Stability analysis of linear systems with a periodical time-varying delay based on an improved non-continuous piecewise Lyapunov functional

Wei Wang; Chang-Xin Li; Ao-Qian Luo; Hui-Qin Xiao; Wei Wang; Chang-Xin Li; Ao-Qian Luo; Hui-Qin Xiao

doi:10.3934/math.2025418

AIMS Mathematics

2025, Volume 10, Issue 4: 9073-9093. doi: 10.3934/math.2025418

Previous Article Next Article

Research article

Stability analysis of linear systems with a periodical time-varying delay based on an improved non-continuous piecewise Lyapunov functional

1.
Department of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou, 412007, Hunan, China
2.
Department of Transportation, Hunan University of Technology, Zhuzhou, 412007, Hunan, China

Received: 30 December 2024 Revised: 28 March 2025 Accepted: 09 April 2025 Published: 21 April 2025
MSC : 34D20, 34K20, 34K25

This paper mainly studies the stability of linear systems with a periodical time-varying delay. An improved delay-segmentation-based non-continuous piecewise Lyapunov–Krasovskii (L–K) functional is proposed. In comparison with the currently available L–K functional, this functional incorporates more delay-segmentation-interval-related information. Additionally, it effectively eases the boundary constraints at each segment point. Consequently, a stability criterion with reduced conservativeness for linear time-delay systems is derived. Finally, two numerical examples and a single-area load frequency control system are given to validate the efficiency of the proposed approach.

Keywords:

Citation: Wei Wang, Chang-Xin Li, Ao-Qian Luo, Hui-Qin Xiao. Stability analysis of linear systems with a periodical time-varying delay based on an improved non-continuous piecewise Lyapunov functional[J]. AIMS Mathematics, 2025, 10(4): 9073-9093. doi: 10.3934/math.2025418

Related Papers:

[1]	Xingyue Liu, Kaibo Shi . Further results on stability analysis of time-varying delay systems via novel integral inequalities and improved Lyapunov-Krasovskii functionals. AIMS Mathematics, 2022, 7(2): 1873-1895. doi: 10.3934/math.2022108
[2]	Huahai Qiu, Li Wan, Zhigang Zhou, Qunjiao Zhang, Qinghua Zhou . Global exponential periodicity of nonlinear neural networks with multiple time-varying delays. AIMS Mathematics, 2023, 8(5): 12472-12485. doi: 10.3934/math.2023626
[3]	Zhigang Zhou, Li Wan, Qunjiao Zhang, Hongbo Fu, Huizhen Li, Qinghua Zhou . Exponential stability of periodic solution for stochastic neural networks involving multiple time-varying delays. AIMS Mathematics, 2024, 9(6): 14932-14948. doi: 10.3934/math.2024723
[4]	Patarawadee Prasertsang, Thongchai Botmart . Improvement of finite-time stability for delayed neural networks via a new Lyapunov-Krasovskii functional. AIMS Mathematics, 2021, 6(1): 998-1023. doi: 10.3934/math.2021060
[5]	Yude Ji, Xitong Ma, Luyao Wang, Yanqing Xing . Novel stability criterion for linear system with two additive time-varying delays using general integral inequalities. AIMS Mathematics, 2021, 6(8): 8667-8680. doi: 10.3934/math.2021504
[6]	Nayika Samorn, Kanit Mukdasai, Issaraporn Khonchaiyaphum . Analysis of finite-time stability in genetic regulatory networks with interval time-varying delays and leakage delay effects. AIMS Mathematics, 2024, 9(9): 25028-25048. doi: 10.3934/math.20241220
[7]	Jenjira Thipcha, Presarin Tangsiridamrong, Thongchai Botmart, Boonyachat Meesuptong, M. Syed Ali, Pantiwa Srisilp, Kanit Mukdasai . Robust stability and passivity analysis for discrete-time neural networks with mixed time-varying delays via a new summation inequality. AIMS Mathematics, 2023, 8(2): 4973-5006. doi: 10.3934/math.2023249
[8]	Seunghoon Lee, Youngjae Kim, Yonggwon Lee, Myeongjin Park, Ohmin Kwon . Relaxed stability conditions for linear systems with time-varying delays via some novel approaches. AIMS Mathematics, 2021, 6(3): 2454-2467. doi: 10.3934/math.2021149
[9]	Wentao Le, Yucai Ding, Wenqing Wu, Hui Liu . New stability criteria for semi-Markov jump linear systems with time-varying delays. AIMS Mathematics, 2021, 6(5): 4447-4462. doi: 10.3934/math.2021263
[10]	Yonggwon Lee, Yeongjae Kim, Seunghoon Lee, Junmin Park, Ohmin Kwon . An improved reachable set estimation for time-delay linear systems with peak-bounded inputs and polytopic uncertainties via augmented zero equality approach. AIMS Mathematics, 2023, 8(3): 5816-5837. doi: 10.3934/math.2023293

Abstract

1. Introduction

The time-delay phenomenon is widely present in various engineering systems, and its existence will affect the stability and performance of the system. Therefore, the investigations into time-delay systems have been widely studied in the control field in recent years ^[1,2,3]. Many achievements have been obtained in the fields involving discrete-time systems ^[4,5,6], fuzzy systems ^[7,8,9], networked control systems ^[10,11], load frequency control systems ^{[12,13,14,15]}, Markovian jump systems ^[16,17], Lur'e systems ^[18,19], and $H_{\infty}$ filtering control ^[20,21,22].

The Lyapunov–Razumikhin approach and the Lyapunov–Krasovskii (L–K) approach are the most widely used methods for addressing stability issues in time-delay systems. The Lyapunov–Razumikhin approach has achieved some advancements in deriving necessary and sufficient conditions for the stability of time-delay systems ^[23]. On the other hand, the L–K approach aims to construct an appropriate L–K functional to obtain less conservative stability criteria in the form of Linear Matrix Inequality (LMI) for time-delay systems, especially for time-varying delay systems ^{[24,25,26,27]}. It is worth mentioning that the L–K approach is frequently employed in combination with some integral inequality methods for bounding the integral terms in the derivatives delay-related L–K functionals ^{[28,29,30,31,32]}. Admittedly, recent research mainly focuses on reducing the conservativeness of the stability criteria by refining the structure of delay-related L–K functionals. An augmented L–K functional incorporating additional state information was presented in ^[21], enhancing the stability criteria's dependence on time delay. The augmented L–K functional constructed in ^[33] further considered the integrity of the information about delay intervals and gave less conservative results. An augmented L–K functional was also introduced in ^[27]. Unlike the L–K functionals in ^[21,33], it avoids the occurrence of high-order terms of variables in the derivative of the functional, thus easing the hardship of the solving process.

In addition to various augmented L–K functionals mentioned above, the delay-segmentation-based piecewise L–K functionals have also been widely discussed in recent research. The delay segmenting method can augment the delay-interval-related information in the functional derivatives. Subsequently, it enables a more in-depth exploration of the functional decrease within each interval instead of merely considering its global decreasing property. Consequently, this functional effectively reduces the conservativeness of the system stability criterion. Delay-segmentation-based piecewise L–K functionals can be categorized into continuous and non-continuous piecewise L–K functionals. A time-delay-partitioning-based L–K functional was constructed in ^[34], which effectively relaxes the constraints on the stability criteria of the system. In ^[35], Han et al. established a non-continuous L–K functional and gave a delay-related stability criterion. Additionally, for time-varying delay systems, some quadratic terms with time-varying delay were introduced in ^[18] in constructing the L–K functional using an improved delay-segmentation method, proposing a novel delay-segmentation-based piecewise functional.

Based on the domain of time-varying delay and its derivatives, the definition of allowable delay set (ADS) was given in ^[1]. Building upon this, an improved ADS was presented in ^[31], which optimized the stability criterion for linear systems with time-varying delay. However, the ADS given in ^[1] and ^[31] exhibit the coverage areas in the form of polygons, as indicated in ^[36]. However, this issue was further explored in ^[37], which gave ADS covering a complete ellipsoidal field by introducing specific periodically varying delays. Subsequently, an ADS partitioning approach was introduced in ^[38], which further refines the ADS through the application of the delay-segmenting method. By integrating this with the non-continuous L–K functional method, a stability criterion with reduced conservativeness was developed. However, the non-continuous piecewise function in ^[38] does not account for information regarding the delay segmentation and hinders potential improvements. Consequently, the existing criterion remains rather conservative.

In recent years, the research for periodical time delay has aroused the interest of researchers, and it exists widely in some mechanical motions^[39,40,41]. Combining delay-related L–K functional and a looped function, a stability criterion based on a periodically varying delay with monotone intervals was given in ^[37]. Based on this, a higher-order free-matrix-based integral inequality was introduced to optimize the stability criteria in ^[42], reducing the conservativeness of the stability criterion of the system. Furthermore, an exponential stability analysis of switching time-delay systems is presented in ^[43], which utilizes the symbolic transformation of delay derivatives as switching information. The stability of periodic time-delay systems is also studied in ^[38] by further partitioning the monotone intervals. In light of this, conducting in-depth research on periodic time-delay systems characterized by monotonic intervals on such a foundation holds significant and promising research prospects.

This paper proposes an innovative delay-segmentation-based non-continuous piecewise L–K functional. In different segments, the construction of different L–K functionals is presented according to the intervals where the delay is located, and information regarding the delay-interval-segmentation is fully exploited in the functional of each segment. Therefore, the derived stability condition is less conservative. Finally, two numerical examples and a single-area load frequency control system are given to demonstrate that the proposed approach significantly reduces the conservativeness of the presented stability criterion.

2. Preliminary

2.1. Notations

For brevity, the notations used in this paper are summarized in Table 1.

Table 1. Notations.

