Fuhong Zhang¹, Xiaolan Liu^2
1College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, China.
²Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things, Zigong, China.
DOI: 10.4236/ojapps.2026.169179   PDF    HTML   XML   0 Downloads   24 Views   Citations

Abstract

In this paper, we investigate the locally practical Mittag-Leffler stability for a class of Caputo fractional-order differential equations subject to state-dependent delays and perturbations. By constructing an appropriate Lyapunov functional and employing key lemmas in fractional calculus, we derive a novel stability criterion under a weakened linear matrix inequality (LMI) condition. This criterion provides sufficient conditions for local practical Mittag-Leffler stability. The system state is bounded by the Mittag-Leffler function and the upper bound of perturbations, which can resist the influences of state-dependent delay and bounded perturbations. Finally, two numerical examples are provided to validate the correctness and effectiveness of the main theoretical results.

Keywords

State-Dependent Delays, Parameter Uncertainties, Lyapunov Functional, LMI

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1. Introduction

The integer-order differential equations have been extended to fractional-order differential equations. Fractional calculus has become a powerful tool for characterizing complex systems with memory and hereditary properties [1] [2]. Fractional differential equations with delay combine the characteristics of fractional calculus with delay differential equations, and can simultaneously describe the dependence of the system on historical states and the influence of time lags [3]. Thus, they have been widely applied in modeling complex dynamic systems such as population biology [4], physiology [5], control theory [6] [7], population dynamics [8], and economics [9]. As the application of fractional differential equations with delay in practical problems has become increasingly widespread, the analysis of their stability, as a prerequisite for the reliable operation and control design of systems, has received extensive attention from the academic community over the past two decades and has become one of the core directions in the theoretical research of fractional-order systems [10]-[13]. Stability, as a key indicator for measuring the dynamic behavior of a system, determines whether the system can maintain the expected performance under external disturbances or initial deviations. Therefore, relevant research not only has theoretical significance but also provides important guiding value for the design and optimization of practical systems.

Among the existing research on the stability of fractional-order systems with delay, systems involving time delays and external disturbances have become a research hotspot in recent years due to their closer proximity to engineering practice. For example, Zhang [14] studied the stability of linear fractional differential systems and their corresponding perturbed systems with respect to the Riemann-Liouville derivative and the Caputo derivative. In 2022, He [15] investigated the stability of Hadamard-type equations, the Caputo-Hadamard fractional derivative, and their related systems in both delay-free and delay-involved cases. Wang [16] conducted a stability analysis of fractional-order nonlinear systems with time delays and proposed the definition of Mittag-Leffler stability for time-delay systems for the first time. Tuan [17] proved the linearized asymptotic stability of nonlinear fractional differential equations with time delays. From a practical perspective, time delays arise from the physical structure of the system or information processing procedures and are almost inevitable in fields such as mechanical control, communication networks, and biological metabolism. The existence of time delays may cause the system to oscillate or even become unstable [18] [19].

External disturbances usually manifest as parameter uncertainties, random noise, or sudden interferences, and these disturbances can also disrupt the steady state of the system and reduce control precision [20] [21]. Li and Wang [22] studied a class of system models that simultaneously consider the effects of state delay and bounded external disturbances, and comprehensively evaluated the dynamic performance of the system under the combined action of external disturbances and event-triggered mechanisms.

The coexistence of time delays and disturbances poses enormous challenges to system stability analysis. In the face of these challenges, robustness research has become the key to breakthroughs [23]-[28]. Regarding the robustness of fractional-order systems, there have been many important explorations: for example, Li [23] studied fractional-order interval nonlinear time-delay systems, expanded the Lyapunov direct method by using new lemmas, and constructed delay-independent stability criteria in the form of linear matrix inequalities by Razumikhin theorem. Zhu [24] focused on the results of uncertain impulsive positive systems with time delays, utilized the impulsive time-varying copositive Lyapunov function, established globally robust exponential stability criteria under different dwell times, and designed controllers. Liao [25] constructed an asymptotic stability test criterion for parameter-uncertain FO-LTI (Fractional-Order Linear Time-Invariant) interval systems with deterministic linear coupling relations through the linear matrix inequality (LMI) method. These achievements have enriched the robustness theory of fractional-order systems and laid an important foundation for dealing with complex uncertainties in practical systems.

The remainder of this paper is organized as follows. In Section 2, we list essential preliminaries, including definitions of fractional calculus, key assumptions, and necessary lemmas. The main results on robust Mittag-Leffler stability are presented in Section 3. In Section 4, we provide two numerical examples to verify the theoretical findings. Finally, in Section 5, we conclude the paper and suggest potential future research directions.

2. Preliminaries

The stability analysis of fractional-order systems with state-dependent delays and perturbations requires a foundational understanding of fractional calculus, Mittag-Leffler functions, and robust control theory. In this section, we introduce the necessary mathematical tools, including definitions of Caputo derivatives, Mittag-Leffler stability, and key assumptions on system dynamics. Additionally, critical lemmas for bounding fractional Lyapunov derivatives and handling perturbations are presented to support the proof of Theorem 3.1 in Section 3.

Next we provide the definitions of the Caputo fractional derivative and Mittag-Leffler stability.

Definition 2.1 ([29]) The Caputo fractional derivative of order $\alpha \in \left(0,1\right)$ for a sufficiently smooth function $x\left(t\right)\in C^1\left(\left[t_0,+\infty \right),{ℝ}^n\right)$ is defined as

${}^C{D}_t^{\alpha }x\left(t\right)=\frac{1}{\Gamma \left(1-\alpha \right)}{\displaystyle {\int }_{t_0}^t\frac{\dot{x}\left(s\right)}{{\left(t-s\right)}^{\alpha }}\text{d}s},$

where $\dot{x}\left(s\right)=\frac{\text{d}x\left(s\right)}{\text{d}s}$, $\Gamma (\cdot )$ is the Gamma function.

Definition 2.2 ([30]) A fractional-order system ${}_{t_0}^C{D}_t^{\alpha }x\left(t\right)=f\left(t,x_t\right)$ is said to be Mittag-Leffler stable if there exist $\lambda \ge 0$ and $b>0$ such that

$\Vert x\left(t,t_0,\phi \right)\Vert \le {\left\{m\left(\phi \right)E_{\alpha }\left(-\lambda {\left(t-t_0\right)}^{\alpha }\right)\right\}}^b,$

where $\alpha \in \left(0,1\right)$ , $b>0$ , $m\left(0\right)=0$ , $m\left(x\right)\ge 0$ , and $m\left(x\right)$ is locally Lipschitz on $x\in ℰ$ with the Lipschitz constant $m_0$ .

In this paper, we focus on a class of fractional-order dynamical systems with state-dependent time delays and perturbations, which is formulated as (1).

$\{\begin{array}{ll}{}^{C}D{}_t^{\alpha }x\left(t\right)=f\left(t,x\left(t\right),x\left(t-\tau \left(x\left(t\right)\right)\right),\text{Δ}A\left(t\right),\text{Δ}f\left(t\right)\right),\hfill & t\ge 0\hfill \\ x\left(t\right)=\varphi \left(t\right),\hfill & t\in \left[-{\tau }_{\text{max}},0\right]\hfill \end{array}$ (1)

where the initial history function $\varphi (\cdot )$ belongs to the admissible initial-history space $\varphi \in C^1\left[-{\tau }_{\text{max}},0;{ℝ}^n\right]$ . The state-dependent delay $\tau \left(x\left(t\right)\right)$ maps ${ℝ}^n$ into $\left[0,{\tau }_{\text{max}}\right]$ . This system incorporates several key characteristics that are prevalent in many real-world applications, such as biological networks, chemical processes, and control systems with memory effects.

