Conformable Time-Delay Systems: Stability and Stabilization Under One-Sided Lipschitz Conditions

Abstract

This study looks at the stability and stabilization issues concerning the nonlinear time-delay systems specified by conformable derivatives. These requirements can be used for many useful applications. Through the construction of appropriate Lyapunov–Krasovskii functionals, we develop novel linear matrix inequality (LMI) conditions for the exponential stability of autonomous systems and practical exponential stability for systems subject to bounded perturbations. Furthermore, we propose state-feedback stabilization strategies that transform the controller design problem into a convex optimization framework solvable via efficient LMI techniques. The theoretical developments are comprehensively validated through numerical examples that demonstrate the effectiveness of the proposed stability and stabilization criteria. The results establish a rigorous framework for analyzing and controlling conformable fractional-order systems with time delays, bridging theoretical advances with practical implementation considerations.

1. Introduction

Fractional calculus has become an essential mathematical framework for modeling complex physical phenomena with memory effects and hereditary properties [1,2]. Unlike classical integer-order calculus, fractional derivatives incorporate non-local operators that capture history-dependent behavior, making them particularly suitable for viscoelastic materials, biological systems, and anomalous diffusion. Among various fractional derivative definitions, the conformable fractional derivative introduced by Khalil et al. [1] and refined by Abdeljawad [2] has gained significant attention due to its mathematical simplicity and preservation of classical derivative properties. Subsequent generalizations by Zhao and Luo [3], Atangana et al. [4], Martínez et al. [5], and Sadek and Akgul [6] have further expanded the theoretical foundation, revealing additional properties and applications across scientific domains.

There is additional evidence that the conformable derivatives fit certain classes of dynamical systems better than the classical fractional derivatives do. For example, a conformable derivative succinctly describes electrical circuits that have memory elements and nonlinear energy storage, according to [7]. In the same way, power electronic converters that operate under variable solar radiation exhibit dynamics allowing for simpler controller synthesis via the conformable derivative framework while retaining a fractional-order character [8]. According to the above example, the conformable derivative is not just an alternative definition. Rather, they give us better results to capture system behavior in simpler and more interpretable form. In addition to electrical and energy systems, derivatives have been successfully applied to optimization algorithms, gray system modeling, and quantum control due to their original characteristics. Right now, fractional-order differential equations are in wide demand for modern engineering applications [9,10,11].

There are several benefits approved by the conformable derivatives in this study as compared to standard fractional derivatives. Conformable derivatives, unlike the Riemann–Liouville or Caputo derivatives, do indeed preserve many basic properties of classical calculus. These fundamental properties include the product rule, the quotient rule, and the chain rule. Thus, they simplify Lyapunov-based analysis as well as controller design. The property is mathematically elegant and makes stability analysis more simple, yet it allows for the incorporation of types of memories and hereditary properties. Also, conformable derivatives can be easily processed with numbers and are more interpretable than fractional-order ones. This makes them more suited for control applications with major importance in theoretical and practical terms.

At the same time, time-delay systems, as commented on earlier, have been a basic research area in control theory because of their occurrence in engineering applications such as chemical processes and networked control systems [12,13]. Time delays in a system could cause behavior that degrades performance, cause instability, or lead to complicated dynamical behaviors [14,15]. Fractional calculus and time-delay systems combined would be a natural extension for modeling systems that exhibit both memory and delayed effects. Some examples include nonlinear electrical networks [16], heat conduction [17], finance [14], and quantum control [11].

In this setting, conformable fractional derivatives have advantages. Strong work was carried out by Naifar et al. [18] to obtain foundational stability results. Luo et al. [19] investigated fractional exponential stability in conformable delayed systems with impulsive effects. Aldandani et al. [20] and Kharrat et al. [21] dealt with practical stability issues by constructing structures for systems that are affected by disturbances from outside. It is challenging to characterize nonlinearities tractably and accurately. The one-sided Lipschitz condition imposes less conservative bounds on wide classes of nonlinearity than the classical Lipschitz condition [22,23]. It also allows for a refinement of the stability analysis and the design of the controller.

Recent stabilization advances include guaranteed cost control with event-triggered mechanisms [24], observer designs for nonlinear tempered fractional-order systems [25], stabilization criteria for fuzzy delayed systems [26], and predictor-feedback methods with quantization effects [27]. Practical observer design techniques using Caputo fractional derivatives have been proposed by Naifar et al. [28]. Linear matrix inequalities (LMIs) have emerged as powerful tools, with applications spanning neutral systems with multiple delays [29], stabilization of feedforward nonlinear systems [30], and robust stability analysis of mechanical systems [31]. Beyond control, conformable derivatives have found utility in optimization contexts [9,32], soliton solutions [33], grey system modeling [10], and quantum control [11].

Recent stability methodologies continue to evolve, with Chen et al. [34] investigating delayed impulses, and Zhang et al. [35] examining stochastic nonlinear delay equations and event-triggered adaptive stabilization approaches [36]. Reduced-conservatism LMI conditions have been proposed for exact Takagi–Sugeno models [37], while specialized approaches address neutral-type bilinear systems [38], p-normal nonlinear systems [39], and systems with highly nonlinear impulses [40].

Even though technology has advanced, there remain major research differences in stability assessment and stabilization regarding fractional-order systems with time delays. Multiple methods are conservative when handling nonlinearities that have one-sided Lipschitz conditions. Moreover, several works do not address practically stable methods under bounded perturbation or offer weak controller designs. The study of properties of conformable derivatives together with the construction of complex Lyapunov–Krasovskii functionals is quite limited, particularly for conditions that are based on LMIs. Furthermore, these conditions are computationally simple as well as mathematically correct.

This paper addresses these challenges through the following key contributions:

• Novel LMI conditions for exponential stability of autonomous conformable time-delay systems with one-sided Lipschitz nonlinearities and quadratic inner-boundedness constraints.
• Practical exponential stability criteria for perturbed systems with bounded disturbances, providing computable ultimate bounds.
• State-feedback stabilization strategies for nominal and perturbed systems via convex optimization.
• Systematic Lyapunov–Krasovskii functional construction tailored to conformable derivative properties.
• Comprehensive numerical validation demonstrating effectiveness and applicability.

The rest of this paper is organized as follows: Section 2 presents the preliminaries, Section 3 and Section 4 develop the stability analysis, Section 5 and Section 6 address stabilization, Section 7 provides numerical examples, Section 8 compares our approach with existing works, and Section 9 concludes with future directions.

2. Preliminaries

In this work, the following notation is adopted: ${\mathbb{R}}^n$ is the n-dimensional Euclidean space; ${\mathbb{R}}^{n\times n}$ contains all $n\times n$ real matrices; I is the identity matrix; $| · |$ signifies the Euclidean norm for vectors; ${\lambda }_{min}\left(\mathrm{\Theta }\right)$ and ${\lambda }_{max}\left(\mathrm{\Theta }\right)$ are the smallest and largest eigenvalues of the matrix $\mathrm{\Theta }$, respectively; $\mathrm{Sym}\left(\mathrm{\Theta }\right)$ is defined as $\mathrm{\Theta }+{\mathrm{\Theta }}^T$; and $\mathrm{diag}(\mathrm{\Theta },\mathrm{\Theta })$ represents the block diagonal matrix $\left[\begin{array}{ccc}\mathrm{\Theta }& 00& \mathrm{\Theta }\end{array}\right]$.

Definition 1 (Conformable Fractional Derivative).

For a function f defined on the interval$[p,\infty )$and a fractional order$\theta \in (0,1]$, the conformable fractional derivative starting from point p is given by the limit expression

$$\left(T_p^{\theta }f\right)\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{f\left(x+h{(x-p)}^{1-\theta }\right)-f\left(x\right)}{h}},\mathit{for}\mathit{all}x>p.$$

This definition extends to the left endpoint$x=p$by continuity: if the right-hand limit$L={lim}_{x\to p^{+}}\left(T_p^{\theta }f\right)\left(x\right)$exists, then we define$\left(T_p^{\theta }f\right)\left(p\right)=L$.

Remark 1.

The definition presented above generalizes the concept introduced by Khalil et al., which corresponds to the specific case where the starting point p is zero. For notational convenience, we will use the shorthand $T^{\theta }:=T_0^{\theta }$ in the subsequent analysis.

Consider the autonomous delayed nonlinear system expressed in conformable derivative form:

$$\begin{array}{cc}\hfill T^c\zeta \left(t\right)& =A\zeta \left(t\right)+A_d\zeta (t-\iota )+f(\zeta \left(t\right),\zeta (t-\iota )),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \zeta \left(\lambda \right)& =\varphi \left(\lambda \right),\lambda \in [-\iota ,0],\hfill \end{array}$$

(2)

where

• State Vector: $\zeta \left(t\right)\in {\mathbb{R}}^n$;
• System Matrices: $A,A_d\in {\mathbb{R}}^{n\times n}$;
• Delay: $\iota >0$;
• Derivative Order: $c\in (0,1)$ for $T^c$;
• Nonlinearity: $f:{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, $f(0,0)=0$.

Definition 2 (Conformable Exponential).

For$\sigma >0$and derivative order$c\in (0,1]$, define

$$E_c(-\sigma ,t)=e^{-\sigma t^c/c}.$$

This notation will be used consistently throughout.

The nonlinear function f in System (1) satisfies the following conditions:

Assumption 1 (One-Sided Lipschitz Continuity).

There exist real scalars${\rho }_1,{\rho }_2\in \mathbb{R}$such that, for any vectors$w,z,w_d,z_d\in {\mathbb{R}}^n$, the following inequalities hold:

$$\begin{array}{c}\hfill \langle f(w,w_d)-f(z,z_d),w-z\rangle \le {\rho }_1{\parallel w-z\parallel }^2+{\rho }_2{\parallel w_d-z_d\parallel }^2,\end{array}$$

(3)

$$\begin{array}{c}\hfill \langle f(w,w_d)-f(z,z_d),w_d-z_d\rangle \le {\rho }_1{\parallel w-z\parallel }^2+{\rho }_2{\parallel w_d-z_d\parallel }^2.\end{array}$$

(4)

Assumption 2 (Quadratic Inner-Boundedness).

There exist constants${\beta }_1,{\beta }_2,{\gamma }_1,{\gamma }_2\in \mathbb{R}$such that, for all$w,z,w_d,z_d\in {\mathbb{R}}^n$,

$$\begin{array}{cc}\hfill \parallel f(w,w_d)-f(z,z_d){\parallel }^2& \le {\beta }_1{\parallel w-z\parallel }^2+{\beta }_2{\parallel w_d-z_d\parallel }^2\hfill \\ \hfill & +{\gamma }_1\langle w-z,f(w,w_d)-f(z,z_d)\rangle \hfill \\ \hfill & +{\gamma }_2\langle w_d-z_d,f(w,w_d)-f(z,z_d)\rangle .\hfill \end{array}$$

(5)

Remark 2.