Symbol	Meaning
$\mathbb{N}$	Natural numbers: $\{1, 2, 3, \cdots\}$
$\mathbb{R}^n$	Euclidean space in $n$ dimensions
$\mathbb{R}^{m \times n}$	All real matrices of dimension $m \times n$
$\mathbb{S}^{n}$	All symmetric matrices of dimension $n \times n$
$\mathbb{S}^{n}_{+}$	All positive definite symmetric matrices of dimension $n \times n$
$\aleph^{-1}$	Inverse of matrix $\aleph$
$\aleph^{T}$	Transpose of matrix $\aleph$
$\text{Sym}\{\aleph\}$	The sum of matrix $\aleph$ and its transpose
*	A symmetric component inside a symmetric matrix
$\text{diag}\{\}$	A matrix with a block diagonal structure

| Show Table

DownLoad: CSV

2.2. System description

Consider the following linear system with time-varying delay:

$\begin{equation} \left\{\begin{aligned} \dot{x}(t)& = Ax(t)+A_{r} x(t-r(t)),\\ x(t)& = \varphi(t),t\in[-r_2,0], \end{aligned}\right.\ \end{equation}$

(2.1)

where $A, A_{r}\in\mathbb{R}^{n\times n}$ are system matrices, $x(t) \in \mathbb{R}^n$ is state vector, and $r(t)$ represents the time-varying delay, which satisfies the following form derived from ^[37,38,41]:

$\begin{equation} r(t) = r_0+\tilde{r} f(\Omega t), \end{equation}$

(2.2)

where $r_0$ is a constant, $f:\mathbb{R}\rightarrow [-1, 1]$ is a differentiable periodic function satisfying $\lvert f\rvert\leqslant1$ , whose each period contains a monotone increasing interval and a monotone decreasing interval, and parameters $\tilde{r}$ and $\Omega$ represent its amplitude and frequency, respectively. For $r(t)$ , $r_1$ is the lower bound, and $r_2$ is the upper bound. In addition, the derivative of $r(t)$ is defined as $\dot{r}(t)$ , $-\mu, \mu$ are its lower and upper bounds. Therefore, $r(t)$ and $\dot{r}(t)$ are satisfied

$\begin{equation} \begin{aligned} r_1\leqslant r(t)\leqslant r_2,\ \ \lvert\dot{r}(t)\rvert\leqslant \mu < 1, \end{aligned} \end{equation}$

(2.3)

with $r_1 = r_0-\tilde{r}$ , $r_2 = r_0+\tilde{r}$ and $\mu = \tilde{r}\Omega$ .

The ADS partitioning approach based on a periodical time-varying delay is discussed in ^[38]. There exists $t_{2k-1}$ , $t_{2k}$ , $t_{2k+1}$ , $k\in\mathbb{N}$ , in the periodic function $r(t)$ , which are extreme values of $r(t)$ satisfying $r(t_{2k-1}) = r(t_{2k+1}) = r_1$ and $r(t_{2k}) = r_2$ . $r(t)$ is monotone increasing in the intervals $t\in[t_{2k-1}, t_{2k})$ and monotone decreasing in the intervals $t\in[t_{2k}, t_{2k+1})$ .

Then, we choose moving points $\varrho_{1k}\in[t_{2k-1}, t_{2k})$ and $\varrho_{2k}\in[t_{2k}, t_{2k+1})$ , ensuring that $r(\varrho_{1k}) = r(\varrho_{2k}) = r_{\varrho}$ . The uncertain delay value $r_{\varrho} = r_1+\varrho(r_2-r_1), \varrho\in[0, 1]$ that satisfies $r_1\leqslant r_{\varrho}\leqslant r_2$ , and its value changes with the change of parameter $\varrho$ . Depending on the value range of $r(t)$ , we obtain two delay-segmentation-based intervals: $r(t)\in[r_1, r_{\varrho}]$ and $r(t)\in[r_{\varrho}, r_2]$ . Thus, the following ADS $\gimel\triangleq\gimel_{\alpha i}\cup\gimel_{\beta i}, i = 1, 2$ are obtained

$\begin{equation} \left\{\begin{aligned} \gimel_{\alpha1} &\triangleq[r_1,r_{\varrho}]\times[0,\mu]\ \ \ \ \ \ \ \ \gimel_{\beta1}\triangleq[r_{\varrho},r_2]\times[0,\mu],\\ \gimel_{\alpha2} &\triangleq[r_1,r_{\varrho}]\times[-\mu,0]\ \ \ \ \ \gimel_{\beta2} \triangleq[r_{\varrho},r_2]\times[-\mu,0]. \end{aligned}\right.\ \end{equation}$

(2.4)

Remark 2.1. The ADS described in formula (2.4) was initially presented in ^[38]. To underscore the efficacy of the enhanced functional put forward in this paper, we opt for the identical ADS to analyze the stability of the considered systems.

2.3. Lemma

To derive the main results, the following lemma needs to be introduced.

Lemma 2.1. ^[31,32] If $x(t)\in[\omega_1, \omega_2]\rightarrow \mathbb{R}^n$ and $\kappa\in\mathbb{R}^{m}$ are continuous and differentiable, for matrices $U\in\mathbb{S}^{n}_{+}$ , $H\in\mathbb{R}^{m\times 2n}$ , the following inequalities hold:

$\begin{align} &-\int^{\omega_2}_{\omega_1} \dot{x}^T(s)U\dot{x}(s)ds\leqslant-\frac{1}{\omega_2-\omega_1}\kappa^T\vartheta^T\hat{U}\vartheta\kappa, \end{align}$

(2.5)

$\begin{align} &-\int^{\omega_2}_{\omega_1}\dot x^T(s)U \dot{x}(s)ds\leqslant(\omega_2-\omega_1)\kappa^TH\hat{U}^{-1}H^T\kappa+2\kappa^TH\vartheta\kappa, \end{align}$

(2.6)

where

$\begin{align*} \kappa& = col\left\{x(\omega_2),x(\omega_1),\frac{1}{\omega_2-\omega_1}\int^{\omega_2}_{\omega_1}x(s)ds\right\},\\ \vartheta& = [\tilde{e}_1^T-\tilde{e}_2^T\ \ \tilde{e}_1^T+\tilde{e}_2^T-2\tilde{e}_3^T]^T,\\ \hat{U}& = diag\{U,3U\},\\ \tilde{e}_{\theta}& = [0_{n\times(\theta-1)n}\ \ I_n\ \ 0_{n\times(3-i)n}],\theta = 1,\cdots,3.\\ \end{align*}$

3. Main results

This section discusses the stability criterion for system (2.1) based on ADS $\gimel$ . The subsequent notations are introduced to represent the vectors and matrices for convenience.

$\begin{align*} \delta_1(t) = &[x^T(t)\ \ \ x^T(t-r_1)\ \ \ x^T(t-r(t))\ \ \ x^T(t-r_2)\ \ \ x^T(t-r_{\varrho})]^T,\\ \delta_{2}(t) = &[\dot{x}^T(t-r_1)\ \ \dot{x}^T(t-r(t))\ \ \dot{x}^T(t-r_2)\ \ \dot{x}^T(t-r_{\varrho})]^T,\\ \delta_{3}(t) = &\int^{t}_{t-r_1}x(s)ds,\ \delta_{4}(t) = \int^{t-r_1}_{t-r(t)}x(s)ds,\ \delta_{5}(t) = \int^{t-r(t)}_{t-r_{\varrho}}x(s)ds,\\ \delta_{6}(t) = &\int^{t-r(t)}_{t-r_2}x(s)ds,\ \delta_{7}(t) = \int^{t-r_1}_{t-r_{\varrho}}x(s)ds,\ \delta_{8}(t) = \int^{t-r_{\varrho}}_{t-r_2}x(s)ds,\\ \varpi_0(t) = &[\delta^T_1(t)\ \ \ \delta^T_3(t)]^T,\ \ \varpi_1(s) = [x^T(s)\ \ \ \dot x^T(s)],\\ \varpi_{2\alpha}(t) = &[\delta^T_1(t)\ \ \delta^T_3(t)\ \ \ \delta^T_4(t)\ \ \delta^T_5(t)\ \ \delta^T_{8}(t)]^T,\\ \varpi_{2\beta}(t) = &[\delta^T_1(t)\ \ \ \delta^T_3(t)\ \ \ \delta^T_{7}(t)\ \ \ -\delta^T_5(t)\ \ \ \delta^T_6(t)]^T,\\ \varpi(t) = &[\delta^T_1(t)\ \ \ \delta^T_2(t)\ \ \ \frac{1}{r_1}\delta^T_{3}(t)\ \ \frac{1}{r(t)-r_1}\delta^T_{4}(t)\ \ \frac{1}{r_{\varrho}-r(t)}\delta^T_{5}(t)\ \ \frac{1}{r_2-r(t)}\delta^T_{6}(t)\ \ \frac{1}{r_{\varrho}-r_1}\delta^T_{7}(t)\ \ \frac{1}{r_2-r_{\varrho}}\delta^T_{8}(t)]^T,\\ e_{\rho} = &[0_{n\times(\rho-1)n}\ \ \ I_n\ \ 0_{n\times(15-\rho)n}],\rho = 1,2,\cdots,15. \end{align*}$

As shown below, Theorem 3.1 provides a stability criterion for the system (2.1) based on ADS $\gimel$ .

