{q}f(\xi ,x(\xi +\theta )))\right]\hfill \\ \hfill & =D^{+}V(\xi ,x(\xi ))<{\displaystyle \frac{C}{{\xi }^q\mathsf{\Gamma }(1-q)}},\hfill \end{array}$$

(16)

The Equation (9) contradicts with Equation (16), hence there is no point $\xi \ge \delta >0$ that satisfies Equation (8). Therefore, we have the inequality

$$V(t,x(t))<C+{\displaystyle \frac{b(\lambda )}{t^{1-q}}},$$

(17)

holds for any $t\in (0,\infty )$.

Therefore, from condition (i) of the Theorem 1 and (17), it follows that

$$a\left(∥x(t)∥\right)\le V(t,x(t))<C+{\displaystyle \frac{b(\lambda )}{t^{1-q}}},t>T,$$

when $t\ge 1$, $t^{1-q}>1$, then

$$∥x(t)∥\le a^{-1}\left(C+b(\lambda )\right),t\in [max\{1,T\},\infty ).$$

Therefore, for the initial state function $\phi (t)$ of the system (1), there exists a pair of positive numbers $(\lambda ,a^{-1}(C+b(\lambda )))$ such that when $(\lambda ,a^{-1}(C+b(\lambda )))$, we have $∥x(t)∥\le a^{-1}\left(C+b(\lambda )\right),t\in [max\{1,T\},\infty )$. According to Definition 2, the system (1) is practically stable with respect to $(\lambda ,a^{-1}(C+b(\lambda )))$. Proof completed. □

4. Illustrative Example

Due to the widespread existence of time delays in dynamic systems, fractional order systems have certain advantages in describing physical systems, the widespread existence of nonlinear phenomena, and the lack of practical stability research in systems coupled with the above factors. These are the motivations for us to study this systems in our article. Consider the following 1-dimensional Riemann-Liouville (R-L) fractional nonlinear time delay system with $q\in (0,1)$,

$${}_0{}^{RL}D{}_t{}^{q}x(t)=-\left(1+{\displaystyle \frac{1}{t^q\mathsf{\Gamma }(1-q)}}+t^{q-1}{\left({\displaystyle \frac{t}{t-1}}\right)}^{1-q}\right)x(t)+2t^{q-1}x\left(t-\tau \right),t>0$$

(18)

and the initial state condition is

$$x(t)=\phi (t),t\in [-1,0],$$

$$\underset{t\to t_0^{+}}{lim}\left[t^{1-q}x(t)\right]={\displaystyle \frac{\phi (0)}{\mathsf{\Gamma }(q)}},$$

where $x(t)\in R$ is the state vector, the time delay is $\tau =1$.

Choose function $V(t,x)=t^{1-q}{x}^2$, then there exists a time $T=1$, when $t\ge T=1$, then $t^{1-q}\ge 1$. Let $a(\nu )={\nu }^2\in M$, then

$$a(\parallel x\parallel )=x^2\le t^{1-q}{x}^2=V(t,x),t\ge 1,$$

Then the (i) in Theorem 1 holds.

Let $\gamma (t)\in C_{1-q}([0,\infty ),R^n)$, from the initial condition, we have

$$\underset{t\to 0^{+}}{lim}\left[t^{1-q}\gamma (t)\right]={\displaystyle \frac{\gamma (0)}{\mathsf{\Gamma }(q)}},$$

Then there exists a $\lambda \ge \gamma (0)$ such that

$$\underset{t\to 0^{+}}{lim}\left[t^{1-q}\gamma (t)\right]\in S_{\lambda },$$

then

$$\underset{t\to 0^{+}}{lim}{t}^{1-q}V(t,\gamma (t))=\underset{t\to 0^{+}}{lim}{\left(t^{1-q}\gamma (t)\right)}^2={\displaystyle \frac{{\gamma }^2(0)}{{\mathsf{\Gamma }}^2(q)}},$$

Let $b(\nu )={\displaystyle \frac{{\nu }^2}{{\mathsf{\Gamma }}^2(q)}}\in M,$ we have

$$\underset{t\to 0^{+}}{lim}{t}^{1-q}V(t,\gamma (t))={\displaystyle \frac{{\gamma }^2(0)}{{\mathsf{\Gamma }}^2(q)}}\le {\displaystyle \frac{{\lambda }^2}{{\mathsf{\Gamma }}^2(q)}}=b(\lambda ),$$

Then (ii) in Theorem 1 holds.

Let $x(t)$ be the solution to (17) with the initial condition, for any $\phi (t)\in S_{\lambda }$, let $t>0$, such that $t^{1-q}V(t,x(t))\ge {(t+s)}^{1-q}V(t+s,x(t+s)),$$s\in [-min\{1,t\},0],$ then

$${\left({(t+s)}^{1-q}x(t+s)\right)}^2\le {\left(t^{1-q}x(t)\right)}^2,$$

This leads to

$$\left|{(t+s)}^{1-q}x(t+s)\right|\le \left|t^{1-q}x(t)\right|,$$

Let $\varphi (s)=x(t+s)$, $s\in [-min\{1,t\},0]$, then

$$\left|\varphi (s)\right|\le {\left({\displaystyle \frac{t}{t+s}}\right)}^{1-q}\left|\varphi (0)\right|\le {\left({\displaystyle \frac{t}{t-1}}\right)}^{1-q}\left|\varphi (0)\right|,s\in (-min\{1,t\},0].$$