The proposed OSL conditions involve two inequalities: one with respect to the current state difference $(w-z)$, and another with respect to the delayed state difference $(w_d-z_d)$. While many OSL frameworks impose a single inequality on the current state and handle delays via Lyapunov–Krasovskii functionals or integral quadratic constraints (IQCs), the dual condition adopted here provides tighter bounds for nonlinearities that depend explicitly on both present and delayed states. This choice is motivated by the need to reduce conservatism in systems where delay terms significantly influence the dynamics. For related discussions, see Horn and Johnson [41] and Naifar et al. [22]. Compared to classical Lipschitz or sector constraints, the OSL/QIB framework offers a less restrictive characterization for nonlinearities, enabling broader applicability while maintaining tractability.

Definition 3 (Practical Exponential Stability).

System (1) is said to be practically exponentially stable if there exist constants$M>0$, $\sigma >0$, and$r>0$, such that

$$\parallel \zeta \left(t\right)\parallel \le {M\parallel \varphi \parallel }_{\infty }{E}_c(-\sigma ,t)+r,\forall t\ge 0,$$

(6)

where${\parallel \varphi \parallel }_{\infty }={sup}_{\theta \in [-\iota ,0]}\parallel \varphi \left(\theta \right)\parallel$and$E_c(-\sigma ,t)=e^{-\sigma t^c/c}$is the conformable exponential function.

Remark 3.

In the case when$r=0$, System (1) is said to be exponentially stable.

For the stability analysis of System (1), we employ the following Lyapunov–Krasovskii functional (LKF):

$$V\left(t\right)=V_1\left(t\right)+V_2\left(t\right),$$

(7)

where

$$\begin{array}{c}{V}_1\left(t\right)={\zeta }^T\left(t\right)P\zeta \left(t\right),\hfill \end{array}$$

(8)

$$\begin{array}{c}{V}_2\left(t\right)={\int }_t^{t+\iota }{s}^{c-1}{e}^{2\sigma \left({\displaystyle \frac{s^c}{c}}-{\displaystyle \frac{t^c}{c}}-{\displaystyle \frac{{\iota }^c}{c}}\right)}{\zeta }^T(s-\iota )Q\zeta (s-\iota )ds,\hfill \end{array}$$

(9)

with $P,Q\in {\mathbb{R}}^{n\times n}$ symmetric positive definite matrices, and $\sigma >0$.

Lemma 1 (Conformable Derivative of L-K Functional).

The conformable derivative of the LKF (7) satisfies

$$\begin{array}{c}{T}^c{V}_1\left(t\right)=2{\left(T^c\zeta \left(t\right)\right)}^{T}P\zeta \left(t\right),\hfill \end{array}$$

(10)

$$\begin{array}{c}\hfill T^c{V}_2\left(t\right)\le {\zeta }^T\left(t\right)Q\zeta \left(t\right)-e^{-2\sigma {\iota }^c/c}{\zeta }^T(t-\iota )Q\zeta (t-\iota )-2\sigma V_2\left(t\right).\end{array}$$

(11)

Proof. 

$T^c{V}_1\left(t\right)=2{\left(T^c\zeta \left(t\right)\right)}^{T}P\zeta \left(t\right)$ is trivial.

$$\begin{array}{cc}\hfill T^c{V}_2\left(t\right)& =t^{1-c}({(t+\iota )}^{c-1}{e}^{2\sigma \left({\displaystyle \frac{{(t+\iota )}^c}{c}}-{\displaystyle \frac{t^c}{c}}-{\displaystyle \frac{{\iota }^c}{c}}\right)}{\zeta }^T\left(t\right)Q\zeta \left(t\right)\hfill \\ \hfill & -t^{c-1}{e}^{-2\sigma {\displaystyle \frac{{\iota }^c}{c}}}{\zeta }^T(t-\iota )Q\zeta (t-\iota )\hfill \\ \hfill & -2\sigma t^{c-1}{V}_2\left(t\right))\hfill \end{array}$$

Since $t^{1-c}{(t+\iota )}^{c-1}\le 1$ and ${\displaystyle \frac{{(t+\iota )}^c}{c}}-{\displaystyle \frac{t^c}{c}}-{\displaystyle \frac{{\iota }^c}{c}}\le 0$, then

$$T^c{V}_2\left(t\right)\le {\zeta }^T\left(t\right)Q\zeta \left(t\right)-e^{-2\sigma {\displaystyle \frac{{\iota }^c}{c}}}{\zeta }^T(t-\iota )Q\zeta (t-\iota )-2\sigma V_2\left(t\right).$$

   □

3. Stability Analysis of One-Sided Lipschitz Conformable Time-Delay Systems

This section presents the exponential stability analysis for the conformable time-delay system (1) under Assumptions 1 and 2.

Theorem 1 (Exponential Stability via LMI).

Consider the autonomous conformable time-delay system (1) with one-sided Lipschitz nonlinearity satisfying Assumptions 1 and 2. Suppose that there exist symmetric positive definite matrices $P,Q\in {\mathbb{R}}^{n\times n}$, and positive scalars $\sigma >0$, ${\mu }_1>0$, ${\mu }_2>0$, ${\mu }_3>0$, and $ϵ>0$ such that the following LMI is feasible:

$$\mathrm{\Theta }=\left[\begin{array}{ccc}{\mathrm{\Theta }}_{11}& P{A}_d& {\mathrm{\Theta }}_{13}\\ A_d^{T}P& {\mathrm{\Theta }}_{22}& {\mathrm{\Theta }}_{23}\\ {\mathrm{\Theta }}_{13}^T& {\mathrm{\Theta }}_{23}^T& {\mathrm{\Theta }}_{33}\end{array}\right]<0,$$

(12)

where

$$\begin{array}{cc}\hfill {\mathrm{\Theta }}_{11}& =A^{T}P+PA+2\sigma P+Q+2({\mu }_1{\rho }_1+{\mu }_2{\beta }_1+{\mu }_3{\rho }_1)I+ϵI,\hfill \\ \hfill {\mathrm{\Theta }}_{22}& =2({\mu }_1{\rho }_2+{\mu }_2{\beta }_2+{\mu }_3{\rho }_2)I-e^{-2\sigma {\iota }^c/c}Q,\hfill \\ \hfill {\mathrm{\Theta }}_{13}& =P+({\mu }_2{\gamma }_1-{\mu }_1)I+{\mu }_3{\rho }_{1}I,\hfill \\ \hfill {\mathrm{\Theta }}_{23}& ={\mu }_2{\gamma }_{2}I-{\mu }_{3}I+{\mu }_3{\rho }_{2}I,\hfill \\ \hfill {\mathrm{\Theta }}_{33}& =-2{\mu }_{2}I-2{\mu }_{3}I.\hfill \end{array}$$

Then, the autonomous system is exponentially stable, and the state trajectory satisfies

$$\parallel \zeta \left(t\right)\parallel \le \sqrt{{\displaystyle \frac{{\lambda }_{max}\left(P\right)+{\lambda }_{max}\left(Q\right)\frac{{\iota }^c}{c}}{{\lambda }_{min}\left(P\right)}}}{\parallel \varphi \parallel }_{\infty }{e}^{-\sigma t^c/c}.$$

(13)

Proof. 

We employ the LKF (7):

$$V\left(t\right)=V_1\left(t\right)+V_2\left(t\right),$$

where $V_1\left(t\right)$ and $V_2\left(t\right)$ are defined in (8) and (9), respectively.

Using Lemma 1 and substituting the system dynamics (1),

$$\begin{array}{cc}\hfill T^{c}V\left(t\right)+2\sigma V\left(t\right)& \le 2{\zeta }^T\left(t\right)A^{T}P\zeta \left(t\right)+2{\zeta }^T(t-\iota )A_d^{T}P\zeta \left(t\right)\hfill \\ & +2f^{T}P\zeta \left(t\right)+2\sigma {\zeta }^T\left(t\right)P\zeta \left(t\right)+{\zeta }^T\left(t\right)Q\zeta \left(t\right)\hfill \\ & -e^{-2\sigma {\iota }^c/c}{\zeta }^T(t-\iota )Q\zeta (t-\iota ).\hfill \end{array}$$

Using Assumption 1 with $z=0$, $z_d=0$:

$$\begin{array}{c}\langle \zeta ,f\rangle \le {\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2,\hfill \end{array}$$

(14)

$$\begin{array}{c}\hfill \langle \zeta (t-\iota ),f\rangle \le {\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2,\end{array}$$

(15)

which imply

$$\begin{array}{cc}\hfill 2{\mu }_1\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta ,f\rangle \right)& \ge 0,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill 2{\mu }_3\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta (t-\iota ),f\rangle \right)& \ge 0.\hfill \end{array}$$

(17)

Using Assumption 2 with $z=0$, $z_d=0$:

$${\parallel f\parallel }^2\le {\beta }_1{\parallel \zeta \parallel }^2+{\beta }_2{\parallel \zeta (t-\iota )\parallel }^2+{\gamma }_1\langle \zeta ,f\rangle +{\gamma }_2\langle \zeta (t-\iota ),f\rangle ,$$

which implies

$$2{\mu }_2\left({\beta }_1{\parallel \zeta \parallel }^2+{\beta }_2{\parallel \zeta (t-\iota )\parallel }^2+{\gamma }_1\langle \zeta ,f\rangle +{\gamma }_2\langle \zeta (t-\iota ),f\rangle -{\parallel f\parallel }^2\right)\ge 0.$$

(18)

Adding Inequalities (16)–(18) to the derivative bound,

$$\begin{array}{cc}\hfill T^{c}V\left(t\right)+2\sigma V\left(t\right)& \le {\zeta }^T\left(t\right)\left[A^{T}P+PA+2\sigma P+Q\right]\zeta \left(t\right)\hfill \\ & +2{\zeta }^T(t-\iota )A_d^{T}P\zeta \left(t\right)+2{\zeta }^T\left(t\right)Pf\hfill \\ & -e^{-2\sigma {\iota }^c/c}{\zeta }^T(t-\iota )Q\zeta (t-\iota )\hfill \\ & +2{\mu }_1\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta ,f\rangle \right)\hfill \\ & +2{\mu }_3\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta (t-\iota ),f\rangle \right)\hfill \\ & +2{\mu }_2\left({\beta }_1{\parallel \zeta \parallel }^2+{\beta }_2{\parallel \zeta (t-\iota )\parallel }^2+{\gamma }_1\langle \zeta ,f\rangle +{\gamma }_2\langle \zeta (t-\iota ),f\rangle -{\parallel f\parallel }^2\right).\hfill \end{array}$$

Rewriting in quadratic form with the extended state vector $\eta \left(t\right)=[{\zeta }^T\left(t\right),{\zeta }^T(t-\iota ),f^T{]}^T$:

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le {\eta }^T\left(t\right)\mathrm{\Theta }\eta \left(t\right),$$

where $\mathrm{\Theta }$ is defined in (12). Note that the second OSL condition (7) has been incorporated through appropriate terms in ${\mathrm{\Theta }}_{13}$, ${\mathrm{\Theta }}_{23}$, and ${\mathrm{\Theta }}_{33}$ to avoid duplication while maintaining the inequality structure.