In order to obtain our results, we make some assumptions.

(H1) The state-dependent delay $\tau :{ℝ}^n\to ℝ$ is locally Lipschitz continuous and satisfies $0\le \tau \left(x\right)\le {\tau }_{\text{max}}$ for all $x\in {ℝ}^n$ . Its composition along system trajectories $\tau \left(x\left(t\right)\right)$ is absolutely continuous, and its time derivative $\dot{\tau }\left(x\left(t\right)\right)$ exists almost everywhere. Moreover,

$\dot{\tau }\left(x\left(t\right)\right)\le \eta <1\text{a}.\text{e}.t\ge 0,$

where ${\tau }_{\text{max}}>0$ and $\eta \in \left(0,1\right)$ are known constants. For locally Lipschitz functions, the time derivative can be interpreted via subgradients: $\dot{\tau }\left(x\left(t\right)\right)={\partial }_x\tau \left(x\left(t\right)\right)\dot{x}\left(t\right)$ holds almost everywhere. This condition prevents rapid delay variations violating causality, and guarantees well-posedness for the state-dependent fractional delay system. If $\eta \ge 1$ , fast state-dependent delay fluctuations may lead to non-uniqueness or oscillatory solutions. This assumption is consistent with existing literature for fractional state-dependent delay systems.

(H2) The perturbations $\text{Δ}A\left(t\right)$ and $\text{Δ}f\left(t\right)$ are bounded such that

$\Vert \Delta A\left(t\right)\Vert \le \rho ,\Vert \Delta f\left(t\right)\Vert \le d,$

where $\rho ,d>0$ are known constants. If the perturbations are unbounded, the system state will diverge directly due to the accumulation of uncertainties, making stability analysis meaningless. Meanwhile, the boundedness conditions provide a quantitative basis for bounding the perturbation terms using Young’s inequality and matrix norm inequalities in subsequent derivations.

(H3) The function $f\left(t,x\left(t\right),x\left(t-\tau \left(x\left(t\right)\right)\right),\text{Δ}A\left(t\right),\text{Δ}f\left(t\right)\right)$ in system (1) is of linear form with respect to the state variables $x$ and $x\left(t-\tau \left(x\left(t\right)\right)\right)$ , i.e.,

$\begin{array}{l}f\left(t,x\left(t\right),x\left(t-\tau \left(x\left(t\right)\right)\right),\text{Δ}A\left(t\right),\text{Δ}f\left(t\right)\right)\\ =\left(A+\text{Δ}A\left(t\right)\right)x\left(t\right)+Bx\left(t-\tau \left(x\left(t\right)\right)\right)+\text{Δ}f\left(t\right),\end{array}$

where $A,B\in {ℝ}^{n\times n}$ are constant matrices, and $\text{Δ}A\left(t\right),\text{Δ}f\left(t\right)$ satisfy the boundedness conditions in (H2).

(H4) When the initial condition satisfies $\Vert x\left(t_0\right)\Vert \le R_0$ ($R_0\le R$ ), the state trajectory of the system remains within the neighborhood $\Omega =\left\{x\in {ℝ}^n|\Vert x\Vert \le R\right\}$ at all times, where $R>0$ is a given constant.

Regarding the well-posedness of system (1), we emphasize that under assumptions (H1)-(H3) and the continuity of the nonlinear function $f\left(t,x\left(t\right),x\left(t-\tau \left(x\left(t\right)\right)\right),\text{Δ}A\left(t\right),\text{Δ}f\left(t\right)\right)$ with respect to all its arguments, the system admits a unique solution on $\left[0,+\infty \right)$ by the existence and uniqueness theorem for fractional differential equations with state-dependent delays. Furthermore, as rigorously proved in Vainikko, for the Caputo fractional derivative of order $\alpha \in \left(0,1\right)$ adopted in this paper, the solution possesses a specific structure: it is continuous on $\left[0,+\infty \right)$ , its Caputo derivative exists almost everywhere, and the growth behavior of the solution is dominated by the $t^{\alpha }$ term (directly related to the initial value $x\left(0\right)$ ). This structural characteristic ensures that the subsequent stability analysis based on Lyapunov-Krasovskii functionals has a rigorous theoretical foundation.

Definition 2.3 ([31]) This definition characterizes that system trajectories converge to a bounded neighborhood of the origin. The size of the neighborhood depends on the disturbance bounds $d,\rho$ . No equilibrium point is required to exist for this definition, its state satisfies

$\Vert x\left(t\right)\Vert \le M\Vert x\left(0\right)\Vert E_{\alpha }\left(-\lambda t^{\alpha }\right)+Nd,$

where $M,N>0$ are constants independent of the initial conditions and disturbances.

Next we provide some necessary lemmas to facilitate the establishment of our main results.

Lemma 1 ([32]) For a Caputo fractional derivative of order $\alpha \in \left(0,1\right)$ , consider a continuously differentiable function $x\left(t\right)\in {ℝ}^n$ and a positive definite matrix $P\in {ℝ}^{n\times n}$ . The Caputo fractional derivative of the quadratic Lyapunov candidate $V\left(t\right)=x^{\top }\left(t\right)Px\left(t\right)$ satisfies

${}^C{D}_t^{\alpha }V\left(t\right)\le 2x^{\top }\left(t\right)P{}^C{D}_t^{\alpha }x\left(t\right).$

Lemma 2 ([33]) Let $X$ and $Y$ be matrices of appropriate dimensions. Then, the following inequality holds, for all $\epsilon >0$

$X^{\top }Y+Y^{\top }X\le \epsilon X^{\top }X+{\epsilon }^{-1}{Y}^{\top }Y.$

Lemma 3 ([34]) Let $\alpha \in \left(0,1\right)$ , and $u\left(t\right)\in C^1\left[t_0,+\infty ,ℝ\right)$ be a nonnegative scalar function. If there exist constants $\lambda >0$ , $\gamma \ge 0$ such that

${}^C{D}_t^{\alpha }u\left(t\right)\le -\lambda u\left(t\right)+\gamma ,t\ge t_0,$

then

$u\left(t\right)\le u\left(t_0\right)E_{\alpha }\left(-\lambda {\left(t-t_0\right)}^{\alpha }\right)+\frac{\gamma }{\lambda }\left(1-E_{\alpha }\left(-\lambda {\left(t-t_0\right)}^{\alpha }\right)\right).$

Next we introduce a special symbol that appears in the proof of a theorem.

$M\prec 0$ means that all eigenvalues of $M$ are strictly less than zero. This notation is widely used in the analysis of linear matrix inequalities (LMI) to characterize the negative definiteness of matrices, which plays a crucial role in the stability analysis of dynamical systems.

3. Main Results

In this section, through a refined Lyapunov functional approach, we rigorously derive sufficient LMI conditions for local practical Mittag-Leffler stability of system (1) subject to state-dependent delay and bounded perturbations.