Theorem 3.1. For scalars $\varrho\in[0, 1], r_2\geqslant r_1\geqslant 0, \mu < 1$ , suppose that there exist matrices $P\in\mathbb{S}^{9n}_{+}$ , $F, X_{\alpha i\varsigma}, X_{\beta i\varsigma}\in\mathbb{S}^{6n}_{+}$ , $Q_\varsigma, S_{\alpha i\varsigma}, S_{\beta i\varsigma}\in\mathbb{S}^{2n}_{+}$ , $Y_\varsigma, T_{\alpha i\varsigma}, T_{\beta i\varsigma}\in\mathbb{S}^{n}_{+}$ satisfying the condition (3.1), $W_{\alpha i1}, W_{\alpha i2}, W_{\beta i1}, W_{\beta i2}\in\mathbb{S}^{n}$ , $G_{\alpha i\varsigma}, G_{\beta i\varsigma}\in\mathbb{R}^{15n\times 2n}$ , $N_{\alpha i1}, N_{\alpha i2}, N_{\beta i1}, N_{\beta i2}\in\mathbb{R}^{15n\times n}, i = 1, 2, \varsigma = 1, \cdots, 3$ . Then, system (2.1) is stable if LMIs (3.2)–(3.3) and (3.4)–(3.5) are satisfied at the vertices of $\gimel_{\alpha i}$ and $\gimel_{\beta i}$ , respectively.

$\begin{equation} \begin{aligned} X_{\alpha 11}&\geqslant X_{\beta11}\geqslant X_{\beta21}\geqslant X_{\alpha 21},\ X_{\beta12}\geqslant X_{\beta22},\\ X_{\beta 23}&\geqslant X_{\alpha 23}\geqslant X_{\alpha 13}\geqslant X_{\beta13},\ X_{\alpha 22}\geqslant X_{\alpha 12},\\ S_{\alpha 11}&\geqslant S_{\beta11}\geqslant S_{\beta21}\geqslant S_{\alpha 21},\ S_{\beta12}\geqslant S_{\beta22},\\ S_{\beta 23}&\geqslant S_{\alpha 23}\geqslant S_{\alpha 13}\geqslant S_{\beta13},\ S_{\alpha 22}\geqslant S_{\alpha 12},\\ T_{\alpha 11}&\geqslant T_{\beta11}\geqslant T_{\beta21}\geqslant T_{\alpha 21},\ T_{\beta12}\geqslant T_{\beta22},\\ T_{\beta 23}&\geqslant T_{\alpha 23}\geqslant T_{\alpha 13}\geqslant T_{\beta13},\ T_{\alpha 22}\geqslant T_{\alpha12},\\ \end{aligned} \end{equation}$

(3.1)

$\begin{equation} W_{YT\alpha i1} > 0,\ W_{YT\alpha i2} > 0, \end{equation}$

(3.2)

$\begin{equation} \left[\begin{matrix}\Upsilon_{\alpha i}(r(t),\dot{r}(t))&\sqrt{r(t)-r_1}G_{\alpha i1}&\sqrt{r_{\varrho}-r(t)}G_{\alpha i2}&\sqrt{r_2-r_1}G_{\alpha i3}\\\ast&-\widehat{W}_{YT\alpha i1}&0&0\\ \ast&\ast&-\widehat{W}_{YT\alpha i2}&0\\ \ast&\ast&\ast&-\widehat{Y}_{T1} \end{matrix}\right] < 0, \end{equation}$

(3.3)

$\begin{equation} \ W_{YT\beta i1} > 0,\ W_{YT\beta i2} > 0, \end{equation}$

(3.4)

$\begin{equation} \left[\begin{matrix}\Upsilon_{\beta i}(r(t),\dot{r}(t))&\sqrt{r_{\varrho}-r_1}G_{\beta i1}&\sqrt{r(t)-r_{\varrho}}G_{\beta i2}&\sqrt{r_2-r(t)}G_{\beta i3}\\\ast&-\widehat{Y}_{T2}&0&0\\ \ast&\ast&-\widehat{W}_{YT\beta i1}&0\\ \ast&\ast&\ast&-\widehat{W}_{YT\beta i2} \end{matrix}\right] < 0, \end{equation}$

(3.5)

where

$\begin{align*} \Upsilon_{\alpha i}(r(&t),\dot{r}(t)) = \Lambda_1+\Lambda_{2\alpha i}+\Lambda_{3\alpha i}+\Lambda_{4\alpha i},&\\ \Upsilon_{\beta i}(r(&t),\dot{r}(t)) = \Lambda_1+\Lambda_{2\beta i}+\Lambda_{3\beta i}+\Lambda_{4\beta i},&\\ \Lambda_1 = &{\mathit{\text{Sym}}}\{h_1\Pi_0^TF\Pi_1\}+\Pi_2^T{Q}_1\Pi_{2}+\Pi_3^T(Q_2-Q_1)\Pi_{3}+\Pi_4^T(Q_3-Q_2)\Pi_{4}\\ &-\Pi_5^TQ_3\Pi_5+r_1^2\Pi_{20}^TY_1\Pi_{20}+(r_{\varrho}-r_1)\Pi_{20}^TY_2\Pi_{20}+(r_2-r_{\varrho})\Pi_{20}^TY_3\Pi_{20},&\\ \Lambda_{2\alpha i} = &{\mathit{\text{Sym}}}\{\Pi_0^T[(r(t)-r_1)X_{\alpha i1}+(r_{\alpha}-r(t))X_{\alpha i2}+(r_2-r_{\varrho})X_{\alpha i3}]\Pi_1+\Pi_7^TP\Pi_8\}\\ &+\dot{r}(t)\Pi_0^T(X_{\alpha i1}-X_{\alpha i2})\Pi_0+\Pi_3^TS_{\alpha i1}\Pi_3+(1-\dot{r}(t))\Pi_6^T(S_{\alpha i2}-S_{\alpha i1})\Pi_6\\ &+\Pi_4^T(S_{\alpha i3}-S_{\alpha i2})\Pi_4-\Pi_5^TS_{\alpha i3}\Pi_5+(r_2-r_{\varrho})e_9^T(T_{\alpha i3}-T_{\alpha i2})e_9\\ &+(r_2-r(t))(1-\dot{r}(t))e_7^T(T_{\alpha i2}-T_{\alpha i1})e_7+(r_2-r_1)e_6^TT_{\alpha i1}e_6,\\ \Lambda_{3\alpha i} = &(r(t)-r_1)((1-\dot{r}(t))e^T_7W_{\alpha i2}e_7-e_9^TW_{\alpha i2}e_9)\\ &-(r_{\varrho}-r(t))(e_6^TW_{\alpha i1}e_6-(1-\dot{r}(t))e^T_7W_{\alpha i1}e_7),&\\ \Lambda_{2\beta i} = &{\mathit{\text{Sym}}}\{\Pi_0^T[(r_{\varrho}-r_1)X_{\beta i1}+(r(t)-r_{\alpha})X_{\beta i2}+(r_2-r(t))X_{\beta i3}]\Pi_1+\Pi_{9}^TP\Pi_{10}\}\\ &+\dot{r}(t)\Pi_0^T(X_{\beta i2}-X_{\beta i3})\Pi_0+\Pi_3^TS_{\beta i1}\Pi_3+(1-\dot{r}(t))\Pi_6^T(S_{\beta i3}-S_{\beta i2})\Pi_6\\ &+\Pi_4^T(S_{\beta i2}-S_{\beta i1})\Pi_4-\Pi_5^TS_{\beta i3}\Pi_5+(r_2-r(t))(1-\dot{r}(t))e_7^T(T_{\beta i3}-T_{\beta i2})e_7\\ &+(r_2-r_{\varrho})e_9^T(T_{\beta i2}-T_{\beta i1})e_9+(r_2-r_1)e_6^TT_{\beta i1}e_6,\\ \Lambda_{3\beta i} = &(r(t)-r_{\varrho})((1-\dot{r}(t))e^T_7W_{\beta i2}e_7-e_8^TW_{\beta i2}e_8)\\ &-(r_2-r(t))(e_9^TW_{\beta i1}e_9-(1-\dot{r}(t))e^T_7W_{\beta i1}e_7),&\\ \Lambda_{4\alpha i} = &{\mathit{\text{Sym}}}\{G_{\alpha i1}\Pi_{12}+G_{\alpha i2}\Pi_{13}+G_{\alpha i3}\Pi_{14}+N_{\alpha i1}\Pi_{18}+N_{\alpha i2}\Pi_{19}\}-\Pi^T_{11}\widehat{Y}_1\Pi_{11},&\\ \Lambda_{4\beta i} = &{\mathit{\text{Sym}}}\{G_{\beta i1}\Pi_{15}+G_{\beta i2}\Pi_{16}+G_{\beta i3}\Pi_{17}+N_{\beta i1}\Pi_{18}+N_{\beta i2}\Pi_{19}\}-\Pi^T_{11}\widehat{Y}_1\Pi_{11},&\\ \widehat{Y}_{T1} = &diag\{{Y}_{T1},3{Y}_{T1}\},\ \ \widehat{Y}_{T2} = diag\{{Y}_{T2},3{Y}_{T2}\},&\\ \widehat{W}_{YT\alpha i1}& = diag\{W_{YT\alpha i1},3W_{YT\alpha i1}\},\ \ \widehat{W}_{YT\alpha i2} = diag\{W_{YT\alpha i2},3W_{YT\alpha i2}\},&\\ \widehat{W}_{YT\beta i1}& = diag\{W_{YT\beta i1},3W_{YT\beta i1}\},\ \ \widehat{W}_{YT\beta i2} = diag\{W_{YT\beta i2},3W_{YT\beta i2}\},& \end{align*}$