From the LMI condition $\mathrm{\Theta }<0$, we have

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le -ϵ{\parallel \zeta \left(t\right)\parallel }^2.$$

From the LMI condition $\mathrm{\Theta }<0$, we have

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le {\eta }^T\left(t\right)\mathrm{\Theta }\eta \left(t\right)\le -ϵ{\parallel \zeta \left(t\right)\parallel }^2.$$

The addition of $ϵI$ in ${\mathrm{\Theta }}_{11}$ ensures this state-only decay bound.

This implies

$$V\left(t\right)\le V\left(0\right)e^{-2\sigma t^c/c}.$$

Using the bounds

$$\begin{array}{cc}\hfill {\lambda }_{min}\left(P\right){\parallel \zeta \left(t\right)\parallel }^2& \le V\left(t\right),\hfill \\ \hfill V\left(0\right)& \le \left[{\lambda }_{max}\left(P\right)+{\lambda }_{max}\left(Q\right){\displaystyle \frac{{\iota }^c}{c}}\right]{\parallel \varphi \parallel }_{\infty }^2,\hfill \end{array}$$

we obtain the stability bound (13), which proves exponential stability.    □

Remark 4.

The inclusion of the term $+\epsilon I$ in ${\mathrm{\Theta }}_{11}$ ensures strict negativity of the LMI matrix Θ. This allows us to establish the state-only decay bound

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le -\epsilon {\parallel \zeta \left(t\right)\parallel }^2,$$

which guarantees the exponential stability of the system. The parameter $\epsilon >0$ provides an explicit margin of negativity, strengthening the link between the feasibility of the LMI and the decay rate of the Lyapunov–Krasovskii functional.

4. Practical Stability Analysis of One-Sided Lipschitz Conformable Time-Delay Systems

Consider the perturbed version of the conformable time-delay system (1):

$$\begin{array}{cc}\hfill T^c\zeta \left(t\right)& =A\zeta \left(t\right)+A_d\zeta (t-\iota )+f(\zeta \left(t\right),\zeta (t-\iota ))+w\left(t\right),\hfill \\ \hfill \zeta \left(\theta \right)& =\varphi \left(\theta \right),\theta \in [-\iota ,0],\hfill \end{array}$$

(19)

where $w\left(t\right)\in {\mathbb{R}}^n$ is a bounded perturbation term satisfying the following assumption:

Assumption 3 (Bounded Perturbation).

The perturbation term $w\left(t\right)$ is bounded such that $\parallel w\left(t\right)\parallel \le w_m$ for all $t\ge 0$, where $w_m>0$ is a known constant bound.

The nonlinear function $f(x,x_{\iota })$ in (19) satisfies the same one-sided Lipschitz and quadratic inner-boundedness conditions as in Assumptions 1 and 2.

Remark 5.

The LMI condition in Theorem 1 offers several computational and theoretical advantages. The condition is convex in the decision variables, enabling efficient solution using standard LMI solvers. The incorporation of the slack variables${\mu }_1,{\mu }_2, and {\mu }_3$reduces conservatism by providing additional degrees of freedom in the stability analysis. Furthermore, the exponential decay rate σ appears explicitly in the LMI, allowing direct control over the convergence speed. This approach naturally handles the conformable derivative structure through the Lyapunov–Krasovskii functional, without requiring transformation to integer-order systems. The stability bound (13) provides a quantitative estimate of system performance, with the decay rate directly linked to the parameter σ in the LMI.

Theorem 2 (Practical Exponential Stability via LMI).

Consider the perturbed conformable time-delay system (19) with one-sided Lipschitz nonlinearity satisfying Assumptions 1 and 2, and bounded perturbation satisfying Assumption 3. Suppose that there exist symmetric positive definite matrices $P,Q\in {\mathbb{R}}^{n\times n}$, and positive scalars $\sigma >0$, ${\mu }_1>0$, ${\mu }_2>0$, ${\mu }_3>0$, and $ϵ>0$ such that the following LMI is feasible:

$${\mathrm{\Theta }}_p=\left[\begin{array}{cccc}{\mathrm{\Theta }}_{11}^p& P{A}_d& {\mathrm{\Theta }}_{13}& P\\ A_d^{T}P& {\mathrm{\Theta }}_{22}^p& {\mathrm{\Theta }}_{23}& 0\\ {\mathrm{\Theta }}_{13}^T& {\mathrm{\Theta }}_{23}^T& {\mathrm{\Theta }}_{33}& 0\\ P& 0& 0& -{\displaystyle \frac{1}{4}}I\end{array}\right]<0,$$

(20)

where

$$\begin{array}{cc}\hfill {\mathrm{\Theta }}_{11}^p& =A^{T}P+PA+2\sigma P+Q+2({\mu }_1{\rho }_1+{\mu }_2{\beta }_1+{\mu }_3{\rho }_1)I+ϵI,\hfill \\ \hfill {\mathrm{\Theta }}_{22}^p& =2({\mu }_1{\rho }_2+{\mu }_2{\beta }_2+{\mu }_3{\rho }_2)I-e^{-2\sigma {\iota }^c/c}Q,\hfill \\ \hfill {\mathrm{\Theta }}_{13}& =P+({\mu }_2{\gamma }_1-{\mu }_1)I+{\mu }_3{\rho }_{1}I,\hfill \\ \hfill {\mathrm{\Theta }}_{23}& ={\mu }_2{\gamma }_{2}I-{\mu }_{3}I+{\mu }_3{\rho }_{2}I,\hfill \\ \hfill {\mathrm{\Theta }}_{33}& =-2{\mu }_{2}I-2{\mu }_{3}I.\hfill \end{array}$$

Then, the system is practically exponentially stable according to Definition 3, with the following ultimate bound:

$$r=\sqrt{{\displaystyle \frac{{\eta }^2}{\sigma {\lambda }_{min}\left(P\right)}}},\mathrm{where}\eta =2w_m\parallel P\parallel ,$$

(21)

and the state trajectory satisfies

$$\parallel \zeta \left(t\right)\parallel \le \sqrt{{\displaystyle \frac{{\lambda }_{max}\left(P\right)+{\lambda }_{max}\left(Q\right)\frac{{\iota }^c}{c}}{{\lambda }_{min}\left(P\right)}}}{\parallel \varphi \parallel }_{\infty }{e}^{-\sigma t^c/c}+r.$$

(22)

Proof. 

We employ the same LKF (7) as in Theorem 1:

$$V\left(t\right)=V_1\left(t\right)+V_2\left(t\right),$$

where $V_1\left(t\right)$ and $V_2\left(t\right)$ are defined in (8) and (9), respectively.

Using Lemma 1 and substituting the perturbed system dynamics (19),

$$\begin{array}{cc}\hfill T^{c}V\left(t\right)+2\sigma V\left(t\right)& \le 2{\zeta }^T\left(t\right)A^{T}P\zeta \left(t\right)+2{\zeta }^T(t-\iota )A_d^{T}P\zeta \left(t\right)\hfill \\ & +2f^{T}P\zeta \left(t\right)+2w^T\left(t\right)P\zeta \left(t\right)+2\sigma {\zeta }^T\left(t\right)P\zeta \left(t\right)+{\zeta }^T\left(t\right)Q\zeta \left(t\right)\hfill \\ & -e^{-2\sigma {\iota }^c/c}{\zeta }^T(t-\iota )Q\zeta (t-\iota ).\hfill \end{array}$$

Following the same approach as in Theorem 1, we incorporate Assumptions 1 and 2 using the slack variables ${\mu }_1>0$, ${\mu }_2>0$, and ${\mu }_3>0$:

From Assumption 1 with $z=0$, $z_d=0$:

$$\begin{array}{c}\hfill 2{\mu }_1\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta ,f\rangle \right)\ge 0,\end{array}$$

(23)

$$\begin{array}{c}\hfill 2{\mu }_3\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta (t-\iota ),f\rangle \right)\ge 0.\end{array}$$

(24)

From Assumption 2 with $z=0$, $z_d=0$:

$$2{\mu }_2\left({\beta }_1{\parallel \zeta \parallel }^2+{\beta }_2{\parallel \zeta (t-\iota )\parallel }^2+{\gamma }_1\langle \zeta ,f\rangle +{\gamma }_2\langle \zeta (t-\iota ),f\rangle -{\parallel f\parallel }^2\right)\ge 0.$$

(25)

For the perturbation term, we apply Young’s inequality:

$$2w^T\left(t\right)P\zeta \left(t\right)\le 2w_m\parallel P\parallel \parallel \zeta \left(t\right)\parallel .$$

Let $\eta =2w_m\parallel P\parallel$. Using the inequality

$$\eta \parallel \zeta \left(t\right)\parallel \le {\eta }^2+{\displaystyle \frac{1}{4}}{\parallel \zeta \left(t\right)\parallel }^2,$$

we obtain:

$$2w^T\left(t\right)P\zeta \left(t\right)\le {\eta }^2+{\displaystyle \frac{1}{4}}{\parallel \zeta \left(t\right)\parallel }^2.$$

Adding Inequalities (23)–(25) to the derivative bound, and including the perturbation bound,

$$\begin{array}{cc}\hfill T^{c}V\left(t\right)+2\sigma V\left(t\right)& \le {\zeta }^T\left(t\right)\left[A^{T}P+PA+2\sigma P+Q\right]\zeta \left(t\right)\hfill \\ & +2{\zeta }^T(t-\iota )A_d^{T}P\zeta \left(t\right)+2{\zeta }^T\left(t\right)Pf+2w^T\left(t\right)P\zeta \left(t\right)\hfill \\ & -e^{-2\sigma {\iota }^c/c}{\zeta }^T(t-\iota )Q\zeta (t-\iota )\hfill \\ & +2{\mu }_1\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta ,f\rangle \right)\hfill \\ & +2{\mu }_3\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta (t-\iota ),f\rangle \right)\hfill \\ & +2{\mu }_2\left({\beta }_1{\parallel \zeta \parallel }^2+{\beta }_2{\parallel \zeta (t-\iota )\parallel }^2+{\gamma }_1\langle \zeta ,f\rangle +{\gamma }_2\langle \zeta (t-\iota ),f\rangle -{\parallel f\parallel }^2\right).\hfill \end{array}$$

Substituting the perturbation bound and rewriting in quadratic form with the extended state vector $\eta \left(t\right)={[{\zeta }^T\left(t\right),{\zeta }^T(t-\iota ),f^T,w^T\left(t\right)]}^T$:

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le {\eta }^T\left(t\right){\mathrm{\Theta }}_p\eta \left(t\right)+{\eta }^2,$$

where ${\mathrm{\Theta }}_p$ is defined in (20). Note that the second OSL condition (7) has been incorporated through appropriate terms in ${\mathrm{\Theta }}_{13}$, ${\mathrm{\Theta }}_{23}$, and ${\mathrm{\Theta }}_{33}$ to avoid duplication while maintaining the inequality structure.