Theorem 3.1 Consider system (1) under assumptions (H1)-(H3). If there exist positive definite matrices $P$ , $Q$ and parameters $\epsilon ,\sigma ,\nu ,\lambda ,\mu >0$ such that the following linear matrix inequalities hold

Current state LMI

$2PA+\left(\epsilon +\sigma \right)P^{\top }P+\left({\epsilon }^{-1}{\rho }^2+\nu \right)I+Q+\lambda I\prec 0$

Delayed state LMI

${\nu }^{-1}{B}^{\top }{P}^{\top }PB-\left(1-\eta \right)Q+\mu I\prec 0,$

then (1) is local robust Mittag-Leffler stable, and the state satisfies

$\Vert x\left(t\right)\Vert \le C_0{E}_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)+C_1\left(1-E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\right),$

where $m=\text{min}\left\{\frac{\lambda }{{\lambda }_{\text{max}}\left(P\right)},\frac{\mu }{{\lambda }_{\text{max}}\left(Q\right)}\right\}$ , $C_0=\sqrt{\frac{{\lambda }_{\text{max}}\left(P\right)+{\lambda }_{\text{max}}\left(Q\right)}{{\lambda }_{\text{min}}\left(P\right)}}\Vert x\left(t_0\right)\Vert$ ,

$C_1=\sqrt{\frac{M}{m{\lambda }_{\text{min}}\left(P\right)}}$ , $M={\sigma }^{-1}{d}^2+{\Vert \Delta A\left(t\right)\Vert }^2+{\Vert \Delta f\left(t\right)\Vert }^2$ .

Proof. Considering the state-dependent time delay $\tau \left(x\left(t\right)\right)$ which makes the system’s memory effect related to state evolution, and the external disturbance $\text{Δ}f\left(t\right)$ that brings uncertain perturbations, we construct the Lyapunov function $V\left(t\right)$ as follows.

The state quadratic term

$V_1\left(t\right)=x^{\top }\left(t\right)Px\left(t\right),$

directly reflects the energy of the current system state.

The time-delay integral term

$V_2\left(t\right)={\displaystyle {\int }_{t-\tau \left(x\left(t\right)\right)}^t{x}^{\top }\left(s\right)Qx\left(s\right)\text{d}s},$

by integrating the state energy over the time-delay interval, it captures the cumulative influence of the time-delay history on the current state; since the time delay is state-dependent, the integration interval varies dynamically with the state.

The disturbance norm integral term

$V_3\left(t\right)={\displaystyle {\int }_{t_0}^t\left({\Vert \Delta A\left(s\right)\Vert }^2+{\Vert \Delta f\left(s\right)\Vert }^2\right)\text{d}s},$

for bounded disturbances, by summing the norms of disturbance-related terms, the uncertainties are transformed into quantifiable accumulated energy of disturbances, thereby enabling a comprehensive analysis of system stability with these factors taken into account. $P,Q>0$ are the positive definite matrix. Compared with existing literature, the functional proposed in this paper simultaneously includes the state term, time-delay integral term, and disturbance integral term. It does not require the introduction of additional complex nonlinear weighting functions, is more easily solvable via LMI, and is thus suitable for engineering application scenarios.

Based on assumption (H3) on the structure of $f\left(\cdot ,\cdot ,\cdot \right)$ , we can rewrite the original system (1) as the following linear fractional-order system with state-dependent delay and perturbations

${}^C{D}_t^{\alpha }x\left(t\right)=\left(A+\text{Δ}A\left(t\right)\right)x\left(t\right)+Bx\left(t-\tau \left(x\left(t\right)\right)\right)+\text{Δ}f\left(t\right).$ (2)

This linearization is valid under the boundedness of perturbations (H2) and the state-dependent delay constraint (H1), which lays the foundation for the subsequent LMI-based stability analysis.

Next, we analyze the derivative of the Lyapunov function $V_1\left(t\right)$ by leveraging the properties of the Caputo fractional derivative and integration by parts, we derive

${}^C{D}_t^{\alpha }{V}_1\left(t\right)=\frac{1}{\Gamma \left(1-\alpha \right)}{\displaystyle {\int }_{t_0}^t\frac{x^{\top }\left(s\right)P\dot{x}\left(s\right)+{\dot{x}}^{\top }\left(s\right)Px\left(s\right)}{{\left(t-s\right)}^{\alpha }}\text{d}s}.$

By the fractional-order inequality Lemma 1, we simplify and obtain a key inequality

${}^C{D}_t^{\alpha }{V}_1\left(t\right)\le 2x^{\top }\left(t\right)P{}^C{D}_t^{\alpha }x\left(t\right).$ (3)

Combining the system dynamics (2) with the Lyapunov derivative inequality (3), we derive

${}^C{D}_t^{\alpha }{V}_1\left(t\right)\le 2x^{\top }\left(t\right)P\left[\left(A+\text{Δ}A\left(t\right)\right)x\left(t\right)+Bx\left(t-\tau \left(x\left(t\right)\right)\right)+\text{Δ}f\left(t\right)\right].$ (4)

By Lemma 2, for sufficiently small $\epsilon >0$ , we bound the uncertainty terms

$\begin{array}{c}2x^{\top }\left(t\right)P\Delta A\left(t\right)x\left(t\right)=x^{\top }\left(t\right)P\Delta A\left(t\right)x\left(t\right)+x^{\top }\left(t\right)\Delta A^{\top }\left(t\right)P^{\top }x\left(t\right)\\ \le \epsilon x^{\top }\left(t\right)P{P}^{\top }x\left(t\right)+{\epsilon }^{-1}{\left[\Delta A\left(t\right)x\left(t\right)\right]}^{\top }\Delta A\left(t\right)x\left(t\right)\\ =\epsilon x^{\top }\left(t\right)P{P}^{\top }x\left(t\right)+{\epsilon }^{-1}{x}^{\top }\left(t\right)\Delta A^{\top }\left(t\right)\Delta A\left(t\right)x\left(t\right)\\ \le \epsilon x^{\top }\left(t\right)P^{\top }Px\left(t\right)+{\epsilon }^{-1}{\rho }^2{x}^{\top }\left(t\right)x\left(t\right).\end{array}$ (5)

For the external disturbance term, there exists $\sigma >0$ such that

$\begin{array}{c}2x^{\top }\left(t\right)P\Delta f\left(t\right)=x^{\top }\left(t\right)P\Delta f\left(t\right)+\Delta f^{\top }\left(t\right)P^{\top }x\left(t\right)\\ \le \sigma x^{\top }\left(t\right)P{P}^{\top }x\left(t\right)+{\sigma }^{-1}\Delta f^{\top }\left(t\right)\Delta f\left(t\right)\\ =\sigma x^{\top }\left(t\right)P^{\top }Px\left(t\right)+{\sigma }^{-1}\Delta f^{\top }\left(t\right)\Delta f\left(t\right)\\ \le \sigma x^{\top }\left(t\right)P^{\top }Px\left(t\right)+{\sigma }^{-1}{d}^2.\end{array}$ (6)

Combining (4), (5) with (6), we have

$\begin{array}{c}C{D}_t^{\alpha }{V}_1\left(t\right)\le 2x^{\top }\left(t\right)PA\left(t\right)x\left(t\right)+2x^{\top }\left(t\right)PBx\left(t-\tau \left(x\left(t\right)\right)\right)\\ +2x^{\top }\left(t\right)P\Delta A\left(t\right)x\left(t\right)+2x^{\top }\left(t\right)P\Delta f\left(t\right)\\ \le 2x^{\top }\left(t\right)PA\left(t\right)x\left(t\right)+2x^{\top }\left(t\right)PBx\left(t-\tau \left(x\left(t\right)\right)\right)\\ +\epsilon x^{\top }\left(t\right)P^{\top }Px\left(t\right)+{\epsilon }^{-1}{\rho }^2{x}^{\top }\left(t\right)x\left(t\right)+\sigma x^{\top }\left(t\right)P^{\top }Px\left(t\right)+{\sigma }^{-1}{d}^2.\end{array}$ (7)

To comprehensively obtain the Caputo fractional derivative of the entire Lyapunov function $V\left(t\right)$ , after analyzing the derivative of $V_1\left(t\right)$ , we now proceed to analyze the second term $V_2\left(t\right)$ .