with

$\begin{align*} \Pi_0 = &[e_1^T\ \ \ \ e_2^T\ \ \ \ e_3^T\ \ \ \ e_{4}^T\ \ \ \ e_{5}^T\ \ \ \ r_1e_{10}^T]^T,\ \ \ \ \Pi_1 = [\Pi_{20}^T\ \ \ \ e_6^T\ \ \ \ (1-\dot{r}(t))e_7^T\ \ \ \ e_{8}^T\ \ \ \ e_{9}^T\ \ \ \ r_1e_{10}^T]^T,&\\ \Pi_{2} = &[e_1^T\ \ \ \ \Pi_{20}^T]^T,\ \ \ \ \Pi_{3} = [e_2^T\ \ \ \ e_6^T]^T,\ \ \ \ \Pi_{4} = [e_5^T\ \ \ \ e_9^T]^T,\ \ \ \ \Pi_{5} = [e_4^T\ \ \ \ e_8^T]^T,\ \ \ \ \Pi_{6} = [e_3^T\ \ \ \ e_7^T]^T,&\\ \Pi_7 = &[e_1^T\ \ \ \ e_2^T\ \ \ \ e_3^T\ \ \ \ e_{4}^T\ \ \ \ e_{5}^T\ \ \ \ r_1e_{10}^T(r(t)-r_1)e_{11}^T\ \ \ \ (r_{\varrho}-r(t))e_{12}^T\ \ \ \ (r_2-r_{\varrho})e_{15}^T]^T,&\\ \Pi_{8} = &[\Pi_{20}^T\ \ \ \ e_6^T\ \ \ \ (1-\dot{r}(t))e_7^T\ \ \ \ e_{8}^T\ \ \ \ e_{9}^T\ \ \ \ e_{1}^T-e_{2}^T\ \ \ \ e_{2}^T-(1-\dot{r}(t))e_{3}^T\ \ \ \ (1-\dot{r}(t))e_{3}^T-e_{5}^T\ \ \ \ e_{5}^T-e^T_{4}]^T,&\\ \Pi_{9} = &[e_1^T\ \ \ \ e_2^T\ \ \ \ e_3^T\ \ \ \ e_{4}^T\ \ \ \ e_{5}^T\ \ \ \ r_1e_{10}^T\ \ \ \ (r_{\varrho}-r_1)(t)e_{14}^T\ \ \ \ (r(t)-r_{\varrho})e_{12}^T\ \ \ \ (r_2-r(t))e_{13}^T]^T,&\\ \Pi_{10} = &[\Pi_{20}^T\ \ \ \ e_6^T\ \ \ \ (1-\dot{r}(t))e_7^T\ \ \ \ e_{8}^T\ \ \ \ e_{9}^T\ \ \ \ e_{1}^T-e_{2}^T\ \ \ \ e_{2}^T-e_{5}^T\ \ \ \ e_{5}^T-(1-\dot{r}(t))e_{3}^T\ \ \ \ (1-\dot{r}(t))e_{3}^T-e^T_{4}]^T,&\\ \Pi_{11} = &[e_1^T-e_2^T\ \ \ \ e_1^T+e_2^T-2e_{10}^T]^T,\ \ \ \ \Pi_{12} = [e_2^T-e_3^T\ \ \ \ e_2^T+e_3^T-2e_{11}^T]^T,&\\ \Pi_{13} = &[e_3^T-e_5^T\ \ \ \ e_3^T+e_5^T-2e_{12}^T]^T,\ \ \ \ \Pi_{14} = [e_5^T-e_4^T\ \ \ \ e_5^T+e_4^T-2e_{15}^T]^T,&\\ \Pi_{15} = &[e_2^T-e_5^T\ \ \ \ e_2^T+e_5^T-2e_{14}^T]^T,\ \ \ \ \Pi_{16} = [e_5^T-e_3^T\ \ \ \ e_5^T+e_3^T-2e_{12}^T]^T,&\\ \Pi_{17} = &[e_3^T-e_4^T\ \ \ \ e_3^T+e_4^T-2e_{13}^T]^T,&\\ \Pi_{18} = &(r(t)-r_1)e_{11}+(r_{\varrho}-r(t))e_{12}-(r_{\varrho}-r_1)e_{14},&\\ \Pi_{19} = &(r(t)-r_{\varrho})e_{12}+(r_2-r(t))e_{13}-(r_2-r_{\varrho})e_{15},&\\ \Pi_{20} = &Ae_{1}+A_{r}e_{3},&\\ {Y}_{T1} = &Y_3+T_{\alpha i3},\ \ {Y}_{T2} = Y_2+T_{\beta i1},\ \ \widehat{Y}_1 = diag\{Y_{1},3Y_{1}\},\\ {W}_{YT\alpha i1}& = Y_2+T_{\alpha i1}-\dot{r}(t)W_{\alpha i1},\ \ {W}_{YT\alpha i2} = Y_2+T_{\alpha i2}-\dot{r}(t)W_{\alpha i2},&\\ {W}_{YT\beta i1}& = Y_3+T_{\beta i2}-\dot{r}(t)W_{\beta i1},\ \ {W}_{YT\beta i2} = Y_3+T_{\beta i3}-\dot{r}(t)W_{\beta i2}.& \end{align*}$

Proof. Choosing the non-continuous piecewise L–K functional defined below:

$\begin{equation} V(t) = \left\{\begin{aligned} &V_0(t)+V_{1\alpha1}(t)+V_{2\alpha1}+V_{l\alpha1}(t),\ t\in[t_{2k-1},\varrho_{1k}),\\ &V_0(t)+V_{1\beta1}(t)+V_{2\beta1}+V_{l\beta1}(t),\ t\in[\varrho_{1k},t_{2k}),\\ &V_0(t)+V_{1\beta2}(t)+V_{2\beta2}+V_{l\beta2}(t),\ t\in[t_{2k},\ \varrho_{2k}),\\ &V_0(t)+V_{1\alpha2}(t)+V_{2\alpha2}+V_{l\alpha2}(t),\ t\in[\varrho_{2k},\ t_{2k+1}), \end{aligned}\right.\ \end{equation}$

(3.6)

where

$\begin{align*} {V_0(t)} = &r_1\varpi_0(t)^TF\varpi_0(t)+\int_{t-r_1}^t\varpi_1^T(s)Q_1\varpi_1(s)ds +\int_{t-r_{\varrho}}^{t-r_1}\varpi_1^T(s)Q_2\varpi_1(s)ds\\ &+\int^{t-r_{\varrho}}_{t-r_2}\varpi_1^T(s)Q_3\varpi_1(s)ds +r_1\int_{-r_1}^0\int_{t+u}^t\dot{x}^T(s)Y_1\dot{x}(s)dsdu\\ &+\int_{-r_{\varrho}}^{-r_1}\int_{t+u}^t\dot{x}^T(s)Y_2\dot{x}(s)dsdu +\int_{-r_2}^{-r_{\varrho}}\int_{t+u}^t\dot{x}^T(s)Y_3\dot{x}(s)dsdu,\\ {V_{1\alpha i}(t)} = &\varpi^T_0(t)[(r(t)-r_1)X_{\alpha i1}+(r_{\alpha}-r(t))X_{\alpha i2} +(r_2-r_{\varrho})X_{\alpha i3}]\varpi_0(t)+\varpi_{2\alpha}^T(t)P\varpi_{2\alpha}(t)\\ &+\int_{t-r(t)}^{t-r_1}\varpi_1^T(s)S_{\alpha i1}\varpi_1(s)ds +\int_{t-r_{\varrho}}^{t-r(t)}\varpi_1^T(s)S_{\alpha i2}\varpi_1(s)ds+\int^{t-r_{\varrho}}_{t-r_2}\varpi_1^T(s)S_{\alpha i3}\varpi_1(s)ds,\\ {V_{1\beta i}(t)} = &\varpi^T_0(t)[(r_{\varrho}-r_1)X_{\beta i1}+(r(t)-r_{\alpha})X_{\beta i2} +(r_2-r(t))X_{\beta i3}]\varpi_0(t)+\varpi_{2\beta}^T(t)P\varpi_{2\beta}(t)\\ &+\int_{t-r_{\varrho}}^{t-r_1}\varpi_1^T(s)S_{\beta i1}\varpi_1(s)ds +\int^{t-r_{\varrho}}_{t-r(t)}\varpi_1^T(s)S_{\beta i2}\varpi_1(s)ds+\int^{t-r(t)}_{t-r_2}\varpi_1^T(s)S_{\beta i3}\varpi_1(s)ds,\\ {V_{2\alpha i}(t)} = &\int_{t-r(t)}^{t-r_1}(r_2+s-t)\dot{x}^T(s)T_{\alpha i1}\dot{x}(s)ds +\int^{t-r(t)}_{t-r_{\varrho}}(r_2+s-t)\dot{x}^T(s)T_{\alpha i2}\dot{x}(s)ds\\ &+\int_{t-r_2}^{t-r_{\varrho}}(r_2+s-t)\dot{x}^T(s)T_{\alpha i3}\dot{x}(s)ds,\\ {V_{2\beta i}(t)} = &\int_{t-r_{\varrho}}^{t-r_1}(r_2+s-t)\dot{x}^T(s)T_{\beta i1}\dot{x}(s)ds +\int_{t-r(t)}^{t-r_{\varrho}}(r_2+s-t)\dot{x}^T(s)T_{\beta i2}\dot{x}(s)ds\\ &+\int_{t-r_2}^{t-r(t)}(r_2+s-t)\dot{x}^T(s)T_{\beta i3}\dot{x}(s)ds,\\ {V_{l\alpha i}(t)} = &(r(t)-r_{\varrho})\int^{t-r_1}_{t-r(t)}\dot{x}^T(s)W_{\alpha i1}\dot{x}(s)ds +(r(t)-r_1)\int_{t-r_{\varrho}}^{t-r(t)}\dot{x}^T(s)W_{\alpha i2}\dot{x}(s)ds,\\ {V_{l\beta i}(t)} = &(r(t)-r_2)\int^{t-r_{\varrho}}_{t-r(t)}\dot{x}^T(s)W_{\beta i1}\dot{x}(s)ds +(r(t)-r_{\varrho})\int_{t-r_{2}}^{t-r(t)}\dot{x}^T(s)W_{\beta i2}\dot{x}(s)ds. \end{align*}$