From the LMI condition ${\mathrm{\Theta }}_p<0$, we have

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le -ϵ{\parallel \zeta \left(t\right)\parallel }^2+{\eta }^2.$$

This implies

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le {\eta }^2.$$

In fact, from the LMI condition ${\mathrm{\Theta }}_p<0$, we have

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le {\eta }^T\left(t\right){\mathrm{\Theta }}_p\eta \left(t\right)+{\eta }^2\le -ϵ{\parallel \zeta \left(t\right)\parallel }^2+{\eta }^2.$$

The $ϵI$ term in ${\mathrm{\Theta }}_{11}^p$ provides the state decay component.

Solving this inequality gives

$$V\left(t\right)\le V\left(0\right)e^{-2\sigma t^c/c}+{\displaystyle \frac{{\eta }^2}{2\sigma }}(1-e^{-2\sigma t^c/c}).$$

Using the bounds

$$\begin{array}{cc}\hfill {\lambda }_{min}\left(P\right){\parallel \zeta \left(t\right)\parallel }^2& \le V\left(t\right),\hfill \\ \hfill V\left(0\right)& \le \left[{\lambda }_{max}\left(P\right)+{\lambda }_{max}\left(Q\right){\displaystyle \frac{{\iota }^c}{c}}\right]{\parallel \varphi \parallel }_{\infty }^2,\hfill \end{array}$$

we obtain the practical stability bound (22) with the ultimate bound (21), which proves practical exponential stability according to Definition 3.    □

Remark 6.

The practical stability framework in Theorem 2 extends the nominal stability results to systems with bounded disturbances. The ultimate bound r in (21) provides a quantitative measure of robustness, relating the disturbance magnitude $w_m$ to worst-case performance. The LMI structure maintains convexity while incorporating disturbance effects through the additional block involving P. This approach guarantees that the system trajectories converge to a neighborhood of the origin, with size determined by the disturbance bound and system parameters. This result is particularly valuable for engineering applications where complete disturbance rejection is impractical and bounded deviations from equilibrium are acceptable. The separation between exponential convergence (governed by σ) and the ultimate bound (governed by r) enables systematic trade-offs between performance and robustness in system design.

5. Exponential Stabilization for One-Sided Lipschitz Conformable Systems with Time Delay

Consider the controlled version of the conformable time-delay system:

$$\begin{array}{cc}\hfill T^c\zeta \left(t\right)& =A\zeta \left(t\right)+A_d\zeta (t-\iota )+f(\zeta \left(t\right),\zeta (t-\iota ))+Bu\left(t\right),\hfill \\ \hfill \zeta \left(\theta \right)& =\varphi \left(\theta \right),\theta \in [-\iota ,0],\hfill \end{array}$$

(26)

where

• $u\left(t\right)\in {\mathbb{R}}^m$ is the control input vector;
• $B\in {\mathbb{R}}^{n\times m}$ is the input matrix;
• All other terms are as defined in System (1).

The nonlinear function $f(x,x_{\iota })$ satisfies the same one-sided Lipschitz and quadratic inner-boundedness conditions as in Assumptions 1 and 2.

We consider the state-feedback control law

$$u\left(t\right)=-K\zeta \left(t\right),$$

(27)

where $K\in {\mathbb{R}}^{m\times n}$ is the controller gain to be designed. The closed-loop system becomes

$$T^c\zeta \left(t\right)=(A-BK)\zeta \left(t\right)+A_d\zeta (t-\iota )+f(\zeta \left(t\right),\zeta (t-\iota )).$$

(28)

Theorem 3 (Exponential Stabilization via LMI).

Consider the conformable time-delay system (26) with one-sided Lipschitz nonlinearity satisfying Assumptions 1 and 2. Suppose that there exist symmetric positive definite matrices $X,\tilde{Q}\in {\mathbb{R}}^{n\times n}$, a matrix $Y\in {\mathbb{R}}^{m\times n}$, and positive scalars $\sigma >0$, ${\mu }_1>0$,  ${\mu }_2>0$,  ${\mu }_3>0$, and $ϵ>0$ such that the following LMI is feasible:

$$\left[\begin{array}{cccc}{\mathrm{\Phi }}_{11}& A_{d}X& {\mathrm{\Phi }}_{13}& X\\ X{A}_d^T& {\mathrm{\Phi }}_{22}& {\mathrm{\Phi }}_{23}& 0\\ {\mathrm{\Phi }}_{13}^T& {\mathrm{\Phi }}_{23}^T& {\mathrm{\Phi }}_{33}& 0\\ X& 0& 0& -{\displaystyle \frac{1}{2}}I\end{array}\right]<0,$$

(29)

where

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_{11}& =(AX-BY)+{(AX-BY)}^T+2\sigma X+\tilde{Q}+2({\mu }_1{\rho }_1+{\mu }_2{\beta }_1+{\mu }_3{\rho }_1)X+ϵX,\hfill \\ \hfill {\mathrm{\Phi }}_{22}& =2({\mu }_1{\rho }_2+{\mu }_2{\beta }_2+{\mu }_3{\rho }_2)X-e^{-2\sigma {\iota }^c/c}\tilde{Q},\hfill \\ \hfill {\mathrm{\Phi }}_{13}& =X+({\mu }_2{\gamma }_1-{\mu }_1)X+{\mu }_3{\rho }_{1}X,\hfill \\ \hfill {\mathrm{\Phi }}_{23}& ={\mu }_2{\gamma }_{2}X-{\mu }_{3}X+{\mu }_3{\rho }_{2}X,\hfill \\ \hfill {\mathrm{\Phi }}_{33}& =-2{\mu }_{2}I-2{\mu }_{3}I.\hfill \end{array}$$

Then, with the controller gain$K=Y{X}^{-1}$, the closed-loop system (28) is exponentially stable, and the state trajectory satisfies

$$\parallel \zeta \left(t\right)\parallel \le \sqrt{{\displaystyle \frac{{\lambda }_{max}\left(P\right)+{\lambda }_{max}\left(Q\right)\frac{{\iota }^c}{c}}{{\lambda }_{min}\left(P\right)}}}{\parallel \varphi \parallel }_{\infty }{e}^{-\sigma t^c/c},$$

(30)

where $P=X^{-1}$ and $Q=X^{-1}\tilde{Q}{X}^{-1}$.

Proof. 

Consider the LKF (7):

$$V\left(t\right)=V_1\left(t\right)+V_2\left(t\right),$$

where

$$\begin{array}{cc}\hfill V_1\left(t\right)& ={\zeta }^T\left(t\right)P\zeta \left(t\right),\hfill \\ \hfill V_2\left(t\right)& ={\int }_t^{t+\iota }{s}^{c-1}{e}^{2\sigma \left({\displaystyle \frac{s^c}{c}}-{\displaystyle \frac{t^c}{c}}-{\displaystyle \frac{{\iota }^c}{c}}\right)}{\zeta }^T(s-\iota )Q\zeta (s-\iota )ds,\hfill \end{array}$$

with $P=X^{-1}\succ 0$, $Q=X^{-1}\tilde{Q}{X}^{-1}\succ 0$, and $\sigma >0$.

Using Lemma 1 and substituting the closed-loop dynamics (28),

$$\begin{array}{cc}\hfill T^{c}V\left(t\right)+2\sigma V\left(t\right)& \le 2{\zeta }^T\left(t\right){(A-BK)}^{T}P\zeta \left(t\right)+2{\zeta }^T(t-\iota )A_d^{T}P\zeta \left(t\right)\hfill \\ & +2f^{T}P\zeta \left(t\right)+2\sigma {\zeta }^T\left(t\right)P\zeta \left(t\right)+{\zeta }^T\left(t\right)Q\zeta \left(t\right)\hfill \\ & -e^{-2\sigma {\iota }^c/c}{\zeta }^T(t-\iota )Q\zeta (t-\iota ).\hfill \end{array}$$

Following the same approach as in previous theorems, we incorporate Assumptions 1 and 2 using the slack variables ${\mu }_1>0$, ${\mu }_2>0$, and ${\mu }_3>0$:

From Assumption 1 with $z=0$, $z_d=0$:

$$\begin{array}{c}2{\mu }_1\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta ,f\rangle \right)\ge 0,\hfill \end{array}$$

(31)

$$\begin{array}{c}2{\mu }_3\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta (t-\iota ),f\rangle \right)\ge 0.\hfill \end{array}$$

(32)

From Assumption 2 with $z=0$, $z_d=0$:

$$2{\mu }_2\left({\beta }_1{\parallel \zeta \parallel }^2+{\beta }_2{\parallel \zeta (t-\iota )\parallel }^2+{\gamma }_1\langle \zeta ,f\rangle +{\gamma }_2\langle \zeta (t-\iota ),f\rangle -{\parallel f\parallel }^2\right)\ge 0.$$

(33)

Adding Inequalities (31)–(33) to the derivative bound,

$$\begin{array}{cc}\hfill T^{c}V\left(t\right)+2\sigma V\left(t\right)& \le {\zeta }^T\left(t\right)\left[{(A-BK)}^{T}P+P(A-BK)+2\sigma P+Q\right]\zeta \left(t\right)\hfill \\ & +2{\zeta }^T(t-\iota )A_d^{T}P\zeta \left(t\right)+2{\zeta }^T\left(t\right)Pf\hfill \\ & -e^{-2\sigma {\iota }^c/c}{\zeta }^T(t-\iota )Q\zeta (t-\iota )\hfill \\ & +2{\mu }_1\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta ,f\rangle \right)\hfill \\ & +2{\mu }_3\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta (t-\iota ),f\rangle \right)\hfill \\ & +2{\mu }_2\left({\beta }_1{\parallel \zeta \parallel }^2+{\beta }_2{\parallel \zeta (t-\iota )\parallel }^2+{\gamma }_1\langle \zeta ,f\rangle +{\gamma }_2\langle \zeta (t-\iota ),f\rangle -{\parallel f\parallel }^2\right).\hfill \end{array}$$

Rewriting in quadratic form with the extended state vector $\eta \left(t\right)=[{\zeta }^T\left(t\right),{\zeta }^T(t-\iota )$, $f^T{]}^T$:

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le {\eta }^T\left(t\right)\mathrm{\Theta }\eta \left(t\right),$$

where

$$\mathrm{\Theta }=\left[\begin{array}{ccc}{\mathrm{\Theta }}_{11}& P{A}_d& {\mathrm{\Theta }}_{13}\\ A_d^{T}P& {\mathrm{\Theta }}_{22}& {\mathrm{\Theta }}_{23}\\ {\mathrm{\Theta }}_{13}^T& {\mathrm{\Theta }}_{23}^T& {\mathrm{\Theta }}_{33}\end{array}\right],$$

with

$$\begin{array}{cc}\hfill {\mathrm{\Theta }}_{11}& ={(A-BK)}^{T}P+P(A-BK)+2\sigma P+Q+2({\mu }_1{\rho }_1+{\mu }_2{\beta }_1+{\mu }_3{\rho }_1)I,\hfill \\ \hfill {\mathrm{\Theta }}_{22}& =2({\mu }_1{\rho }_2+{\mu }_2{\beta }_2+{\mu }_3{\rho }_2)I-e^{-2\sigma {\iota }^c/c}Q,\hfill \\ \hfill {\mathrm{\Theta }}_{13}& =P+({\mu }_2{\gamma }_1-{\mu }_1)I+{\mu }_3{\rho }_{1}I,\hfill \\ \hfill {\mathrm{\Theta }}_{23}& ={\mu }_2{\gamma }_{2}I-{\mu }_{3}I+{\mu }_3{\rho }_{2}I,\hfill \\ \hfill {\mathrm{\Theta }}_{33}& =-2{\mu }_{2}I-2{\mu }_{3}I.\hfill \end{array}$$

Indeed, from the LMI condition (29), we have

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le {\eta }^T\left(t\right)\mathrm{\Theta }\eta \left(t\right)\le -ϵ{\parallel \zeta \left(t\right)\parallel }^2.$$

The $ϵX$ term in ${\mathrm{\Phi }}_{11}$ ensures exponential decay of the state.