Utilizing properties of Caputo fractional derivatives for integral terms, we compute ${}^C{D}_t^{\alpha }{V}_2\left(t\right)$ . Assume $\dot{\tau }\left(x\left(s\right)\right)\le \eta <1$ , since the time delay $\tau \left(x\left(t\right)\right)$ is state-dependent, its derivative $\dot{\tau }\left(x\left(t\right)\right)$ reflects the rate of change of the delay with respect to the state. The condition $\dot{\tau }\left(x\left(s\right)\right)\le \eta <1$ ensures that the delay does not change too rapidly, which is a common and reasonable assumption to guarantee the solvability and stability analysis feasibility of the system with state-dependent delays. It restricts the delay’s variation rate to be within a range that our subsequent fractional-order derivative analysis can handle. Then

$\begin{array}{c}{}^{C}D{}_t^{\alpha }{V}_2\left(t\right)=\frac{1}{\Gamma \left(1-\alpha \right)}{\displaystyle {\int }_{t_0}^t{\dot{V}}_2\left(s\right){\left(t-s\right)}^{-\alpha }\text{d}s}\\ =\frac{1}{\Gamma \left(1-\alpha \right)}{\displaystyle {\int }_{t_0}^t\left[x^{\top }\left(s\right)Qx\left(s\right)-x^{\top }\left(s-\tau \left(x\left(s\right)\right)\right)Qx\left(s-\tau \left(x\left(s\right)\right)\right)\left(1-\dot{\tau }\left(x\left(s\right)\right)\right)\right]{\left(t-s\right)}^{-\alpha }\text{d}s}\\ \le \frac{1}{\Gamma \left(1-\alpha \right)}{\displaystyle {\int }_{t_0}^t\left[x^{\top }\left(s\right)Qx\left(s\right)-\left(1-\eta \right)x^{\top }\left(s-\tau \left(x\left(s\right)\right)\right)Qx\left(s-\tau \left(x\left(s\right)\right)\right)\right]{\left(t-s\right)}^{-\alpha }\text{d}s}.\end{array}$

For constant or slowly-varying delays, this simplifies to a key inequality

$\begin{array}{c}{}^{C}D{}_t^{\alpha }{V}_2\left(t\right)\le x^{\top }\left(t\right)Qx\left(t\right)-\left(1-\eta \right)x^{\top }\left(t-\tau \left(x\left(t\right)\right)\right)Qx\left(t-\tau \left(x\left(t\right)\right)\right)\frac{1}{\Gamma \left(1-\alpha \right)}{\displaystyle {\int }_{t_0}^t\frac{1}{{\left(t-s\right)}^{\alpha }}\text{d}s}\\ =x^{\top }\left(t\right)Qx\left(t\right)-\left(1-\eta \right)x^{\top }\left(t-\tau \left(x\left(t\right)\right)\right)Qx\left(t-\tau \left(x\left(t\right)\right)\right)\frac{{\left(t-t_0\right)}^{1-\alpha }}{\Gamma \left(1-\alpha \right)\left(1-\alpha \right)}\\ \le x^{\top }\left(t\right)Qx\left(t\right)-\left(1-\eta \right)x^{\top }\left(t-\tau \left(x\left(t\right)\right)\right)Qx\left(t-\tau \left(x\left(t\right)\right)\right).\end{array}$ (8)

Having analyzed the derivatives of $V_1\left(t\right)$ and $V_2\left(t\right)$ , which account for the state energy and time-delay energy respectively, we now move on to the final term $V_3\left(t\right)$ of the Lyapunov function. Analyzing $V_3\left(t\right)$ will help us capture the influence of disturbances on the system’s energy from the perspective of fractional-order calculus, so as to obtain the complete derivative of $V\left(t\right)$ . For the Caputo fractional derivative ${}^C{D}_t^{\alpha }{V}_3\left(t\right)$ , we use properties of fractional calculus

$\begin{array}{c}{}^C{D}_t^{\alpha }{V}_3\left(t\right)=\frac{1}{\Gamma \left(1-\alpha \right)}{\displaystyle {\int }_{t_0}^t\frac{{\dot{V}}_3\left(s\right)}{{\left(t-s\right)}^{\alpha }}\text{d}s}\\ =\frac{1}{\Gamma \left(1-\alpha \right)}{\displaystyle {\int }_{t_0}^t\frac{{\Vert \Delta A\left(s\right)\Vert }^2+{\Vert \Delta f\left(s\right)\Vert }^2}{{\left(t-s\right)}^{\alpha }}\text{d}s}\\ \le \frac{1}{\Gamma \left(1-\alpha \right)}\left(\underset{\xi \in \left[t_0,t\right]}{\sup }\left[{\Vert \Delta A\left(\xi \right)\Vert }^2+{\Vert \Delta f\left(\xi \right)\Vert }^2\right]\right){\displaystyle {\int }_{t_0}^t\frac{1}{{\left(t-s\right)}^{\alpha }}\text{d}s}\\ =\frac{{\left(t-t_0\right)}^{1-\alpha }}{\Gamma \left(1-\alpha \right)\left(1-\alpha \right)}\cdot \left(\underset{\xi \in \left[t_0,t\right]}{\sup }\left[{\Vert \Delta A\left(\xi \right)\Vert }^2+{\Vert \Delta f\left(\xi \right)\Vert }^2\right]\right).\end{array}$

Considering the singularity of the fractional-order integral kernel ${\left(t-s\right)}^{-\alpha }$ , its value is mainly dominated by the behavior of the integrand in the neighborhood near the upper limit $t$ of the integral. Therefore, the global supremum ${\text{sup}}_{\xi \in \left[t_0,t\right]}(\cdot )$ can be replaced by the instantaneous value ${\Vert \Delta A\left(t\right)\Vert }^2+{\Vert \Delta f\left(t\right)\Vert }^2$ at the current moment, forming a simpler but conservative upper bound.

At the same time, precisely because the factor $\frac{{\left(t-t_0\right)}^{1-\alpha }}{\Gamma \left(1-\alpha \right)\left(1-\alpha \right)}$ does not affect the direction of the fractional-order derivative inequality. In order to obtain a form consistent with the integer-order theory, this factor will be absorbed into the design of Lyapunov function coefficients or control gains in subsequent stability analysis, and its specific value does not affect the establishment of the stability conclusion.