Differentiating ${V}(t)$ yields

$\begin{equation} \dot{V}(t) = \left\{\begin{aligned} &\dot{V}_0(t)+\dot{V}_{1\alpha1}(t)++\dot{V}_{2\alpha1}(t)+\dot{V}_{l\alpha1}(t),\ t\in[t_{2k-1},\varrho_{1k}),\\ &\dot{V}_0(t)+\dot{V}_{1\beta1}(t)+\dot{V}_{2\beta1}(t)+\dot{V}_{l\beta1}(t),\ t\in[\varrho_{1k},t_{2k}),\\ &\dot{V}_0(t)+\dot{V}_{1\beta2}(t)+\dot{V}_{2\beta2}(t)+\dot{V}_{l\beta2}(t),\ t\in[t_{2k},\ \varrho_{2k}),\\ &\dot{V}_0(t)+\dot{V}_{1\alpha2}(t)+\dot{V}_{2\alpha2}(t)+\dot{V}_{l\alpha2}(t),\ t\in[\varrho_{2k},\ t_{2k+1}), \end{aligned}\right.\ \end{equation}$

(3.7)

where

$\begin{align*} {\dot{V}_{0}(t)} = &2r_1\varpi_0^T(t)F\dot{\varpi}_0(t)+\varpi_1^T(t){Q}_1\varpi_1(t)+\varpi_1^T(t-r_1)(Q_2 -Q_1)\varpi_1(t-r_1)\\ &+\varpi_1^T(t-r_{\varrho})(Q_3-Q_2)\varpi_1(t-r_{\varrho}) -\varpi_1^T(t-r_2)Q_3\varpi_1(t-r_2)+r_1^2\dot{x}^T(t)Y_1\dot{x}(t)\\ &+(r_{\varrho}-r_1)\dot{x}^T(t)Y_2\dot{x}(t)+(r_2-r_{\varrho})\dot{x}^T(t)Y_3\dot{x}(t) +\digamma_{c1}+\digamma_{c2}+\digamma_{c3},&\\ {\dot{V}_{1\alpha i}(t)} = &2\varpi_0^T(t)[(r(t)-r_1)X_{\alpha i1}+(r_{\alpha}-r(t))X_{\alpha i2}+(r_2-r_{\varrho})X_{\alpha i3}]\dot{\varpi}_0(t)\\ &+\dot{r}(t)\varpi_0^T(t)(X_{\alpha i1}-X_{\alpha i2})\varpi_0(t)+2\varpi_{2\alpha}^T(t)P\dot{\varpi}_{2\alpha}(t) +\varpi_1^T(t-r_1)S_{\alpha i1}\varpi_1(t-r_1)\\ &+(1-\dot{r}(t))\varpi_1^T(t-r(t))(S_{\alpha i2}-S_{\alpha i1})\varpi_1(t-r(t)) +\varpi_1^T(t-r_{\varrho})(S_{\alpha i3}-S_{\alpha i2})\varpi_1(t-r_{\varrho})\\ &-\varpi_1^T(t-r_2)S_{\alpha i3}\varpi_1(t-r_2),\\ {\dot{V}_{1\beta i}(t)} = &2\varpi_0^T(t)[(r_{\varrho}-r_1)X_{\beta i1}+(r(t)-r_{\alpha})X_{\beta i2}+(r_2-r(t))X_{\beta i3}]\dot{\varpi}_0(t)\\ &+\dot{r}(t)\varpi_0^T(t)(X_{\beta i2}-X_{\beta i3})\varpi_0(t)+2\varpi_{2\beta}^T(t)P\dot{\varpi}_{2\beta}(t) +\varpi_1^T(t-r_1)S_{\beta i1}\varpi_1(t-r_1)\\ &+(1-\dot{r}(t))\varpi_1^T(t-r(t))(S_{\beta i3}-S_{\beta i2})\varpi_1(t-r(t)) +\varpi_1^T(t-r_{\varrho})(S_{\beta i2}-S_{\beta i1})\varpi_1(t-r_{\varrho})\\ &-\varpi_1^T(t-r_2)S_{\beta i3}\varpi_1(t-r_2),\\ {\dot{V}_{2\alpha i}(t)} = &(r_2-r_{\varrho})\dot{x}^T(t-r_{\varrho})(T_{\alpha i3}-T_{\alpha i2})\dot{x}(t-r_{\varrho})\\ &+(r_2-r(t))(1-\dot{r}(t))\dot{x}^T(t-r(t))^T(T_{\alpha i2}-T_{\alpha i1})\dot{x}(t-r(t))\\ &+(r_2-r_1)\dot{x}^T(t-r_1)T_{\alpha i1}\dot{x}(t-r_1)+\digamma_{T\alpha i1}+\digamma_{T\alpha i2}+\digamma_{T\alpha i3},\\ {\dot{V}_{2\beta i}(t)} = &(r_2-r(t))(1-\dot{r}(t))\dot{x}^T(t-r(t))(T_{\beta i3}-T_{\beta i2})\dot{x}(t-r(t))\\ &+(r_2-r_{\varrho})\dot{x}^T(t-r_{\varrho})(T_{\beta i2}-T_{\beta i1})\dot{x}(t-r_{\varrho}) +(r_2-r_1)\dot{x}^T(t-r_1)T_{\beta i1}\dot{x}(t-r_1)&\\ &+\digamma_{T\beta i1}+\digamma_{T\beta i2}+\digamma_{T\beta i3},&\\ {\dot{V}_{l\alpha i}(t)} = &(r(t)-r_1)[(1-\dot{r}(t))\dot{x}^T(t-r(t))W_{\alpha i2}\dot{x}(t-r(t)) -\dot{x}^T(t-r_{\varrho})W_{\alpha i2}\dot{x}(t-r_{\varrho})]\\ &+(r(t)-r_{\varrho})[\dot{x}^T(t-r_1)W_{\alpha i1} \dot{x}(t-r_1)-(1-\dot{r}(t))\dot{x}^T(t-r(t))W_{\alpha i1}\dot{x}(t-r(t))]\\ &+\digamma_{W\alpha i1}+\digamma_{W\alpha i2},&\\ {\dot{V}_{l\beta i}(t)} = &(r(t)-r_{\varrho})[(1-\dot{r}(t))\dot{x}^T(t-r(t))W_{\beta i2}\dot{x}(t-r(t)) -\dot{x}^T(t-r_2)W_{\beta i2}\dot{x}(t-r_2)]\\ &+(r(t)-r_2)[\dot{x}^T(t-r_{\varrho})W_{\beta i1} \dot{x}(t-r_{\varrho})-(1-\dot{r}(t))\dot{x}^T(t-r(t))W_{\beta i1}\dot{x}(t-r(t))]\\ &+\digamma_{W\beta i1}+\digamma_{W\beta i2},& \end{align*}$

with

$\begin{align*} \digamma_{c1}& = -r_1\int^{t}_{t-r_1}\dot{x}^T(s)Y_1\dot{x}(s)ds,\ \ \digamma_{c2} = -\int_{t-r_{\varrho}}^{t-r_1}\dot{x}^T(s)Y_2\dot{x}(s)ds,\ \ \digamma_{c3} = -\int^{t-r_{\varrho}}_{t-r_2}\dot{x}^T(s)Y_3\dot{x}(s)ds,\\ \digamma_{T\alpha i1}& = -\int^{t-r_1}_{t-r(t)}\dot{x}^T(s)T_{\alpha i1}\dot{x}(s)ds,\ \ \ \ \digamma_{T\alpha i2} = -\int_{t-r_{\varrho}}^{t-r(t)}\dot{x}^T(s)T_{\alpha i2}\dot{x}(s)ds,\\ \digamma_{T\alpha i3}& = -\int_{t-r_2}^{t-r_{\varrho}}\dot{x}^T(s)T_{\alpha i3}\dot{x}(s)ds,\ \ \ \ \digamma_{T\beta i1} = -\int^{t-r_1}_{t-r_{\varrho}}\dot{x}^T(s)T_{\beta i1}\dot{x}(s)ds,\\ \digamma_{T\beta i2}& = -\int^{t-r_{\varrho}}_{t-r(t)}\dot{x}^T(s)T_{\beta i2}\dot{x}(s)ds,\ \ \ \ \digamma_{T\beta i3} = -\int_{t-r_2}^{t-r(t)}\dot{x}^T(s)T_{\beta i3}\dot{x}(s)ds,\\ \digamma_{W\alpha i1}& = \dot{r}(t)\int^{t-r_1}_{t-r(t)}\dot{x}^T(s)W_{\alpha i1}\dot{x}(s)ds,\ \ \ \ \digamma_{W\alpha i2} = \dot{r}(t)\int_{t-r_{\varrho}}^{t-r(t)}\dot{x}^T(s)W_{\alpha i2}\dot{x}(s)ds,\\ \digamma_{W\beta i1}& = \dot{r}(t)\int^{t-r_{\varrho}}_{t-r(t)}\dot{x}^T(s)W_{\beta i1}\dot{x}(s)ds,\ \ \ \ \digamma_{W\beta i2} = \dot{r}(t)\int_{t-r_2}^{t-r(t)}\dot{x}^T(s)W_{\beta i2}\dot{x}(s)ds. \end{align*}$

Applying (2.5) in Lemma 2.1 to estimate $\digamma_{c1}$ , we obtain

$\begin{equation} \begin{aligned} \digamma_{c1}\leqslant-&\varpi^T(t)E_{24}^TY_{1}E_{24}\varpi(t). \end{aligned} \end{equation}$

(3.8)

Then, merging the integrals that have the same integral intervals and applying (2.6) in Lemma 2.1 to estimate the various combinations of $\sum_{\iota = 2}^3\digamma_{c\iota}+\sum_{\iota = 1}^3\digamma_{T\alpha i\iota}+\sum_{\iota = 1}^2\digamma_{W\alpha i\iota}$ .