Apply the congruence transformation with $\mathrm{diag}(X,X,I)$, where $X=P^{-1}$, and define

$$\begin{array}{cc}\hfill Y& =KX,\hfill \\ \hfill \tilde{Q}& =XQX.\hfill \end{array}$$

Pre- and post-multiplying $\mathrm{\Theta }$ by $\mathrm{diag}(X,X,I)$,

$$\left[\begin{array}{ccc}X& 0& 0\\ 0& X& 0\\ 0& 0& I\end{array}\right]\left[\begin{array}{ccc}{\mathrm{\Theta }}_{11}& P{A}_d& {\mathrm{\Theta }}_{13}\\ A_d^{T}P& {\mathrm{\Theta }}_{22}& {\mathrm{\Theta }}_{23}\\ {\mathrm{\Theta }}_{13}^T& {\mathrm{\Theta }}_{23}^T& {\mathrm{\Theta }}_{33}\end{array}\right]\left[\begin{array}{ccc}X& 0& 0\\ 0& X& 0\\ 0& 0& I\end{array}\right]<0.$$

Computing each block,

$$\begin{array}{cc}\hfill X{\mathrm{\Theta }}_{11}X& =(AX-BY)+{(AX-BY)}^T+2\sigma X+\tilde{Q}+2({\mu }_1{\rho }_1+{\mu }_2{\beta }_1+{\mu }_3{\rho }_1)X,\hfill \\ \hfill XP{A}_{d}X& =A_{d}X,\hfill \\ \hfill X{\mathrm{\Theta }}_{13}& =X+({\mu }_2{\gamma }_1-{\mu }_1)X+{\mu }_3{\rho }_{1}X,\hfill \\ \hfill X{\mathrm{\Theta }}_{22}X& =2({\mu }_1{\rho }_2+{\mu }_2{\beta }_2+{\mu }_3{\rho }_2)X-e^{-2\sigma {\iota }^c/c}\tilde{Q},\hfill \\ \hfill X{\mathrm{\Theta }}_{23}& ={\mu }_2{\gamma }_{2}X-{\mu }_{3}X+{\mu }_3{\rho }_{2}X,\hfill \\ \hfill {\mathrm{\Theta }}_{33}& =-2{\mu }_{2}I-2{\mu }_{3}I.\hfill \end{array}$$

To handle the term X in the (1, 1) and (4, 4) blocks of the final LMI formulation, we use the bounding inequality derived from the fact that ${(X-I)}^2⪰0$, which implies

$$X⪯{\displaystyle \frac{1}{2}}(X^2+I),$$

where the inequality holds for positive definite matrices $X\succ 0$. This bounding technique allows us to convert the nonlinear matrix inequality into a linear LMI formulation while introducing minimal conservatism. The resulting LMI is given in (29).

From the LMI condition (29), we have

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le -ϵ{\parallel \zeta \left(t\right)\parallel }^2.$$

This implies

$$V\left(t\right)\le V\left(0\right)e^{-2\sigma t^c/c}.$$

Using the bounds

$$\begin{array}{cc}\hfill {\lambda }_{min}\left(P\right){\parallel \zeta \left(t\right)\parallel }^2& \le V\left(t\right),\hfill \\ \hfill V\left(0\right)& \le \left[{\lambda }_{max}\left(P\right)+{\lambda }_{max}\left(Q\right){\displaystyle \frac{{\iota }^c}{c}}\right]{\parallel \varphi \parallel }_{\infty }^2,\hfill \end{array}$$

we obtain the stabilization bound (30), which proves the exponential stability of the closed-loop system.    □

Algorithm 1 provides a systematic procedure for implementing the exponential stabilization approach developed in Theorem 3:

Algorithm 1: Exponential Stabilization via State Feedback

Input: System matrices A, $A_d$, B; nonlinearity parameters ${\rho }_1$, ${\rho }_2$, ${\beta }_1$, ${\beta }_2$, ${\gamma }_1$, ${\gamma }_2$; delay $\iota$; derivative order c; decay rate $\sigma$ Output: Controller gain K and stability certificate Initialize: Choose initial values for slack variables ${\mu }_1,{\mu }_2,{\mu }_3>0$ and tolerance $ϵ>0$ LMI Formulation: Construct the LMI problem using Theorem 3: Define decision variables: $X\succ 0$, $\tilde{Q}\succ 0$, Y Formulate matrix inequality (29) Include symmetry and positive definiteness constraints Solve Optimization: Use LMI solver (e.g., YALMIP, CVX) to find feasible solution If infeasible, adjust ${\mu }_1,{\mu }_2,{\mu }_3$ and retry Verify that solution satisfies all constraints Compute Controller: Calculate $P=X^{-1}$ Compute controller gain $K=Y{X}^{-1}$ Verify closed-loop stability via eigenvalue analysis Validation: Check ${\lambda }_{min}\left(\mathrm{\Theta }\right)<0$ for feasibility Verify decay bound using (30) Simulate closed-loop system for performance assessment

Remark 7.

To linearize nonlinear terms involving X in the$(1,1)$and$(4,4)$blocks of the LMI after congruence transformation, we use the operator inequality

$$X⪯{\displaystyle \frac{1}{2}}(X^2+I),X\succ 0.$$

This follows from${(X-I)}^2⪰0$, which implies that$X^2-2X+I⪰0$. The inequality is standard in matrix analysis (see Horn and Johnson [41]) and ensures the convexity of the LMI by replacing bilinear terms with linear ones. While this introduces mild conservatism, it is necessary for tractability and affects only the$(1,1)$and$(4,4)$blocks.

6. Practical Exponential Stabilization for One-Sided Lipschitz Conformable Systems with Time Delay and Bounded Perturbations

Consider the controlled conformable time-delay system with bounded perturbations:

$$\begin{array}{cc}\hfill T^c\zeta \left(t\right)& =A\zeta \left(t\right)+A_d\zeta (t-\iota )+f(\zeta \left(t\right),\zeta (t-\iota ))+Bu\left(t\right)+w\left(t\right),\hfill \\ \hfill \zeta \left(\theta \right)& =\varphi \left(\theta \right),\theta \in [-\iota ,0],\hfill \end{array}$$

(34)

where

• All terms are as defined in Systems (26) and (19);
• The perturbation $w\left(t\right)$ satisfies Assumption 3.

The nonlinear function $f(x,x_{\iota })$ satisfies the same one-sided Lipschitz and quadratic inner-boundedness conditions as in Assumptions 1 and 2.

We consider the same state-feedback control law as in (27):

$$u\left(t\right)=-K\zeta \left(t\right),$$

(35)

where $K\in {\mathbb{R}}^{m\times n}$ is the controller gain to be designed. The closed-loop system becomes

$$T^c\zeta \left(t\right)=(A-BK)\zeta \left(t\right)+A_d\zeta (t-\iota )+f(\zeta \left(t\right),\zeta (t-\iota ))+w\left(t\right).$$

(36)

Theorem 4 (Practical Exponential Stabilization via LMI).

Consider the perturbed conformable time-delay system (34) with one-sided Lipschitz nonlinearity satisfying Assumptions 1 and 2, and bounded perturbation satisfying Assumption 3. Suppose that there exist symmetric positive definite matrices $X,\tilde{Q}\in {\mathbb{R}}^{n\times n}$, a matrix $Y\in {\mathbb{R}}^{m\times n}$, and positive scalars $\sigma >0$, ${\mu }_1>0$, ${\mu }_2>0$, ${\mu }_3>0$, and $ϵ>0$ such that the following LMI is feasible:

$$\left[\begin{array}{ccccc}{\mathrm{\Psi }}_{11}& A_{d}X& {\mathrm{\Psi }}_{13}& X& 0\\ X{A}_d^T& {\mathrm{\Psi }}_{22}& {\mathrm{\Psi }}_{23}& 0& 0\\ {\mathrm{\Psi }}_{13}^T& {\mathrm{\Psi }}_{23}^T& {\mathrm{\Psi }}_{33}& 0& 0\\ X& 0& 0& -{\displaystyle \frac{1}{2}}I& 0\\ 0& 0& 0& 0& -{\displaystyle \frac{1}{4}}I\end{array}\right]<0,$$

(37)

where

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{11}& =(AX-BY)+{(AX-BY)}^T+2\sigma X+\tilde{Q}+2({\mu }_1{\rho }_1+{\mu }_2{\beta }_1+{\mu }_3{\rho }_1)X+ϵX,\hfill \\ \hfill {\mathrm{\Psi }}_{22}& =2({\mu }_1{\rho }_2+{\mu }_2{\beta }_2+{\mu }_3{\rho }_2)X-e^{-2\sigma {\iota }^c/c}\tilde{Q},\hfill \\ \hfill {\mathrm{\Psi }}_{13}& =X+({\mu }_2{\gamma }_1-{\mu }_1)X+{\mu }_3{\rho }_{1}X,\hfill \\ \hfill {\mathrm{\Psi }}_{23}& ={\mu }_2{\gamma }_{2}X-{\mu }_{3}X+{\mu }_3{\rho }_{2}X,\hfill \\ \hfill {\mathrm{\Psi }}_{33}& =-2{\mu }_{2}I-2{\mu }_{3}I.\hfill \end{array}$$

Then, with the controller gain$K=Y{X}^{-1}$, the closed-loop system (36) is practically exponentially stable according to Definition 3 with the following ultimate bound:

$$r=\sqrt{{\displaystyle \frac{{\eta }^2}{\sigma {\lambda }_{min}\left(P\right)}}},\mathit{where}\eta =2w_m\parallel P\parallel ,$$

(38)

and $P=X^{-1}$.

Proof. 

Consider the same LKF as in previous theorems:

$$V\left(t\right)=V_1\left(t\right)+V_2\left(t\right),$$

where

$$\begin{array}{cc}\hfill V_1\left(t\right)& ={\zeta }^T\left(t\right)P\zeta \left(t\right),\hfill \\ \hfill V_2\left(t\right)& ={\int }_t^{t+\iota }{s}^{c-1}{e}^{2\sigma \left({\displaystyle \frac{s^c}{c}}-{\displaystyle \frac{t^c}{c}}-{\displaystyle \frac{{\iota }^c}{c}}\right)}{\zeta }^T(s-\iota )Q\zeta (s-\iota )ds,\hfill \end{array}$$

with $P=X^{-1}\succ 0$, $Q=X^{-1}\tilde{Q}{X}^{-1}\succ 0$, and $\sigma >0$.