Based on the above processing, we obtain the simplified form for the derivation of the stability criterion

$\begin{array}{c}{}^C{D}_t^{\alpha }{V}_3(t)\le \left({\Vert \Delta A\left(t\right)\Vert }^2+{\Vert \Delta f\left(t\right)\Vert }^2\right)\frac{{\left(t-t_0\right)}^{1-\alpha }}{\Gamma \left(1-\alpha \right)\left(1-\alpha \right)}\\ \le {\Vert \Delta A\left(t\right)\Vert }^2+{\Vert \Delta f\left(t\right)\Vert }^2.\end{array}$ (9)

Combining (7), (8) with (9), we derive the final Lyapunov derivative bound

$\begin{array}{c}{}^{C}D{}_t^{\alpha }V\left(t\right)\le 2x^{\top }\left(t\right)PA\left(t\right)x\left(t\right)+2x^{\top }\left(t\right)PBx\left(t-\tau \left(x\left(t\right)\right)\right)+\epsilon x^{\top }\left(t\right)P^{\top }Px\left(t\right)\\ +{\epsilon }^{-1}{\rho }^2{x}^{\top }\left(t\right)x\left(t\right)+\sigma x^{\top }\left(t\right)P^{\top }Px\left(t\right)+{\sigma }^{-1}{d}^2+x^{\top }\left(t\right)Qx\left(t\right)\\ -\left(1-\eta \right)x^{\top }\left(t-\tau \left(x\left(t\right)\right)\right)Qx\left(t-\tau \left(x\left(t\right)\right)\right)+{\Vert \Delta A\left(t\right)\Vert }^2+{\Vert \Delta f\left(t\right)\Vert }^2.\end{array}$ (10)

By LMI, we transform the quadratic form in ${}^C{D}_t^{\alpha }V\left(t\right)$ into a negative definite form. The key derivations are as follows.

Let $z\left(t\right)=x\left(t-\tau \left(x\left(t\right)\right)\right)$ . Then for the cross term $2x^{\top }\left(t\right)PBz\left(t\right)$ , we apply Young Inequality

$2x^{\top }\left(t\right)PBz\left(t\right)\le \nu x^{\top }\left(t\right)x\left(t\right)+{\nu }^{-1}{z}^{\top }\left(t\right)B^{\top }{P}^{\top }PBz\left(t\right),$ (11)

where $\nu >0$ is a tuning parameter.

Substituting (11) into (10), we have

$\begin{array}{c}{}^{C}D{}_t^{\alpha }V\left(t\right)\le 2x^{\top }\left(t\right)PA\left(t\right)x\left(t\right)+\nu x^{\top }\left(t\right)x\left(t\right)+{\nu }^{-1}{z}^{\top }\left(t\right)B^{\top }{P}^{\top }PBz\left(t\right)\\ +\epsilon x^{\top }\left(t\right)P^{\top }Px\left(t\right)+{\epsilon }^{-1}{\rho }^2{x}^{\top }\left(t\right)x\left(t\right)+\sigma x^{\top }\left(t\right)P^{\top }Px\left(t\right)\\ +{\sigma }^{-1}{d}^2+x^{\top }\left(t\right)Qx\left(t\right)-\left(1-\eta \right)z^{\top }\left(t\right)Qz\left(t\right)+{\Vert \Delta A\left(t\right)\Vert }^2+{\Vert \Delta f\left(t\right)\Vert }^2.\end{array}$ (12)

To achieve the stability of the current state, we need to verify

$\begin{array}{l}2x^{\top }\left(t\right)PA\left(t\right)x\left(t\right)+\left(\epsilon +\sigma \right)x^{\top }\left(t\right)P^{\top }Px\left(t\right)+\left({\epsilon }^{-1}{\rho }^2+\nu \right)x^{\top }\left(t\right)x\left(t\right)+x^{\top }\left(t\right)Qx\left(t\right)\\ \le -\lambda x^{\top }\left(t\right)x\left(t\right),\end{array}$ (13)

that is,

$x^{\top }\left(t\right)\left[2PA+\left(\epsilon +\sigma \right)P^{\top }P+\left({\epsilon }^{-1}{\rho }^2+\nu \right)I+Q+\lambda I\right]x\left(t\right)\le 0.$ (14)

Since (14) holds for all non-zero vectors $x\left(t\right)$ , the matrix must be negative definite. By choosing appropriate parameters via LMI, we have

$2PA+\left(\epsilon +\sigma \right)P^{\top }P+\left({\epsilon }^{-1}{\rho }^2+\nu \right)I+Q+\lambda I\prec 0.$

To achieve the stability of the delayed state, we need to verify

${\nu }^{-1}{z}^{\top }\left(t\right)B^{\top }{P}^{\top }PBz\left(t\right)-\left(1-\eta \right)z^{\top }\left(t\right)Qz\left(t\right)\le -\mu z^{\top }\left(t\right)z\left(t\right),$ (15)

that is,

$z^{\top }\left(t\right)\left[{\nu }^{-1}{B}^{\top }{P}^{\top }PB-\left(1-\eta \right)Q+\mu I\right]z\left(t\right)\le 0.$ (16)

Since (16) holds for all $x\left(t\right),z\left(t\right)\ne 0$ , the quadratic form of $z\left(t\right)$ must be negative definite. That is

${\nu }^{-1}{B}^{\top }{P}^{\top }PB-\left(1-\eta \right)Q+\mu I\prec 0.$

Substitute (13) and (15) into (12). We derive the key fractional Lyapunov derivative bound

$\begin{array}{c}{}^{C}D{}_t^{\alpha }V\left(t\right)\le -\lambda x^{\top }\left(t\right)x\left(t\right)-\mu z^{\top }\left(t\right)z\left(t\right)+{\sigma }^{-1}{d}^2+{\Vert \Delta A\left(t\right)\Vert }^2+{\Vert \Delta f\left(t\right)\Vert }^2\\ =-\lambda x^{\top }\left(t\right)x\left(t\right)-\mu z^{\top }\left(t\right)z\left(t\right)+M\\ \le -\frac{\lambda }{{\lambda }_{\max }\left(P\right)}{V}_1\left(t\right)-\frac{\mu }{{\lambda }_{\max }\left(Q\right)}{V}_2\left(t\right)+M\\ \le -m\left(V_1\left(t\right)+V_2\left(t\right)\right)+M\\ \le -mV\left(t\right)+M,\end{array}$ (17)

where $m=\text{min}\left\{\frac{\lambda }{{\lambda }_{\text{max}}\left(P\right)},\frac{\mu }{{\lambda }_{\text{max}}\left(Q\right)}\right\}$ , $M={\sigma }^{-1}{d}^2+{\Vert \Delta A\left(t\right)\Vert }^2+{\Vert \Delta f\left(t\right)\Vert }^2$ .

By Lemma 3, we solve the bound of $V\left(t\right)$ . For $t\ge t_0$ , assume $V\left(t_0\right)$ is the initial value. The solution satisfies

$\begin{array}{c}V\left(t\right)\le V\left(t_0\right)E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)+\frac{M}{m}\left(1-E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\right)\\ \le \left[x^{\top }\left(t_0\right)Px\left(t_0\right)+{\displaystyle {\int }_{t_0-\tau \left(x\left(t_0\right)\right)}^{t_0}{x}^{\top }\left(s\right)Qx\left(s\right)\text{d}s}\right]E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\\ +\frac{M}{m}\left(1-E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\right),\end{array}$ (18)

where $E_{\alpha }(\cdot )$ is the Mittag-Leffler function.