For $\digamma_{c2}+\digamma_{T\alpha i1}+\digamma_{T\alpha i2}+\digamma_{W\alpha i1}+\digamma_{W\alpha i2}$ and $\digamma_{c3}+\digamma_{T\alpha i3}$ , suppose that $Y_2+T_{\alpha i1}+\dot{r}(t)W_{\alpha i1} > 0$ and $Y_2+T_{\alpha i2}+\dot{r}(t)W_{\alpha i2} > 0$ at the vertices of $\gimel_{\alpha i}$ , we obtain

$\begin{align} \digamma_{c3}+\digamma_{T\alpha i3}& = -\int^{t-r_{\varrho}}_{t-r_2}\dot{x}^T(s)(Y_3+T_{\alpha i3})\dot{x}(s)ds\\ &\leqslant\varpi^T(t)[(r_2-r_{\varrho})G_{\alpha i3}(Y_3+T_{\alpha i3})^{-1}G_{\alpha i3}^T +{\text{Sym}}\{G_{\alpha i3}E_{27}\}]\varpi(t), \\ \end{align}$

(3.9)

$\begin{align} &\digamma_{c2}+\digamma_{T\alpha i1}+\digamma_{T\alpha i2}+\digamma_{W\alpha i1}+\digamma_{W\alpha i2}\\ = &-\int^{t-r_1}_{t-r(t)}\dot{x}^T(s)(Y_2+T_{\alpha i1}+\dot{r}(t)W_{\alpha i1})\dot{x}(s)ds-\int_{t-r_{\varrho}}^{t-r(t)}\dot{x}^T(s)(Y_2+T_{\alpha i2}+\dot{r}(t)W_{\alpha i2})\dot{x}(s)ds\\ \leqslant&\varpi^T(t)[(r(t)-r_1)G_{\alpha i1}(Y_2+T_{\alpha i1}+\dot{r}(t)W_{\alpha i1})^{-1}G_{\alpha i1}^T\\ &+(r_{\varrho}-r(t))G_{\alpha i2}(Y_2+T_{\alpha i2}+\dot{r}(t)W_{\alpha i2})^{-1}G_{\alpha i2}^T +{\text{Sym}}\{G_{\alpha i1}E_{25}+G_{\alpha i2}E_{26}\}]\varpi(t). \end{align}$

(3.10)

Similarly, suppose that $Y_3+T_{\beta i2}+\dot{r}(t)W_{\beta i1} > 0$ and $Y_3+T_{\beta i3}+\dot{r}(t)W_{\beta i2} > 0$ at the vertices of $\gimel_{\beta i}$ . Then, $\digamma_{c3}+\digamma_{T\beta i2}+\digamma_{T\beta i3}+\digamma_{W\beta i1}+\digamma_{W\beta i2}$ and $\digamma_{c2}+\digamma_{T\beta i1}$ satisfy the following inequalities:

$\begin{align} \digamma_{c2}+\digamma_{T\beta i1}& = -\int_{t-r_{\varrho}}^{t-r_1}\dot{x}^T(s)(Y_2+T_{\beta i1})\dot{x}(s)ds\\ &\leqslant\varpi^T(t)[(r_{\varrho}-r_1)G_{\beta i1}(Y_2+T_{\beta i1})^{-1}G_{\beta i1}^T+{\text{Sym}}\{G_{\beta i1}E_{28}\}]\varpi(t), \end{align}$

(3.11)

$\begin{align} &\digamma_{c3}+\digamma_{T\beta i2}+\digamma_{T\beta i3}+\digamma_{W\beta i1}+\digamma_{W\beta i2}\\ = &-\int^{t-r_{\varrho}}_{t-r(t)}\dot{x}^T(s)(Y_3+T_{\beta i2}+\dot{r}(t)W_{\beta i1})\dot{x}(s)ds -\int_{t-r_2}^{t-r(t)}\dot{x}^T(s)(Y_3+T_{\beta i3}+\dot{r}(t)W_{\beta i2})\dot{x}(s)ds\\ \leqslant&\varpi^T(t)[(r(t)-r_{\varrho})G_{\beta i2}(Y_3+T_{\beta i2}+\dot{r}(t)W_{\beta i1})^{-1}G_{\beta i2}^T\\ &+(r_2-r(t))G_{\beta i3}(Y_3+T_{\beta i3}+\dot{r}(t)W_{\beta i2})^{-1}G_{\beta i3}^T +{\text{Sym}}\{G_{\beta i2}E_{29}+G_{\beta i3}E_{30}\}]\varpi(t). \end{align}$

(3.12)

According to the relationship between the internal elements of the vector, the subsequent equations are valid.

$\begin{equation} \begin{aligned} 0& = 2\varpi(t)^TN_{\alpha i1}[\delta_4(t)+\delta_5(t)-\delta_{7}(t)],\\ 0& = 2\varpi(t)^TN_{\alpha i2}[\delta_6(t)-\delta_5(t)-\delta_{8}(t)],\\ 0& = 2\varpi(t)^TN_{\beta i1}[\delta_4(t)+\delta_5(t)-\delta_{7}(t)],\\ 0& = 2\varpi(t)^TN_{\beta i2}[\delta_6(t)-\delta_5(t)-\delta_{8}(t)].\\ \end{aligned} \end{equation}$

(3.13)

Combining (3.7)–(3.10) and (3.13), for $t\in[t_{2k-1}, \varrho_{1k})\cup[\varrho_{2k}, t_{2k+1})$ , we derive

$\begin{equation} \begin{aligned} \dot{V}(t)\leqslant&\varpi^T(t)[\Upsilon_{\alpha i}(r(t),\dot{r}(t))+(r(t)-r_1)G_{\alpha i1}(Y_2+T_{\alpha i1} +{\dot{r}(t)W_{\alpha i1})}^{-1}{\alpha i1}^T\\ &+(r_{\varrho}-r(t))G_{\alpha i2}(Y_2+T_{\alpha i2}+{\dot{r}(t)W_{\alpha i2}})^{-1} G_{\alpha i2}^T\\ &+(r_2-r_{\varrho})G_{\alpha i3}(Y_3+T_{\alpha i3})^{-1}G_{\alpha i3}^T]\varpi(t)\\ = &\varpi^T(t)\Xi_{\alpha i}(r(t),\dot{r}(t))\varpi(t), \end{aligned} \end{equation}$

(3.14)

where $\Upsilon_{\alpha i}(r(t), \dot{r}(t))$ is defined in Theorem 3.1. From the Schur complement lemma, $\Xi_{\alpha i}(r(t), \dot{r}(t)) < 0$ is equivalent to LMIs (3.3) at the vertices of $\gimel_{\alpha i}$ . Therefore, there exist scalars $\gamma_{\alpha i}$ satisfying $\dot{V}(t) < -\gamma_{\alpha i}\lvert x(t)\rvert^2$ for $t\in[t_{2k-1}, \varrho_{1k})\cup[\varrho_{2k}, t_{2k+1})$ if (3.3) is satisfied.

Similarly, for $t\in[\varrho_{1k}, \varrho_{2k})$ , combining (3.7), (3.8), (3.11)–(3.13), we derive

$\begin{equation} \begin{aligned} \dot{V}(t)\leqslant&\varpi^T(t)[\Upsilon_{\beta i}(r(t),\dot{r}(t))+(r_{\varrho}-r_1)G_{\beta i1}(Y_2+T_{\beta i1})^{-1} G_{b\beta i1}^T\\ &+(r(t)-r_{\varrho})G_{\beta i2}(Y_3+T_{\beta i2}+{\dot{r}(t)W_{\beta i1}})^{-1}G_{\beta i2}^T\\ &+(r_2-r(t))G_{\beta i3}(Y_3+T_{\beta i3}+{\dot{r}(t)W_{\beta i2}})^{-1}G_{\beta i3}^T]\varpi(t)\\ = &\varpi^T(t)\Xi_{\beta i}(r(t),\dot{r}(t))\varpi(t), \end{aligned} \end{equation}$

(3.15)

where $\varpi^T(t)\Xi_{\beta i}(r(t), \dot{r}(t))\varpi(t) < 0$ corresponds to LMIs (3.5) at the vertices of $\gimel_{\beta i}$ and there exist scalars $\gamma_{\beta i}$ satisfying $\dot{V}(t) < -\gamma_{\beta i}\lvert x(t)\rvert^2$ for $t\in[\varrho_{1k}, \varrho_{2k})$ if (3.5) are satisfied.

To ensure the overall decrement of the functional, some boundary conditions shown below need to be satisfied at each segmented point:

$\begin{equation} \left\{\begin{aligned} \lim\limits_{t \to \varrho_{1k}^{\ -}}[V_{1\alpha1}(t)+V_{2\alpha1}(t)]&\geqslant [V_{1\beta1}(\varrho_{1k})+V_{2\beta1}(\varrho_{1k})],\\ \lim\limits_{t \to t_{2k}^{\ -}}[V_{1\beta1}(t)+V_{2\beta1}(t)]&\geqslant [V_{1\beta2}(t_{2k})+V_{2\beta2}(t_{2k})], \\ \lim\limits_{t \to \varrho_{2k}^{\ -}}[V_{1\beta2}(t)+V_{2\beta2}(t)]&\geqslant [V_{1\alpha2}(\varrho_{2k})+V_{2\alpha2}(\varrho_{2k})],\\ \lim\limits_{t \to t_{2k+1}^{\ -}}[V_{1\alpha2}(t)+V_{2\alpha2}(t)]&\geqslant [V_{1\alpha1}(t_{2k-1})+V_{2\alpha1}(t_{2k-1})]. \end{aligned}\right. \end{equation}$

(3.16)

From which we obtain the restriction in (3.1). Thus, the proof is completed. □

Remark 3.1. An ADS based on the delay segmenting method is proposed in ^[38]. However, the L–K functional established therein does not match the ADS appropriately, i.e., the delay-segmentation-related information is not directly reflected in the established L–K functional. Undoubtedly, this makes the stability criterion of the system have a relatively high conservativeness in an intuitive way. Therefore, an improved L–K functional is established in this paper, which enables the direct manifestation of the segmented delay intervals. Compared with the functional established in ^[38], this functional increases the weight of delay derivatives in the criterion, thereby enhancing the correlation between the system stability conditions and the delay-related information. Moreover, as shown in $V_{1\alpha i}$ and $V_{1\beta i}$ , the number of relevant Lyapunov matrices it encompasses has increased by $50\%$ , while the number of constraint conditions between Lyapunov matrices has only increased by 33.33%. As shown in constraint conditions (3.1) of Theorem 3.1, $X_{\alpha 12}\geqslant X_{\beta12}, \ X_{\beta 22}\geqslant X_{\alpha22}$ , $S_{\alpha 12}\geqslant S_{\beta12}$ , and $S_{\beta 22}\geqslant S_{\alpha22}$ are not required. Consequently, the function presented in this paper will further reduce the conservativeness of the results.