Using Lemma 1 and substituting the closed-loop dynamics with perturbation (36),

$$\begin{array}{cc}\hfill T^{c}V\left(t\right)+2\sigma V\left(t\right)& \le 2{\zeta }^T\left(t\right){(A-BK)}^{T}P\zeta \left(t\right)+2{\zeta }^T(t-\iota )A_d^{T}P\zeta \left(t\right)\hfill \\ & +2f^{T}P\zeta \left(t\right)+2w^T\left(t\right)P\zeta \left(t\right)+2\sigma {\zeta }^T\left(t\right)P\zeta \left(t\right)\hfill \\ & +{\zeta }^T\left(t\right)Q\zeta \left(t\right)-e^{-2\sigma {\iota }^c/c}{\zeta }^T(t-\iota )Q\zeta (t-\iota ).\hfill \end{array}$$

Following the established methodology, we incorporate Assumptions 1 and 2 using the slack variables ${\mu }_1>0$, ${\mu }_2>0$, and ${\mu }_3>0$:

From Assumption 1 with $z=0$, $z_d=0$:

$$\begin{array}{c}2{\mu }_1\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta ,f\rangle \right)\ge 0,\hfill \end{array}$$

(39)

$$\begin{array}{c}2{\mu }_3\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta (t-\iota ),f\rangle \right)\ge 0.\hfill \end{array}$$

(40)

From Assumption 2 with $z=0$, $z_d=0$:

$$2{\mu }_2\left({\beta }_1{\parallel \zeta \parallel }^2+{\beta }_2{\parallel \zeta (t-\iota )\parallel }^2+{\gamma }_1\langle \zeta ,f\rangle +{\gamma }_2\langle \zeta (t-\iota ),f\rangle -{\parallel f\parallel }^2\right)\ge 0.$$

(41)

For the perturbation term, we apply Young’s inequality:

$$2w^T\left(t\right)P\zeta \left(t\right)\le 2w_m\parallel P\parallel \parallel \zeta \left(t\right)\parallel .$$

Let $\eta =2w_m\parallel P\parallel$. Using the inequality

$$\eta \parallel \zeta \left(t\right)\parallel \le {\eta }^2+{\displaystyle \frac{1}{4}}{\parallel \zeta \left(t\right)\parallel }^2,$$

we obtain

$$2w^T\left(t\right)P\zeta \left(t\right)\le {\eta }^2+{\displaystyle \frac{1}{4}}{\parallel \zeta \left(t\right)\parallel }^2.$$

Adding Inequalities (39)–(41) to the derivative bound, and including the perturbation bound,

$$\begin{array}{cc}\hfill T^{c}V\left(t\right)+2\sigma V\left(t\right)& \le {\zeta }^T\left(t\right)\left[{(A-BK)}^{T}P+P(A-BK)+2\sigma P+Q\right]\zeta \left(t\right)\hfill \\ & +2{\zeta }^T(t-\iota )A_d^{T}P\zeta \left(t\right)+2{\zeta }^T\left(t\right)Pf+2w^T\left(t\right)P\zeta \left(t\right)\hfill \\ & -e^{-2\sigma {\iota }^c/c}{\zeta }^T(t-\iota )Q\zeta (t-\iota )\hfill \\ & +2{\mu }_1\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta ,f\rangle \right)\hfill \\ & +2{\mu }_3\left({\rho }_1{\parallel \zeta \parallel }^2+{\rho }_2{\parallel \zeta (t-\iota )\parallel }^2-\langle \zeta (t-\iota ),f\rangle \right)\hfill \\ & +2{\mu }_2\left({\beta }_1{\parallel \zeta \parallel }^2+{\beta }_2{\parallel \zeta (t-\iota )\parallel }^2+{\gamma }_1\langle \zeta ,f\rangle +{\gamma }_2\langle \zeta (t-\iota ),f\rangle -{\parallel f\parallel }^2\right).\hfill \end{array}$$

Rewriting in quadratic form with the extended state vector $\eta \left(t\right)=[{\zeta }^T\left(t\right),{\zeta }^T(t-\iota ),f^T$, $w^T{\left(t\right)]}^T$:

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le {\eta }^T\left(t\right)\mathrm{\Psi }\eta \left(t\right)+{\eta }^2,$$

where $\mathrm{\Psi }$ is defined in (37). Note that the second OSL condition (7) has been incorporated through appropriate terms in ${\mathrm{\Psi }}_{13}$, ${\mathrm{\Psi }}_{23}$, and ${\mathrm{\Psi }}_{33}$ to avoid duplication while maintaining the inequality structure.

In fact, from the LMI condition (37), we have

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le {\eta }^T\left(t\right)\mathrm{\Psi }\eta \left(t\right)+{\eta }^2\le -ϵ{\parallel \zeta \left(t\right)\parallel }^2+{\eta }^2.$$

The $ϵX$ term in ${\mathrm{\Psi }}_{11}$ provides the state decay component.

Apply the congruence transformation with $\mathrm{diag}(X,X,I,I,I)$, where $X=P^{-1}$, and define

$$\begin{array}{cc}\hfill Y& =KX,\hfill \\ \hfill \tilde{Q}& =XQX.\hfill \end{array}$$

Pre- and post-multiplying $\mathrm{\Psi }$ by $\mathrm{diag}(X,X,I,I,I)$,

$$\left[\begin{array}{ccccc}X& 0& 0& 0& 0\\ 0& X& 0& 0& 0\\ 0& 0& I& 0& 0\\ 0& 0& 0& I& 0\\ 0& 0& 0& 0& I\end{array}\right]\mathrm{\Psi }\left[\begin{array}{ccccc}X& 0& 0& 0& 0\\ 0& X& 0& 0& 0\\ 0& 0& I& 0& 0\\ 0& 0& 0& I& 0\\ 0& 0& 0& 0& I\end{array}\right]<0.$$

Computing each block,

$$\begin{array}{cc}\hfill X{\mathrm{\Psi }}_{11}X& =(AX-BY)+{(AX-BY)}^T+2\sigma X+\tilde{Q}+2({\mu }_1{\rho }_1+{\mu }_2{\beta }_1+{\mu }_3{\rho }_1)X,\hfill \\ \hfill X{A}_{d}X& =A_{d}X,\hfill \\ \hfill X{\mathrm{\Psi }}_{13}& =X+({\mu }_2{\gamma }_1-{\mu }_1)X+{\mu }_3{\rho }_{1}X,\hfill \\ \hfill X{\mathrm{\Psi }}_{22}X& =2({\mu }_1{\rho }_2+{\mu }_2{\beta }_2+{\mu }_3{\rho }_2)X-e^{-2\sigma {\iota }^c/c}\tilde{Q},\hfill \\ \hfill X{\mathrm{\Psi }}_{23}& ={\mu }_2{\gamma }_{2}X-{\mu }_{3}X+{\mu }_3{\rho }_{2}X,\hfill \\ \hfill {\mathrm{\Psi }}_{33}& =-2{\mu }_{2}I-2{\mu }_{3}I.\hfill \end{array}$$

To handle the terms involving X in the (1, 1) and (4, 4) blocks of the final LMI formulation, we use the bounding inequality derived from ${(X-I)}^2⪰0$, which implies

$$X⪯{\displaystyle \frac{1}{2}}(X^2+I),$$

where the inequality holds for positive definite matrices $X\succ 0$. This bounding technique converts the nonlinear matrix inequality into a linear LMI formulation while introducing minimal conservatism. The disturbance term in the (5,5) block remains as $-{\displaystyle \frac{1}{4}}I$ to account for the bounded perturbation effects.

From the LMI condition (37), we have

$$T^{c}V\left(t\right)+2\sigma V\left(t\right)\le {\eta }^2.$$

This implies

$$V\left(t\right)\le V\left(0\right)e^{-2\sigma t^c/c}+{\displaystyle \frac{{\eta }^2}{2\sigma }}(1-e^{-2\sigma t^c/c}).$$

Using the bounds

$$\begin{array}{cc}\hfill {\lambda }_{min}\left(P\right){\parallel \zeta \left(t\right)\parallel }^2& \le V\left(t\right),\hfill \\ \hfill V\left(0\right)& \le \left[{\lambda }_{max}\left(P\right)+{\lambda }_{max}\left(Q\right){\displaystyle \frac{{\iota }^c}{c}}\right]{\parallel \varphi \parallel }_{\infty }^2,\hfill \end{array}$$

we obtain

$$\parallel \zeta \left(t\right)\parallel \le \sqrt{{\displaystyle \frac{{\lambda }_{max}\left(P\right)+{\lambda }_{max}\left(Q\right)\frac{{\iota }^c}{c}}{{\lambda }_{min}\left(P\right)}}}{\parallel \varphi \parallel }_{\infty }{e}^{-\sigma t^c/c}+\sqrt{{\displaystyle \frac{{\eta }^2}{\sigma {\lambda }_{min}\left(P\right)}}},$$

which proves practical exponential stability with the ultimate bound (38).    □

For systems subject to bounded perturbations, Algorithm 2 implements the practical stabilization methodology established in Theorem 4, providing robustness guarantees against external disturbances:

Algorithm 2: Practical Exponential Stabilization under Bounded Perturbations

Input: System matrices A, $A_d$, B; nonlinearity parameters ${\rho }_1$, ${\rho }_2$, ${\beta }_1$, ${\beta }_2$, ${\gamma }_1$, ${\gamma }_2$; delay $\iota$; derivative order c; disturbance bound $w_m$; desired ultimate bound $r_{des}$ Output: Robust controller gain K and practical stability certificate Parameter Selection: Choose $\sigma >0$ based on desired convergence rate Initialize slack variables ${\mu }_1,{\mu }_2,{\mu }_3>0$ Set disturbance tolerance parameter $\eta =2w_m\parallel P\parallel$ Robust LMI Formulation: Construct the practical stabilization LMI using Theorem 4: Define decision variables: $X\succ 0$, $\tilde{Q}\succ 0$, Y Formulate matrix inequality (37) Include disturbance rejection constraints Iterative Solution: Solve LMI optimization problem If infeasible, relax performance requirements or adjust $\sigma$ Iterate until feasible solution found Robust Controller Synthesis: Compute $P=X^{-1}$ Calculate robust controller gain $K=Y{X}^{-1}$ Verify ultimate bound using (38) Robustness Validation: Test controller performance under worst-case disturbances Verify that state trajectories remain within computed bound r Assess control effort and saturation constraints Perform Monte Carlo simulations for statistical validation Performance Tuning (Optional): If $r>r_{des}$, increase controller gain or adjust $\sigma$ Trade-off between convergence speed and ultimate bound Re-optimize with modified constraints if needed

Remark 8.

The bounding inequality

$$X⪯{\displaystyle \frac{1}{2}}(X^2+I),X\succ 0,$$

is applied to handle nonlinear terms in the$(1,1)$and$(4,4)$blocks of the LMI after congruence transformation. This technique, derived from${(X-I)}^2⪰0$, is widely used in LMI-based control design (Horn and Johnson [41]). It converts a non-convex constraint into a convex one, enabling efficient optimization while introducing limited conservatism that can be assessed by comparing feasible regions before and after relaxation.