By positive definiteness of $P$ , there exists ${\lambda }_{\text{min}}\left(P\right)>0$ such that

$V\left(t\right)\ge x^{\top }\left(t\right)Px\left(t\right)\ge {\lambda }_{\text{min}}\left(P\right){\Vert x\left(t\right)\Vert }^2.$ (19)

Then

${\Vert x\left(t\right)\Vert }^2\le \frac{1}{{\lambda }_{\text{min}}\left(P\right)}V\left(t\right).$

Taking square roots (for $\Vert x\left(t\right)\Vert \ge 0$ ), we derive

$\begin{array}{c}\Vert x\left(t\right)\Vert \le \sqrt{\frac{1}{{\lambda }_{\min }\left(P\right)}}\sqrt{V\left(t\right)}\\ \le \sqrt{\frac{1}{{\lambda }_{\min }\left(P\right)}}\left[\sqrt{V\left(t_0\right)E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)+\frac{M}{m}\left(1-E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\right)}\right]\\ \le \sqrt{\frac{1}{{\lambda }_{\min }\left(P\right)}}\left[\sqrt{V\left(t_0\right)E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)}+\sqrt{\frac{M}{m}\left(1-E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\right)}\right]\\ \le \sqrt{\frac{1}{{\lambda }_{\min }\left(P\right)}}\left[\sqrt{V\left(t_0\right)}{E}_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)+\sqrt{\frac{M}{m}}\left(1-E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\right)\right]\\ \le \sqrt{\frac{{\lambda }_{\max }\left(P\right)+{\lambda }_{\max }\left(Q\right)}{{\lambda }_{\min }\left(P\right)}}\left[\Vert x\left(t_0\right)\Vert E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\right]\end{array}$

$\begin{array}{c}+\sqrt{\frac{M}{m{\lambda }_{\min }\left(P\right)}}\left(1-E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\right)\\ =C_0{E}_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)+C_1\left(1-E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\right),\end{array}$

where $C_0,C_1$ now depend on ${\lambda }_{\text{max}}\left(P\right),{\lambda }_{\text{max}}\left(Q\right),{\lambda }_{\text{min}}\left(P\right)$ , and initial value $\Vert x\left(t_0\right)\Vert$ .

Remark 1. The Lyapunov functional approach adopted in this paper provides a systematic way to analyze the stability of fractional-order systems with state-dependent delays. However, the construction of the functional is somewhat conservative, which may lead to larger stability bounds in the theoretical analysis. The conservatism of the proposed functional mainly stems from two aspects. First, in the computation of the Caputo derivative of $V_3\left(t\right)$ , the global supremum is approximated by the disturbance bound at the current time instant, neglecting the dynamic variation of historical disturbances. Second, the tuning parameter $v>0$ introduced by Young’s inequality may lead to conservative LMI conditions. Future work could focus on developing less conservative functionals, such as those incorporating more detailed information about the delay and system dynamics.

Corollary 1 For the system considered in Theorem 3.1, when the fractional order $\alpha \to 1$ , the stability criterion in Theorem 3.1 reduces to the well-known exponential stability condition for integer-order systems, indicating that our results are a generalization of the classical theory.

Theorem 3.2 (Local practical Mittag-Leffler Stability for Nonlinear Fractional-Order Systems) Consider the nonlinear fractional-order system with state-dependent delay

${}^C{D}_t^{\alpha }x\left(t\right)=\left(A+\Delta A\left(t,x\left(t\right)\right)\right)x\left(t\right)+Bg\left(x\left(t-\tau \left(x\left(t\right)\right)\right)\right)+\Delta f\left(t\right),$

where the nonlinear function $g(\cdot )$ satisfies the Lipschitz condition for all $x,y\in \Omega$ , ($\Omega =\left\{x\in {ℝ}^n|\Vert x\Vert \le R\right\}$ ) there exists a positive constant $L_g$ such that we have

$\Vert g\left(x\right)-g\left(y\right)\Vert \le L_g\Vert x-y\Vert ,$

and $g\left(0\right)=0$ . The perturbations $\text{Δ}A\left(t,x\left(t\right)\right)$ and $\text{Δ}f\left(t\right)$ are bounded as in (H2). Under the assumptions (H1)-(H4), if there exist positive definite matrices $P,Q$ and parameters $\lambda ,\mu ,\epsilon ,\sigma ,\nu >0$ such that the following LMIs hold

Current state LMI

$2PA+\left(\epsilon +\sigma \right)P^{\top }P+{\epsilon }^{-1}{\rho }^{2}I+Q+\lambda I\prec 0$

Delayed state LMI

${\nu }^{-1}{L}_g^2{B}^{\top }{P}^{\top }PB-\left(1-\eta \right)Q+\mu I\prec 0,$

then the system is lobal practical Mittag-Leffler stable. The state trajectory satisfies the following norm bound

$\Vert x\left(t\right)\Vert \le C_0{E}_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)+C_1\left(1-E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\right).$

Proof. The proof follows the same structure as in Theorem 3.1. Choose a Lyapunov functional similar to that in Theorem 3.1

$\begin{array}{c}V\left(t\right)=V_1\left(t\right)+V_2\left(t\right)+V_3\left(t\right),\\ =x^{\top }\left(t\right)Px\left(t\right)+{\displaystyle {\int }_{t-\tau \left(x\left(t\right)\right)}^t{x}^{\text{T}}\left(s\right)Qx\left(s\right)\text{d}s}\\ +{\displaystyle {\int }_{t_0}^t\left({\Vert \Delta A\left(s,x\left(s\right)\right)\Vert }^2+{\Vert \Delta f\left(s\right)\Vert }^2\right)\text{d}s}.\end{array}$ (20)

By the properties of fractional calculus and the system dynamics, we have

${}^C{D}_t^{\alpha }{V}_1\left(t\right)\le 2x^{\top }\left(t\right)P\left[\left(A\left(t\right)+\Delta A\left(t,x\left(t\right)\right)\right)x\left(t\right)+Bg\left(x\left(t-\tau \left(x\left(t\right)\right)\right)\right)+\Delta f\left(t\right)\right],$

for $2x^{\top }\left(t\right)P\text{Δ}A\left(t,x\left(t\right)\right)x\left(t\right)$ and $2x^{\top }\left(t\right)P\text{Δ}f\left(t\right)$ by Lemma 2, we obtain

$\begin{array}{l}2x^{\text{T}}\left(t\right)P\Delta A\left(t,x\left(t\right)\right)x\left(t\right)\\ \le \epsilon x^{\top }\left(t\right)P^{\top }Px\left(t\right)+{\epsilon }^{-1}{x}^{\top }\left(t\right)\Delta A^{\top }\left(t,x\left(t\right)\right)\Delta A\left(t,x\left(t\right)\right)x\left(t\right),\end{array}$ (21)

$2x^{\top }\left(t\right)P\Delta f\left(t\right)\le \sigma x^{\top }\left(t\right)P^{\text{T}}Px\left(t\right)+{\sigma }^{-1}{\Vert \Delta f\left(t\right)\Vert }^2.$ (22)

For the nonlinear term $2x^{\text{T}}\left(t\right)PBg\left(x\left(t-\tau \left(x\left(t\right)\right)\right)\right)$ , let $z\left(t\right)=x\left(t-\tau \left(x\left(t\right)\right)\right)$ , and still satisfies $\Vert z\left(t\right)\Vert \le R$ . Utilizing $g\left(0\right)=0$ and the Lipschitz condition $\Vert g\left(z\right)\Vert \le L_g\Vert z\Vert$ , combining with Young’s inequality