Remark 3.2. In ^[38], only two delay-product terms were incorporated in the established functional, aside from loop-like functions. This suggests that the delay-dependent stability conditions across different intervals will rely on a fixed set of Lyapunov matrices. Consequently, the stability criterion derived from ^[38] tends to produce overly conservative results. In contrast, the functional introduced in this paper incorporates $V_{2\alpha i}$ and $V_{2\beta i}$ based on the specific intervals associated with the delay. This approach allows for the Lyapunov matrices used in the directly delay-dependent stability conditions across different intervals to differ from one another. As a result, the proposed function grants the stability criteria of the system greater flexibility and significantly diminishes its conservativeness.

Remark 3.3. A loop function based on periodic time delay was originally proposed in ^[37]. This function exhibits the property of being identically zero at the vertices of the domain of the delay function. Consequently, when integrated with the L–K functional, it inherently satisfies the requirements of Lyapunov functions. Moreover, within distinct intervals of the functional, the Lyapunov matrices associated with this function are not required to be identical. However, the ADS partitioning approach introduced in ^[38] leads to the derivation of stability conditions of more meticulous segmentation from the proposed functional. As a result, constraints are imposed on the Lyapunov matrices within the loop functions. This imposition undermines the definition of the loop functions and increases the conservativeness of the system stability criteria. In contrast, in the functional presented herein, a set of loop functions corresponding to the delay-belonging intervals within each segment of the functional is established. These loop functions are valid on every individual segment. For $t\in[t_{2k-1}, \varrho_{1k})$ , given that $V(t_{2k-1})\geqslant0$ , $V(\varrho_{1k})\geqslant0$ , and $\dot{V}(t) < -\gamma_{\alpha 1}\lvert x(t)\rvert^2$ , then $V(t_{2k-1})\geqslant V(t)\geqslant V(\varrho_{1k})$ holds true. For other intervals, similar conditions also hold. Therefore, there are no restrictive relationships among their respective Lyapunov matrices. Evidently, this configuration further diminishes the conservativeness of the derived stability conditions.

Remark 3.4. As evident from LMIs (3.3) and (3.5), the stability criterion derived from Theorem 3.1 encompasses a substantial number of variables (NoVs). To minimize computational complexity to the greatest extent possible, conserve computational resources, and investigate the correlation between the increment in computational complexity and the results enhancement, we reduce the number of free matrices in Eq (3.13). Subsequently, we substitute Eq (3.13) with the following equations and apply them to Theorem 3.1,

$\begin{equation} \begin{aligned} 0& = 2\varpi(t)^TN_{\alpha}[\delta_4(t)+\delta_5(t)-\delta_{7}(t)],\\ 0& = 2\varpi(t)^TN_{\beta }[\delta_6(t)-\delta_5(t)-\delta_{8}(t)]. \end{aligned} \end{equation}$

(3.17)

The results obtained from these modifications will be compared and analyzed in the section dedicated to numerical examples.

Then, the following corollary is derived based on a continuous L–K functional simplified by (3.6).

Corollary 3.1. For scalars $\varrho\in[0, 1], r_2 \geqslant r_1 \geqslant 0, \mu < 1$ , if there exist matrices $P\in\mathbb{S}^{9n}_{+}$ , $F, \bar{X}_{\varsigma}\in\mathbb{S}^{6n}_{+}$ , $Q_\varsigma, \bar{S}_{\varsigma}\in\mathbb{S}^{2n}_{+}$ , $Y_\varsigma, \bar{T}_{\varsigma}\in\mathbb{S}^{n}_{+}$ , $W_{\alpha i1}, W_{\alpha i2}, W_{\beta i1}, W_{\beta i2}\in\mathbb{S}^{n}$ , $G_{\alpha i\varsigma}, G_{\beta i\varsigma}\in\mathbb{R}^{15n\times 2n}$ , $N_{\alpha i1}, N_{\alpha i2}, N_{\beta i1}, N_{\beta i2}\in\mathbb{R}^{15n\times n}, i = 1, 2, \varsigma = 1, \cdots, 3$ , such that LMIs (3.18)–(3.19) and (3.20)–(3.21) are satisfied at the vertices of $\gimel_{\alpha i}$ and $\gimel_{\beta i}$ , respectively, then system (2.1) is stable,

$\begin{equation} W_{Y\bar{T}\alpha i1} > 0,\ W_{Y\bar{T}\alpha i2} > 0, \end{equation}$

(3.18)

$\begin{equation} \left[\begin{matrix}\bar{\Upsilon}_{\alpha i}(r(t),\dot{r}(t))&\sqrt{r(t)-r_1}G_{\alpha i1}&\sqrt{r_{\varrho}-r(t)}G_{\alpha i2}&\sqrt{r_2-r_1}G_{\alpha i3}\\\ast&-\widehat{W}_{Y\bar{T}\alpha i1}&0&0\\ \ast&\ast&-\widehat{W}_{Y\bar{T}\alpha i2}&0\\ \ast&\ast&\ast&-\widehat{Y}_{\bar{T}1} \end{matrix}\right] < 0, \end{equation}$

(3.19)

$\begin{equation} W_{Y\bar{T}\beta i1} > 0,\ W_{Y\bar{T}\beta i2} > 0, \end{equation}$

(3.20)

$\begin{equation} \left[\begin{matrix}\bar{\Upsilon}_{\beta i}(r(t),\dot{r}(t))&\sqrt{r_{\varrho}-r_1}G_{\beta i1}&\sqrt{r(t)-r_{\varrho}}G_{\beta i2}&\sqrt{r_2-r(t)}G_{\beta i3}\\\ast&-\widehat{Y}_{\bar{T}2}&0&0\\ \ast&\ast&-\widehat{W}_{Y\bar{T}\beta i1}&0\\ \ast&\ast&\ast&-\widehat{W}_{Y\bar{T}\beta i2} \end{matrix}\right] < 0, \end{equation}$

(3.21)