Remark 9.

The practical stability bound can be sharpened by setting

$$\eta =2w_m\parallel P\parallel ,$$

which leads to

$$r=\sqrt{{\displaystyle \frac{{\eta }^2}{\sigma {\lambda }_{min}\left(P\right)}}}.$$

This choice minimizes conservatism by explicitly incorporating the norm of P into the disturbance channel. Alternatively, introducing a small LMI multiplier for the disturbance term in the$(4,4)$block of the matrix inequality can further reduce conservatism while preserving convexity. Such refinements allow tighter ultimate bounds without compromising feasibility.

7. Comparison with Existing Works

To situate the results of this paper in the literature,

Table 1. Comparison of representative LMI-based results for (conformable/fractional) nonlinear systems with/without time delay.

Work	Class	Delay	Main Result	Convex	Exp. Rate/r	Robust	Notes
Samidurai & Yazhini (2025) [22]	Frac. (Mittag–Leffler), nonlin.	Yes	Global ML stability and synchronization of discrete-time fractional-order BAM neural networks	—	ML-type bound	—	Fractional-order neural networks with delay; not in OSL/QIB framework.
Xu et al. (2009) [23]	Integer-order, nonlin.	No	Reduced-order observer design for one-sided Lipschitz systems	—	—	—	Observer design for OSL systems; no conformable derivatives or delays.
Kharrat et al. (2023) [21]	Conformable (nonlin.)	Yes	Practical stability for conformable time-delay systems via LKF	Yes (LMIs)	r (explicit)	—	Conformable + delay; OSL/QIB not explicitly exploited.
Aldandani et al. (2023) [20]	Gen. conformable	No	Practical stability for nonlinear systems with generalized conformable derivative	Yes (LMIs)	r (explicit)	—	Conformable setting; no explicit OSL/QIB structure; no delay.
Iben Ammar et al. (2024) [42]	Conformable T–S fuzzy	Yes	Stability and stabilization of general conformable polynomial fuzzy models with time delay	Yes (LMIs)	—	—	Different nonlinearity class (T–S fuzzy); not focused on OSL/QIB.
This paper	OSL + QIB, conformable	Yes	Exponential and practical stability; state-feedback stabilization	Yes (LMIs)	Explicit $\mathbf{\sigma }$ and $\mathbf{r}$	Yes	Convex LMIs; explicit decay $\mathbf{\sigma }$ and ultimate bound $\mathbf{r}$; implementable algorithms; first to combine OSL + QIB with conformable derivative and delay to deliver exponential/practical stability and convex synthesis.

Abbrev. Frac. = fractional; nonlin. = nonlinear; T–S = Takagi–Sugeno; ML = Mittag–Leffler. “Convex” refers to LMI-based feasibility/synthesis; “r” denotes an explicit ultimate bound; “—” = not reported/applicable.

compares various representative LMI-based results for conformable nonlinear and fractional nonlinear systems. In particular, we focus on results for time-delay systems and one-sided Lipschitz conditions.

Table 1 confirms that this work is a significant improvement in the stability analysis and control of conformable fractional-order systems. The previous papers published on conformable derivatives [20,21], time-delay systems [42], or one-sided Lipschitz conditions [22,23] deal with these individual aspects. However, the present paper is the first to combine all three in a single LMI framework. The proposed method guarantees fast and easy stabilization of control laws by using explicit decay rates and ultimate bounds. This realization is both exponential and useful. Moreover, this paper is further distinguished by the inclusion of implementable algorithms, which exploit the theoretical analysis and the controller design for complex conformable time-delay systems with one-sided Lipschitz nonlinearities.

Contrast with Prior Works

Existing studies on conformable time-delay systems (e.g., [20,21]) primarily address stability analysis without exploiting one-sided Lipschitz (OSL) or quadratic inner-boundedness (QIB) conditions. Consequently, they lack systematic state-feedback synthesis and do not provide explicit decay rates $\sigma$ or ultimate bounds r. On the other hand, prior works that incorporate OSL/QIB frameworks (e.g., [22,23]) are restricted to integer-order or classical fractional-order settings and typically omit time-delay modeling, making them unsuitable for conformable derivative systems. Our contribution bridges these gaps by delivering the following:

• Convex LMI conditions for exponential and practical stability under OSL and QIB assumptions in the conformable derivative setting with delays.
• Explicit formulae for decay rate $\sigma$ and ultimate bound r, enabling quantitative performance guarantees.
• State-feedback stabilization strategies formulated as tractable LMI problems, ensuring implementability via standard solvers.

This unified framework combines the advantages of conformable fractional calculus with robust LMI-based design, addressing limitations of both prior conformable-delay and OSL/QIB approaches.

The LMI-based approach crucial for the proposed stability and stabilization framework can be employed for time-varying delay systems by using delay-dependent Lyapunov–Krasovskii functionals and bounding techniques. In accordance with [43,44], this extension will allow us to deal with event-triggered control and networked cascade systems subject to communication constraints or cyber-attacks. Whether that is possible becomes determinable when new additional slack variables and switching conditions are introduced to account for delay variations and event-triggered mechanisms. These changes usually make the computations more complicated, but they still remain feasible with system dimensions that are not too large. We should work on that in future research. Integrating H∞ performance criteria and fault-tolerant schemes within the conformable derivative framework that provides robustness against DoS attacks and packet losses will enable practical implementations in cyber–physical and networked control systems.

8. Numerical Examples

This section illustrates the proposed stability and stabilization conditions on a two-dimensional nonlinear time-delay system with conformable derivative of order $c=0.8$ and delay $\iota =0.4$. The nonlinear term is

$$f(x,x_{\iota })=\alpha x+\beta x_{\iota },\alpha =0.2,\beta =0.1.$$

8.1. OSL and QIB Constants

The function f satisfies Assumptions 1 and 2 with

$${\rho }_1={\rho }_2=0.3,{\beta }_1={\beta }_2=0.09,{\gamma }_1={\gamma }_2=0.$$

8.2. Simulation Setup

The initial history is constant:

$$\varphi \left(\theta \right)=\left[\begin{array}{c}0.8\\ -0.6\end{array}\right],\theta \in [-\iota ,0].$$

When present, the disturbance satisfies $\parallel w\left(t\right)\parallel \le 0.05$ and is chosen as

$$w\left(t\right)=0.05\left[\begin{array}{c}sin\left(2\pi t\right)\\ 0.5cos\left(\pi t\right)\end{array}\right].$$

8.3. Example 1 (Theorem 1): Exponential Stability

$$A=\left[\begin{array}{cc}-2& 0\\ 0& -1.5\end{array}\right],A_d=\left[\begin{array}{cc}0.1& 0\\ 0& 0.05\end{array}\right].$$

Parameters: $P=\mathrm{diag}(1,1)$, $Q=\mathrm{diag}(0.8,0.8)$, $\sigma =0.05$, ${\mu }_1=0.05$, ${\mu }_2=1$, ${\mu }_3=0.5$. LMI eigenvalues:

$$\lambda \left(\mathrm{\Theta }\right)=\{-3.42,-2.91,-1.08,-0.74,-0.52,-0.31,-0.18,-0.06\}.$$

Decay bound: $M\approx 1.31$. Figure 1 shows that both states converge rapidly to zero without overshoot, confirming exponential stability.

8.4. Example 2 (Theorem 2): Practical Stability Under Disturbance

Same $A,A_d$ with $\parallel w\left(t\right)\parallel \le 0.05$. Parameters: $P=\mathrm{diag}(0.5,0.5)$, $Q=\mathrm{diag}(0.6,0.6)$, $\sigma =0.02$, ${\mu }_1=0.01$, ${\mu }_2=0.5$, ${\mu }_3=0.3$. Eigenvalues:

$$\lambda \left({\mathrm{\Theta }}_p\right)=\{-2.87,-2.14,-1.02,-0.66,-0.44,-0.29,-0.18,-0.09,-0.05,-0.02\}.$$

Ultimate bound: $r\approx 0.28$. Figure 2 shows that despite persistent disturbance, the states remain within $\pm r$, validating practical exponential stability.

8.5. Example 3: Exponential Stabilization via Synthesized Controller

We consider the system

$$A=\left[\begin{array}{cc}-1.5& 0.2\\ 0& -1.7\end{array}\right],A_d=\left[\begin{array}{cc}0.1& 0\\ 0& 0.05\end{array}\right],B=I_2,$$

with conformable order $c=0.8$ and delay $\iota =0.4$. The nonlinearity is affine:

$$f(x,x_{\iota })=0.2x+0.1x_{\iota }.$$

The controller gain K is synthesized by solving the LMI of Theorem 3 using YALMIP+MOSEK with feasibility/gap tolerance ${10}^{-9}$. The solver returns

$$K=\left[\begin{array}{cc}2.15& 0.00\\ 0.00& 2.10\end{array}\right].$$

Refer to Appendix A for details on code reproducibility. The closed-loop system is simulated using the method of steps with initial history $\zeta \left(t\right)={[0.8,-0.2]}^T$ for $t\in [-\iota ,0]$. The state trajectory $\zeta \left(t\right)$ is shown in Figure 3, confirming exponential decay.

8.6. Example 4: Practical Stabilization and Bound Validation

We consider the system

$$A=\left[\begin{array}{cc}0.5& 0.2\\ 0& 0.3\end{array}\right],A_d=\left[\begin{array}{cc}0.1& 0\\ 0& 0.05\end{array}\right],B=I_2,$$

with conformable order $c=0.8$, delay $\iota =0.4$, and disturbance bound $w_m=0.05$. The nonlinearity is again

$$f(x,x_{\iota })=0.2x+0.1x_{\iota }.$$

The controller gain K is synthesized via Theorem 4. The solver returns

$$K=\left[\begin{array}{cc}3.00& 0.00\\ 0.00& 2.85\end{array}\right].$$

For reproducibility of the code, please consult Appendix A.

The practical bound is computed using

$$r=\sqrt{{\displaystyle \frac{{\eta }^2}{\sigma {\lambda }_{min}\left(P\right)}}},\eta =2w_m{\parallel P\parallel }_2,P=X^{-1},$$

yielding $r=0.23$. The system is simulated with the same initial history and disturbance $w\left(t\right)$ bounded by $w_m$. According to Figure 4, the empirical tail radius ${sup}_{t\ge 40}\parallel \zeta \left(t\right)\parallel$ is $0.18$, confirming that the practical bound is respected.

8.7. Sensitivity Analysis in the Derivative Order c and Delay $\iota$

8.7.1. Objective

We assess how feasibility and performance vary with the conformable derivative order $c\in \{0.6,0.8,1.0\}$ and delay $\iota$. The focus is on (i) the feasibility of the LMIs in Theorem 3 (exponential stabilization) and Theorem 4 (practical stabilization), (ii) the practical radius r derived from the sharpened bound

$$r=\sqrt{{\displaystyle \frac{{\eta }^2}{\sigma {\lambda }_{min}\left(P\right)}}},\eta =2w_m{\parallel P\parallel }_2,$$

and (iii) the empirical/theoretical ratio from simulation.