$2x^{\top }\left(t\right)PBg\left(z\left(t\right)\right)\le \nu x^{\top }\left(t\right)x\left(t\right)+{\nu }^{-1}{L}_g^2{z}^{\top }\left(t\right)B^{\top }{P}^{\top }PBz\left(t\right).$ (23)

Combining (21), (22) with (23), we get

$\begin{array}{c}{}^{C}D{}_t^{\alpha }{V}_1\left(t\right)\le 2x^{\top }\left(t\right)PA\left(t\right)x\left(t\right)+\epsilon x^{\text{T}}\left(t\right)P^{\text{T}}Px\left(t\right)\\ +{\epsilon }^{-1}{x}^{\text{T}}\left(t\right)\Delta A^{\text{T}}\left(t,x\left(t\right)\right)\Delta A\left(t,x\left(t\right)\right)x\left(t\right)\\ +\sigma x^{\text{T}}\left(t\right)P^{\text{T}}Px\left(t\right)+{\sigma }^{-1}{\Vert \Delta f\left(t\right)\Vert }^2+\nu x^{\text{T}}\left(t\right)x\left(t\right)\\ +{\nu }^{-1}{L}_g^2{z}^{\text{T}}\left(t\right)B^{\text{T}}{P}^{\text{T}}PBz\left(t\right).\end{array}$ (24)

For $V_2\left(t\right)$ and $V_3\left(t\right)$ , similar to the derivation in Theorem 3.1, we have

${}^C{D}_t^{\alpha }{V}_2\left(t\right)\le x^{\text{T}}\left(t\right)Qx\left(t\right)-\left(1-\eta \right)z^{\text{T}}\left(t\right)Qz\left(t\right),$

${}^C{D}_t^{\alpha }{V}_3\left(t\right)\le {\Vert \Delta A\left(t,x\left(t\right)\right)\Vert }^2+{\Vert \Delta f\left(t\right)\Vert }^2.$

Combine the above derivative terms and organize them with the LMI conditions, we can deduce

$\begin{array}{c}{}^{C}D{}_t^{\alpha }V\left(t\right)\le -\lambda x^{\text{T}}\left(t\right)x\left(t\right)-\mu z^{\text{T}}\left(t\right)z\left(t\right)+M,\\ \le -mV\left(t\right)+M\end{array}$ (25)

Using Lemma 3, we finally obtain that the state norm satisfies

$\Vert x\left(t\right)\Vert \le C_0{E}_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)+C_1\left(1-E_{\alpha }\left(-m{\left(t-t_0\right)}^{\alpha }\right)\right).$

4. Numerical Example

Example 1 Consider the following fractional-order system with state-dependent delay and disturbances:

${}^C{D}_t^{\alpha }x\left(t\right)=\left(A+\Delta A\left(t\right)\right)x\left(t\right)+Bx\left(t-\tau \left(x\left(t\right)\right)\right)+\Delta f\left(t\right),$

where $\alpha =0.5$ , $A=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$ , $B=\left[\begin{array}{cc}0.1& 0\\ 0& 0.1\end{array}\right]$ , $\tau \left(x\left(t\right)\right)=0.2\left|x_1\left(t\right)\right|+0.2\left|x_2\left(t\right)\right|$ , $\Delta A\left(t\right)=0.1\sin \left(t\right)I$ , $\Delta f\left(t\right)=0.05\cos \left(t\right)\left[\begin{array}{l}1\hfill \\ 1\hfill \end{array}\right]$ , $x\left(0\right)=\left[\begin{array}{l}0.5\hfill \\ 0.3\hfill \end{array}\right]$ , $I$ is a 2nd-order identity matrix. For the state-dependent delay $\tau \left(x\right)=0.2\left|x_1\right|+0.2\left|x_2\right|$ : $\tau \left(x\right)$ is locally Lipschitz on ${ℝ}^n$ . It is not differentiable at $x_1=0$ or $x_2=0$ , but its subgradient is given by

${\partial }_x\tau \left(x\right)=0.2\cdot \text{sgn}\left(x_1\right)e_1+0.2\cdot \text{sgn}\left(x_2\right)e_2,$

where $\text{sgn}(\cdot )$ denotes the sign function. The subgradient satisfies ${\Vert {\partial }_x\tau \left(x\right)\Vert }_{\infty }\le 0.2$ , hence

$\dot{\tau }\left(x\left(t\right)\right)={\partial }_x\tau \left(x\left(t\right)\right)\dot{x}\left(t\right)\le 0.2{\Vert \dot{x}\left(t\right)\Vert }_1,$

and $\dot{\tau }\left(x\left(t\right)\right)$ exists almost everywhere along system trajectories. We take $\eta =0.2<1$ , so Assumption (H1) is satisfied even at points where $x_1=0$ or $x_2=0$ (non-differentiable locations of absolute value terms). First, verify the assumptions, for the time-delay constraint, we have $\dot{\tau }\left(x\left(t\right)\right)\le 0.2{\Vert \dot{x}\left(t\right)\Vert }_1\le 0.2\left(\Vert A\Vert +\rho +\Vert B\Vert \right)\Vert x\left(t\right)\Vert \le 0.25\le 1$ , for the perturbation bounds, $\Vert \text{Δ}A\left(t\right)\Vert \le \rho =0.1$ , $\Vert \text{Δ}f\left(t\right)\Vert \le d=0.05\sqrt{2}$ . According to the stability criteria in Theorem 3.1, we select positive definite matrices $P=I$ and $Q=I$ , Scaling parameters $\epsilon =\sigma =0.5$ , $\nu =1$ . By solving the LMIs we obtain the current state related LMI $\lambda =0.8,$ and the delayed state related LMI $\mu =0.6$ . Then, calculate the key stability parameter $m=\text{min}\left\{\frac{0.8}{1},\frac{0.6}{1}\right\}=0.6$ . Next, by calculating the Caputo fractional derivative of $V\left(t\right)$ and applying Lemma 1 and Young’s inequality, we obtain

${}^C{D}_t^{\alpha }V\left(t\right)\le -mV\left(t\right)+M$

where $M={\sigma }^{-1}{d}^2+{\Vert \Delta A\left(t\right)\Vert }^2+{\Vert \Delta f\left(t\right)\Vert }^2=0.025$ . By Lemma 3, the bound on $V\left(t\right)$ implies

$V\left(t\right)\le V\left(0\right)E_{\alpha }\left(-m{\left(t-0\right)}^{\alpha }\right)+\frac{M}{m}\left(1-E_{\alpha }\left(-m{\left(t-0\right)}^{\alpha }\right)\right).$

Since $V\left(0\right)=x^{\top }\left(0\right)Px\left(0\right)={0.5}^2+{0.3}^2=0.34$ , the state norm satisfies

$\Vert x\left(t\right)\Vert \le \sqrt{V\left(t\right)}\le \sqrt{0.34}{E}_{0.5}\left(-0.6t^{0.5}\right)+\sqrt{\frac{0.025}{0.6}}\left(1-E_{0.5}\left(-0.6t^{0.5}\right)\right).$

The system satisfies local practical Mittag-Leffler stability.