(19)

From (18) and the following inequality

$${\varphi }^2(0)+{\varphi }^2(-1)\ge 2\varphi (0)\varphi (-1),$$

we have

$${\varphi }^2(0)+{\left({\displaystyle \frac{t}{t+1}}\right)}^{1-q}{\varphi }^2(0)\ge 2\varphi (0)\varphi (-1).$$

(20)

from (6), the Dini fractional derivative of Lyapunov function $D^{+}V(t,\varphi )$ for (17) is:

$$\begin{array}{cc}\hfill D^{+}V(t,\varphi )& =\underset{h\to 0}{lim}sup{\displaystyle \frac{1}{h^q}}\left[V(t,\varphi (0))-\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^{r}V(t-rh,\varphi (0)-h^{q}f(t,\varphi ))\right]\hfill \\ \hfill & =\underset{h\to 0}{lim}sup{\displaystyle \frac{1}{h^q}}\left[t^{1-q}{\varphi }^2(0)-\sum _{r=1}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^{r+1}{C}_q^r{t}^{1-q}{\left(\varphi (0)-h^{q}f(t,\varphi )\right)}^2\right]\hfill \\ \hfill & =\underset{h\to 0}{lim}sup{\displaystyle \frac{1}{h^q}}\left[t^{1-q}\left({\varphi }^2(0)-{\left(\varphi (0)-h^{q}f(t,\varphi )\right)}^2\right)\right.\hfill \\ \hfill & +\left.{\left(\varphi (0)-h^{q}f(t,\varphi )\right)}^2\sum _{r=0}^{\left[{\displaystyle \frac{t}{h}}\right]}{(-1)}^r{C}_q^r{t}^{1-q}\right]\hfill \\ \hfill & =2t^{1-q}\varphi (0)f(t,\varphi )+{\displaystyle \frac{t^{1-q}{\varphi }^2(0)}{t^q\mathsf{\Gamma }(1-q)}}\hfill \\ \hfill & =2t^{1-q}\varphi (0)\left[-\left({\displaystyle \frac{1}{2t^q\mathsf{\Gamma }(1-q)}}+1+t^{q-1}{\left({\displaystyle \frac{t}{t-1}}\right)}^{1-q}\right)\varphi (0)+2t^{q-1}\varphi (0-1)\right]\hfill \\ \hfill & +{\displaystyle \frac{t^{1-q}{\varphi }^2(0)}{t^q\mathsf{\Gamma }(1-q)}}\hfill \\ \hfill & =-{\displaystyle \frac{t^{1-q}{\varphi }^2(0)}{t^q\mathsf{\Gamma }(1-q)}}+{\displaystyle \frac{t^{1-q}{\varphi }^2(0)}{t^q\mathsf{\Gamma }(1-q)}}+2\left[2\varphi (0)\varphi (-1)-{\varphi }^2(0)-{\left({\displaystyle \frac{t}{t-1}}\right)}^{1-q}{\varphi }^2(0)\right]\hfill \\ \hfill & =2\left[2\varphi (0)\varphi (-1)-{\varphi }^2(0)-{\left({\displaystyle \frac{t}{t-1}}\right)}^{1-q}{\varphi }^2(0)\right],\hfill \end{array}$$

from (19), we have $D^{+}V(t,\varphi )\le 0$.

Therefore, the condition (iii) of Theorem 1 is satisfied. For the system (17), there exists a positive pair $(\lambda ,\lambda {(\mathsf{\Gamma }(q))}^{-1})$ such that $∥x(t)∥\le \lambda {(\mathsf{\Gamma }(q))}^{-1}$ when ${∥\varphi (t)∥}_0\le \lambda$, where $t\in [1,\infty )$. The system (1) is practically stable with respect to the positive pair.

5. Conclusions

This study has systematically investigated the practical stability of Riemann-Liouville fractional-order nonlinear systems with time-varying delays. By establishing a mathematically rigorous framework for initial condition specification that respects the non-local property of Riemann-Liouville derivatives, we have introduced a generalized concept of practical stability in the $L_p$-norm sense. Leveraging this foundation, we derived novel sufficient conditions through the construction of Lyapunov-Krasovskii functionals combined with an enhanced Razumikhin-type approach. The theoretical advancements are substantiated by numerical experiments on fractional-order neural networks, demonstrating superior convergence performance compared to classical integer-order methods. Notably, the proposed criteria provide a unified verification mechanism applicable to both nonlinear and linear systems with time delays, offering valuable insights into the memory-dependent stability analysis of fractional dynamical systems. These results pave the way for developing robust control strategies for networked systems with communication delays and fractional-order dynamics. While Lyapunov functions and the improved Razumikhin method have significantly advanced stability analysis, key challenges remain in constructiveness, scalability, and adaptability to linear or nonlinear systems. Future work should focus on data-driven methods, hybrid techniques, and control synthesis frameworks to bridge the gap between theoretical guarantees and practical implementation. Addressing these gaps will unlock new applications in networked control, biological systems, and cyber-physical security, where delays and nonlinearities are ubiquitous.