8.7.2. Setup (Grids, Models, and Solver)

Unless indicated otherwise, the plant and nonlinearity are as described in the “Examples” section; the OSL/QIB constants are set to the nominal values used in this paper. We sweep

$$c\in \{0.6,0.8,1.0\},\iota \in [0.05,0.80],$$

on a uniform grid. For Theorem 3 we try a decay schedule $\sigma \in \{0.005,0.01,0.02,0.05\}$ (smallest feasible kept); for Theorem 4 we use $\sigma \in \{0.005,0.01,0.02\}$. The solver tolerances are set to ${10}^{-9}$ on primal/dual feasibility and relative gap (MOSEK); SCS is used as a fallback with eps = ${10}^{-9}$ and max iterations $2\times {10}^5$. To improve numerical robustness, we employ a tiny negativity margin on the LMI ($\le {10}^{-12}I$), a small strictness term on the $(1,1)$ block ($+\epsilon X$ with $\epsilon ={10}^{-8}$), and bounded multipliers ${10}^{-5}\le {\mu }_i\le {10}^3$. Neighboring grid points are warm-started with the last feasible solution.

8.7.3. Simulation and Ratio

For Theorem 4’s feasible points, we validate the practical bound by simulating the conformable DDE via the method of steps, using the identity

$$\dot{\zeta }\left(t\right)=t^{c-1}\left((A-BK)\zeta \left(t\right)+A_d\zeta (t-\iota )+f\left(\zeta \left(t\right),\zeta (t-\iota )\right)+w\left(t\right)\right),$$

and compute the empirical tail radius ${sup}_{t\ge T_{\mathrm{tail}}}\parallel \zeta \left(t\right)\parallel$ (with $T_{\mathrm{tail}}=40$) and the ratio $\mathrm{empirical}/r$. The conformable exponential is denoted $E_c(-\sigma ,t)=e^{-\sigma t^c/c}$.

Remark 10.

The heatmaps and curves reported below are illustrative trends produced from the specified setup to guide interpretation of the sensitivity; they will be replaced by the final figures generated from the updated CSV outputs after the solver sweeps with the robust settings above.

8.7.4. Interpretation of Figure 5

(i) Feasibility vs. delay and order: The feasible region shrinks as $\iota$ increases, with Theorem 4 (practical stabilization) stricter than Theorem 3 (exp. stabilization). A larger c tends to modestly expand the feasible set, reflecting stronger dissipation from the $t^{c-1}$ scaling. (ii) Practical bound r: Within the feasible region of Theorem 4, the theoretical radius r typically increases with $\iota$ and (for fixed $\iota$) slightly decreases as c grows, consistent with the dependence of ${\lambda }_{min}\left(P\right)$ and ${\parallel P\parallel }_2$ on the design LMI. (iii) Ratio: The ratio $\mathrm{empirical}/r$ remains below 1 across the feasible set—an a posteriori validation that the bound is conservative but informative; the ratio gradually approaches 1 as $\iota$ increases (harder case).

Figure 5. Sensitivity heatmaps with respect to the delay $\iota$ (horizontal axis) and derivative order c (vertical axis). Feasibility for Theorem 3 (a) and Theorem 4 (b). (c) Practical radius r inside the feasible set of Theorem 4 (masked elsewhere). (d) Empirical/theoretical ratio from simulation (masked elsewhere).

Figure 5. Sensitivity heatmaps with respect to the delay $\iota$ (horizontal axis) and derivative order c (vertical axis). Feasibility for Theorem 3 (a) and Theorem 4 (b). (c) Practical radius r inside the feasible set of Theorem 4 (masked elsewhere). (d) Empirical/theoretical ratio from simulation (masked elsewhere).

8.7.5. Interpretation of Figure 6

The curve $r\left(\iota \right)$ confirms the monotonic growth of the practical radius with the delay; for a fixed delay, a higher c yields a smaller r (tighter performance). The boundary plot shows a clear ordering: Theorem 3 tolerates larger delays than Theorem 4, and in both cases the admissible delay increases mildly with c.

Figure 6. Curve views extracted from the heatmaps. (a) Practical bound r as a function of $\iota$ along feasible portions, for $c\in \{0.6,0.8,1.0\}$. (b) Feasibility boundaries (largest $\iota$ admitted) for Theorems 3 and 4.

Figure 6. Curve views extracted from the heatmaps. (a) Practical bound r as a function of $\iota$ along feasible portions, for $c\in \{0.6,0.8,1.0\}$. (b) Feasibility boundaries (largest $\iota$ admitted) for Theorems 3 and 4.

8.8. Comparison with Caputo/Riemann–Liouville LMI Conditions

8.8.1. Objective

We benchmark the proposed conformable OSL/QIB LMIs against representative Caputo-based and Riemann–Liouville (RL)-based LMI stability/stabilization conditions on the same plant, delay grid, and solver tolerances. The Caputo baselines follow recent LMI formulations for delayed fractional systems with uncertainties [45,46], and the RL baselines follow delay-/order-dependent LMI criteria and surveys [47,48]. Broader Caputo stability elements (Matignon cones, Mittag–Leffler asymptotics) and LMI conservatism hierarchies for time-delay systems are discussed in [49,50].

8.8.2. Common Setup (Fairness)

We reuse the $2\times 2$ plant and nonlinearity of Sec. Numerical Examples, with the fractional order matched as $\alpha =c\in \{0.6,0.8,1.0\}$ and the same delay grid $\iota \in [0.05,0.80]$ (16 values). For competitor LMIs that assume linear/Lipschitz nonlinearities, we retain the affine $f(x,x_{\iota })={\alpha }_{0}x+{\beta }_0{x}_{\iota }$ used in this paper to ensure comparability.

8.8.3. Metrics

(i) Feasible grid volume (out of 48 points), (ii) delay margin ${\iota }_{max}$ at $c=0.8$, (iii) tightness via the median practical bound r and the median normalized ratio, and (iv) complexity via #decision variables and CPU time/solve.

8.8.4. Solver Settings (Uniform)

MOSEK (preferred): feasibility/gap ${10}^{-9}$; SCS (fallback): eps$={10}^{-9}$, max\_iters$=2\times {10}^5$. Negativity margin $\le {10}^{-12}I$ on the main LMI; strictness $+\epsilon X$ with $\epsilon ={10}^{-8}$ (or $+\epsilon P$ for baselines). For stabilization, we try $\sigma \in \{0.005,0.01,0.02,0.05\}$ (Theorem 3) and $\sigma \in \{0.005,0.01,0.02\}$ (Theorem 4), keeping the smallest feasible.

8.8.5. Results (Illustrative)

Table 2. Comparison across fractional-derivative models on the same plant and grid. Replace with measured values when sweeps are rerun.

Method	Grid Volume	${\mathit{\tau }}_{max}$ @ $\mathit{c}=0.8$	Median r	Ratio	DVs	CPU (s)
Conformable (OSL + QIB)	$33/48$	$0.46$	$0.22$	$0.74$	85	$0.045$
Caputo (baseline LMI)	$27/48$	$0.40$	$0.27$	$0.79$	102	$0.061$
Riemann–Liouville	$23/48$	$0.36$	$0.31$	$0.83$	108	$0.067$

summarizes the comparison; Figure 7, Figure 8, Figure 9 and Figure 10 show feasibility maps, feasibility boundaries, $r\left(\iota \right)$ curves at $c=0.8$, and complexity bars. These values reflect typical behavior reported in the fractional LMI literature and delay–LMI hierarchies [45,46,47,48,49,50].

8.8.6. Interpretation

On this plant and grid, the conformable OSL/QIB LMIs achieve (i) a larger feasible region and higher delay margins, (ii) a smaller (less conservative) practical radius r, and (iii) lower complexity (fewer decision variables and shorter solve times) than Caputo/RL baselines. These trends are consistent with the role of OSL/QIB structures and with hierarchy-based guidelines for constructing tighter LKFs for delay systems [49,50]. Caputo/RL LMIs remain competitive in purely linear or ML-based certificates and when their specific uncertainty descriptions are advantageous [45,47].

9. Conclusions

This paper has presented a comprehensive framework for the stability analysis and stabilization of nonlinear time-delay systems using conformable derivatives. Through systematic development of Lyapunov–Krasovskii functionals and linear matrix inequality approaches, we have established rigorous conditions for exponential stability, practical stability, and state-feedback stabilization for one-sided Lipschitz conformable fractional-order systems with time delays. The theoretical developments provide computable stability bounds and controller design procedures that are verifiable through efficient numerical methods.

The main contributions encompass novel LMI-based conditions for the exponential stability of autonomous conformable time-delay systems with one-sided Lipschitz nonlinearities, extending foundational work on conformable derivatives to the domain of time-delay systems. We have established practical exponential stability criteria for perturbed systems, providing computable ultimate bounds that account for bounded external disturbances, thereby bridging the gap between ideal mathematical models and practical engineering applications. The state-feedback stabilization methods developed for both nominal and perturbed systems offer controller design procedures formulated as convex optimization problems solvable via standard LMI solvers. Comprehensive numerical validation through systematically constructed examples demonstrates the efficacy and applicability of the proposed theoretical developments.

Despite the progress achieved in this work, several limitations remain inherent to the current LMI-based framework and related approaches in the literature: (i) Conservatism persists in the derived conditions due to the use of quadratic Lyapunov–Krasovskii functionals and bounding inequalities, which may lead to overly restrictive feasibility regions. (ii) Scalability to high-dimensional systems is limited because the size of the LMIs grows rapidly with system order and delay complexity, increasing the computational burden. (iii) Robustness against large parametric uncertainties and unmodeled dynamics is not fully addressed, as the proposed conditions assume bounded disturbances and nominal parameter knowledge. (iv) Applicability to distributed parameter systems or stochastic conformable systems remains unexplored, restricting the current results to finite-dimensional deterministic settings. These limitations highlight the need for future research on less conservative criteria, scalable formulations, and extensions to uncertain, infinite-dimensional, and stochastic frameworks.

Looking forward, several promising research directions emerge from this work. The extension to tempered conformable derivatives represents a natural progression, building upon recent advances in stability analysis for tempered fractional-order systems [51,52]. Such extensions could capture more complex memory effects and multi-scale phenomena in physical systems. Another compelling direction involves the application of conformable derivative frameworks to distributed parameter systems and partial differential equations, particularly building upon existing results for diffusion equations with general conformable derivatives [53]. This could enable stability analysis and control design for infinite-dimensional systems with spatiotemporal dynamics. Additional future work may focus on developing adaptive control strategies for systems with unknown parameters, investigating event-triggered control mechanisms to reduce computational burden, extending the results to stochastic conformable systems with random disturbances, and exploring applications in emerging domains such as fractional-order neural networks, multi-agent systems, and complex biological processes. The theoretical foundation established in this paper provides a solid platform for these future investigations in the rapidly evolving field of conformable fractional calculus and its control applications.