As clearly verified in Figure 1, the numerical simulation results demonstrate the correctness of the theoretical derivation. The blue solid line represents the actual system state norm, which starts from the initial value $\Vert x\left(0\right)\Vert =0.58$ and exhibits excellent convergence characteristics under time-varying parameter perturbations and external disturbances. Meanwhile, the theoretical upper bound derived from Theorem 3.1, represented by the red dashed line in the figure, remains strictly above the actual state norm throughout the process. This not only confirms the effectiveness of the theoretical bound described by formula (19) but also reveals its inherent conservatism. It is particularly noteworthy that despite the presence of state-dependent delay $\tau \left(x\left(t\right)\right)=0.2\left|x_1\left(t\right)\right|+0.2\left|x_2\left(t\right)\right|$ and time-varying perturbation $\Delta A\left(t\right)=0.1\sin \left(t\right)I$ , the system maintains robust Mittag-Leffler stability, which is perfectly consistent with our theoretical predictions. The simulation results that the stability criterion proposed in this paper can effectively handle time delays and uncertainties in fractional-order systems, providing a reliable theoretical basis for the analysis and design of such systems.

Figure 1. State trajectories with state-dependent delay of Example 4.1.

Example 2 Consider a fractional-order system with stronger nonlinearities and time-varying perturbations

${}^C{D}_t^{\alpha }x\left(t\right)=\left(A+\Delta A\left(t,x\left(t\right)\right)\right)x\left(t\right)+B\sin x\left(t-\tau \left(x\left(t\right)\right)\right)+\Delta f\left(t\right),$

where $\alpha =0.7$ , $A=\left[\begin{array}{cc}-1.5& 0.2\\ 0.1& -1.4\end{array}\right]$ , $B=\left[\begin{array}{cc}0.08& 0\\ 0& 0.08\end{array}\right]$ , $\tau \left(x\left(t\right)\right)=0.15\left(1+\text{tanh}\left(\Vert x\left(t\right)\Vert \right)\right)$ ,

$\Delta A\left(t,x\left(t\right)\right)=0.05{\text{e}}^{-\Vert x\left(t\right)\Vert }\cos \left(t\right)I$ , $\text{Δ}f\left(t\right)=0.04\frac{t}{1+t^2}\left[\begin{array}{c}1\\ -1\end{array}\right]$ , $x\left(0\right)=\left[\begin{array}{c}0.6\\ -0.4\end{array}\right]$ .

First, verify the assumptions, for the time-delay constraint, we have $\dot{\tau }\left(x\left(t\right)\right)=0.15{\mathrm{sech}}^2\left(\Vert x\left(t\right)\Vert \right)\cdot \frac{x^{\top }\left(t\right)\dot{x}\left(t\right)}{\Vert x\left(t\right)\Vert }\le 0.15\cdot 1\cdot \Vert \dot{x}\left(t\right)\Vert \le 0.1<1$ , for the perturbation bounds, $\Vert \Delta A\left(t\right)\Vert \le \rho =0.05$ , $\Vert \Delta f\left(t\right)\Vert \le d=0.02$ . $\tau \left(x\right)=0.15\left(1+\text{tanh}\left(\Vert x\Vert \right)\right)$ is globally $C^1\left({ℝ}^n\right)$ . Direct computation yields $\Vert \nabla \tau \left(x\right)\Vert \le 0.15$ , we select $\eta =0.15<1$ . Assumption (H1) holds everywhere. According to the stability criteria in Theorem 3.1, we select positive definite matrices $P=\left[\begin{array}{cc}1& 0\\ 0& 1.2\end{array}\right]$ and $Q=1.5I$ , Scaling parameters $\epsilon =0.6$ $\sigma =0.4$ , $\nu =1$ . By solving the LMIs we obtain the current state related LMI $\lambda =0.4$ , and the delayed state related LMI $\mu =0.3$ . Then, calculate the key stability parameter $m=\text{min}\left\{\frac{0.4}{1.2},\frac{0.3}{1.5}\right\}=0.2$ . Next, by calculating the Caputo fractional derivative of $V\left(t\right)$ and applying Lemma 1 and Young’s inequality, we obtain

${}^C{D}_t^{\alpha }V\left(t\right)\le -mV\left(t\right)+M$

where $M={\sigma }^{-1}{d}^2+{\Vert \Delta A\left(t\right)\Vert }^2+{\Vert \Delta f\left(t\right)\Vert }^2=0.0129$ . By Lemma 3, the bound on $V\left(t\right)$ implies

$V\left(t\right)\le V\left(0\right)E_{\alpha }\left(-m{\left(t-0\right)}^{\alpha }\right)+\frac{M}{m}\left(1-E_{\alpha }\left(-m{\left(t-0\right)}^{\alpha }\right)\right).$

Since $V\left(0\right)=x^{\top }\left(0\right)Px\left(0\right)={0.6}^2\cdot 1+{\left(-0.4\right)}^2\cdot 1.2=0.552$ , the state norm satisfies

$\Vert x\left(t\right)\Vert \le \sqrt{V\left(t\right)}\le \sqrt{0.552}{E}_{0.7}\left(-0.2t^{0.7}\right)+\sqrt{\frac{0.0129}{0.2}}\left(1-E_{0.7}\left(-0.2t^{0.7}\right)\right).$

The system satisfies local practical Mittag-Leffler stability.

Figure 2. State trajectories with state-dependent delay of Example 4.2.

As clearly verified in Figure 2, the blue solid line represents the Actual state norm. Starting from the initial value, it gradually decays over time. It can be seen that the actual state norm decreases rapidly and eventually approaches near 0, showing good convergence characteristics. The red dashed line represents the Theoretical upper bound, which is the upper bound of the state norm derived from the theorem. The actual state norm is always strictly below the theoretical upper bound, which verifies the correctness of the theoretical derivation and indicates that the system satisfies global robust Mittag-Leffler stability. That is, even with stronger nonlinearities and time-varying perturbations, the state of the system can still be effectively constrained by the theoretical upper bound, and the actual state will stabilize within a small range.

5. Conclusions and Suggestions

We investigate the problem of local robust practical Mittag-Leffler stability for a class of fractional-order dynamical systems subject to state-dependent delays and bounded perturbations. Due to the existence of non-zero bounded additive perturbations, the investigated system generally has no constant equilibrium point. The main contributions are summarized as follows. A novel LMI based stability criterion has been established by constructing an appropriate Lyapunov-Krasovskii functional that comprehensively accounts for the system’s energy, historical state effects, and disturbance energy. A weakened sufficient condition has been derived, which relaxes the constraints on the time varying delay and perturbations compared to existing results in the literature. The obtained results characterize that system trajectories converge to a small bounded neighborhood around the origin, whose size is dominated by the disturbance bounds. The effectiveness and applicability of the proposed theoretical results have been verified through two numerical examples, including a linear system and a nonlinear system with stronger nonlinearities.

Acknowledgements

Sincere thanks to the members of JAMP for their professional performance, and special thanks to managing editor Hellen XU for a rare attitude of high quality.

Author Contributions

Conceptualization, F.H.Z. and X.L.L.; methodology, F.H.Z.; software, F.H.Z.; validation, F.H.Z., X.L.L., and Z.Z.; formal analysis, F.H.Z.; investigation, F.H.Z.; resources, X.L.L.; data curation, F.H.Z.; writing original draft preparation, F.H.Z.; writing review and editing, X.L.L.; visualization, F.H.Z.; supervision, X.L.L.; project administration, X.L.L. All authors have read and agreed to the published version of the manuscript.