where

$\begin{align*} \bar{\Upsilon}_{\alpha i}(r(&t),\dot{r}(t)) = \Lambda_1+\bar{\Lambda}_{2\alpha}+\Lambda_{3\alpha i}+\Lambda_{4\alpha i},&\\ \bar{\Upsilon}_{\beta i}(r(&t),\dot{r}(t)) = \Lambda_1+\bar{\Lambda}_{2\beta}+\Lambda_{3\beta i}+\Lambda_{4\beta i},&\\ \Lambda_1 = &{\mathit{\text{Sym}}}\{h_1\Pi_0^TF\Pi_1\}+\Pi_2^T{Q}_1\Pi_{2}+\Pi_3^T(Q_2-Q_1)\Pi_{3}+\Pi_4^T(Q_3-Q_2)\Pi_{4}\\ &-\Pi_5^TQ_3\Pi_5+r_1^2\Pi_{20}^TY_1\Pi_{20}+(r_{\varrho}-r_1)\Pi_{20}^TY_2\Pi_{20}+(r_2-r_{\varrho})\Pi_{20}^TY_3\Pi_{20},&\\ \bar{\Lambda}_{2\alpha} = &{\mathit{\text{Sym}}}\{\Pi_0^T[(r(t)-r_1)\bar{X}_{1}+(r_{\alpha}-r(t))\bar{X}_{2}+(r_2-r_{\varrho})\bar{X}_{3}]\Pi_1+\Pi_7^TP\Pi_8\}\\ &+\dot{r}(t)\Pi_0^T(\bar{X}_{1}-\bar{X}_{2})\Pi_0+\Pi_3^T\bar{S}_{1}\Pi_3+(1-\dot{r}(t))\Pi_6^T(\bar{S}_{2}-\bar{S}_{1})\Pi_6\\ &+\Pi_4^T(\bar{S}_{3}-\bar{S}_{2})\Pi_4-\Pi_5^T\bar{S}_{3}\Pi_5+(r_2-r_{\varrho})e_9^T(\bar{T}_{3}-\bar{T}_{2})e_9\\ &+(r_2-r(t))(1-\dot{r}(t))e_7^T(\bar{T}_{2}-\bar{T}_{1})e_7+(r_2-r_1)e_6^T\bar{T}_{1}e_6,\\ \bar{\Lambda}_{2\beta} = &{\mathit{\text{Sym}}}\{\Pi_0^T[(r_{\varrho}-r_1)\bar{X}_{1}+(r(t)-r_{\alpha})\bar{X}_{2}+(r_2-r(t))\bar{X}_{3}]\Pi_1+\Pi_{9}^TP\Pi_{10}\}\\ &+\dot{r}(t)\Pi_0^T(\bar{X}_{2}-\bar{X}_{3})\Pi_0+\Pi_3^T\bar{S}_{1}\Pi_3+(1-\dot{r}(t))\Pi_6^T(\bar{S}_{3}-\bar{S}_{2})\Pi_6\\ &+\Pi_4^T(\bar{S}_{2}-\bar{S}_{1})\Pi_4-\Pi_5^T\bar{S}_{3}\Pi_5+(r_2-r(t))(1-\dot{r}(t))e_7^T(\bar{T}_{3}-\bar{T}_{2})e_7\\ &+(r_2-r_{\varrho})e_9^T(\bar{T}_{2}-\bar{T}_{1})e_9+(r_2-r_1)e_6^T\bar{T}_{1}e_6,\\ \Lambda_{3\alpha i} = &(r(t)-r_1)((1-\dot{r}(t))e^T_7W_{\alpha i2}e_7-e_9^TW_{\alpha i2}e_9) -(r_{\varrho}-r(t))(e_6^TW_{\alpha i1}e_6-(1-\dot{r}(t))e^T_7W_{\alpha i1}e_7),&\\ \Lambda_{3\beta i} = &(r(t)-r_{\varrho})((1-\dot{r}(t))e^T_7W_{\beta i2}e_7-e_8^TW_{\beta i2}e_8) -(r_2-r(t))(e_9^TW_{\beta i1}e_9-(1-\dot{r}(t))e^T_7W_{\beta i1}e_7),&\\ \Lambda_{4\alpha i} = &{\mathit{\text{Sym}}}\{G_{\alpha i1}\Pi_{12}+G_{\alpha i2}\Pi_{13}+G_{\alpha i3}\Pi_{14}+N_{\alpha i1}\Pi_{18}+N_{\alpha i2}\Pi_{19}\}-\Pi^T_{11}\widehat{Y}_1\Pi_{11},&\\ \Lambda_{4\beta i} = &{\mathit{\text{Sym}}}\{G_{\beta i1}\Pi_{15}+G_{\beta i2}\Pi_{16}+G_{\beta i3}\Pi_{17}+N_{\beta i1}\Pi_{18}+N_{\beta i2}\Pi_{19}\}-\Pi^T_{11}\widehat{Y}_1\Pi_{11},&\\ \widehat{Y}_{\bar{T}1} = &diag\{{Y}_{\bar{T}1},3{Y}_{\bar{T}1}\},\ \ \widehat{Y}_{\bar{T}2} = diag\{{Y}_{\bar{T}2},3{Y}_{\bar{T}2}\},&\\ \widehat{W}_{Y\bar{T}\alpha i1}& = diag\{W_{Y\bar{T}\alpha i1},3W_{Y\bar{T}\alpha i1}\},\ \ \widehat{W}_{Y\bar{T}\alpha i2} = diag\{W_{Y\bar{T}\alpha i2},3W_{Y\bar{T}\alpha i2}\},&\\ \widehat{W}_{Y\bar{T}\beta i1}& = diag\{W_{Y\bar{T}\beta i1},3W_{Y\bar{T}\beta i1}\},\ \ \widehat{W}_{Y\bar{T}\beta i2} = diag\{W_{Y\bar{T}\beta i2},3W_{Y\bar{T}\beta i2}\},& \end{align*}$

with

$\begin{align*} {Y}_{\bar{T}1} = &Y_3+\bar{T}_{3},\ \ {Y}_{\bar{T}2} = Y_2+\bar{T}_{1},\\ {W}_{Y\bar{T}\alpha i1}& = Y_2+\bar{T}_{1}-\dot{r}(t)W_{\alpha i1},\ \ {W}_{Y\bar{T}\alpha i2} = Y_2+\bar{T}_{2}-\dot{r}(t)W_{\alpha i2},&\\ {W}_{Y\bar{T}\beta i1}& = Y_3+\bar{T}_{2}-\dot{r}(t)W_{\beta i1},\ \ {W}_{Y\bar{T}\beta i2} = Y_3+\bar{T}_{3}-\dot{r}(t)W_{\beta i2},& \end{align*}$

and $\Pi_{\lambda}(\lambda = 0, \cdots, 20)$ , $\widehat{Y}_1$ are defined in Theorem 3.1.

Proof. Setting $\bar{X}_{\varsigma} = X_{\alpha i\varsigma} = X_{\beta i\varsigma}$ , $\bar{S}_{\varsigma} = S_{\alpha i\varsigma} = S_{\beta i\varsigma}$ and $\bar{T}_{\varsigma} = T_{\alpha i\varsigma} = T_{\beta i\varsigma}$ in L–K functional (3.6), we obtain a continuous piecewise L–K functional satisfying the following equations:

$\begin{equation} \left\{\begin{aligned} \lim\limits_{t \to \varrho_{1k}^{\ -}}V_{l\alpha1}(t)& = V_{l\beta1}(\varrho_{1k}),\\ \lim\limits_{t \to t_{2k}^{\ -}}V_{l\beta1}(t)& = V_{l\beta2}(t_{2k}), \\ \lim\limits_{t \to \varrho_{2k}^{\ -}}V_{l\beta2}(t)& = V_{l\alpha2}(\varrho_{2k}),\\ \lim\limits_{t \to t_{2k+1}^{\ -}}V_{l\alpha2}(t)& = V_{l\alpha1}(t_{2k-1}). \end{aligned}\right . \end{equation}$

(3.22)

The proof employs a procedure analogous to that in Theorem 3.1, and the detail will not be repeated here. □

Remark 3.5. To show the advantage of the non-continuous functional (3.6), Corollary 3.1 provides a stability condition that is derived by using the continuous functional (3.22). It will be shown in the section of numerical examples that the non-continuous functional (3.6) plays an important role in the reduction of conservativeness.

4. Numerical examples

In this section, we will verify the validity and superiority of the new results obtained in the previous section through two numerical examples and a single-area load frequency control system provided.

Example 4.1 Consider the system (2.1) with

$\begin{align*} A& = \left[\begin{array}{cc}{-2.0}&{0.0}\\{0.0}&{-0.9}\end{array}\right],\ \ \ A_{r} = \left[\begin{array}{cc}{-1.0}&{0.0}\\{-1.0}&{-1.0}\end{array}\right]. \end{align*}$

To highlight the effectiveness of our novel function for improving the results, we choose ^[37,38] to compare with our results in this example. Both studies adopt a similar bounding method to ours, utilizing the first-order Bessel-Legendre inequality along with the free-matrix-based inequality, and we adopt the optimal solutions for the results of ^[38] under the $\beta$ changes. From , for various $\mu$ with $r_1 = 0$ , we obtain larger allowable upper bounds (AUBs) of $r_2$ under the appropriate $\varrho$ (the value of $\varrho$ is accurate to the percentile) by applying Theorem 3.1 and corresponding NoVs.

Table 2. The AUBs of

$r_2$ for various

$\mu$ with

$r_1 = 0$ .

Methods	$\mu=0.1$	$\mu=0.2$	$\mu=0.5$	$\mu=0.8$	NoVs
^[37] Theorem 1	5.10	4.57	3.78	3.38	$131.5n^2+9.5n$
^[38] Corollary 7	4.82	4.13	3.13	2.70	$76.5n^2+11.5n$
^[38] Theorem 3	5.44	5.00	4.18	3.66	$259.5n^2+32.5n$
Corollary 3.1	4.95 $(\varrho=0.21)$	4.30( $\varrho=0.05$ )	3.37( $\varrho=0.95$ )	2.98( $\varrho=0.89$ )	$317.5n^2+25.5n$
Theorem 3.1	5.61( $\varrho=0.55$ )	5.29( $\varrho=0.56$ )	4.40( $\varrho=0.40$ )	3.86( $\varrho=0.18$ )	$556n^2+70n$

| Show Table

DownLoad: CSV

Further, when $\mu = 0.2$ , the result of Theorem 3.1 exhibits a $5.8\%$ improvement compared to the result of Theorem 3 in ^[38]. Additionally, when $\mu = 0.5$ , the result of Theorem 3.1 shows a $16.4\%$ enhancement over the result reported in ^[37]. On the other hand, we note that the result of Theorem 3.1 yields a larger NoVs than those presented in the listed literature. Indeed, it is inescapable that the computational complexity escalates as the conservativeness of the results diminishes. To further emphasize the reduction in the conservativeness of the results yielded by our proposed method, we will conduct verifications using other examples in the subsequent contents.

In addition, compared with the results of Corollary 7 in ^[38] obtained by a continuous functional, Corollary 1 in this paper also gives higher AUBs in . When $\mu = 0.8$ , Corollary 3.1 presents a $10.3\%$ improvement compared to the result of Corollary 7 in ^[38].

Next, we assign the initial value of the system state as $x_0(t) = [1\ \ -1]^T$ and choose $r(t) = 5.29/2+(5.29/2)cos(0.4t/5.29)$ , which satisfies $r(t)\in[0, 5.29]$ and $\dot{r}(t)\in[-0.2, 0.2]$ . As shown in Figure 1, the state response depiction of the system illustrated in Example 4.1 demonstrates that each state component converges towards zero. This outcome further reinforces the validity of the derived conclusions.

Figure 1. State responses of the single-area load frequency control system.

DownLoad: Full-Size Img PowerPoint

Example 4.2. Consider the system (2.1) with

$\begin{align*} A& = \left[\begin{array}{cc}{0.0}&{1.0}\\{-1.0}&{-2.0}\end{array}\right],\ \ \ A_{r} = \left[\begin{array}{cc}{0.0}&{0.0}\\{-1.0}&{1.0}\end{array}\right]. \end{align*}$

In this example, we further present the simulation results of Theorem 3.1 and Remark 3.4 for various $\mu$ and $r_1$ in and compare them with ^[38,43]. Generally speaking, it is readily apparent that the results derived from Theorem 3.1 are notably superior to those presented in the existing literature, showcasing a substantial improvement. For $r_1 = 0$ , the result of Theorem 3.1 is $92\%$ better than that of Theorem 3 in ^[38] for the case of slow delay $(\mu = 0.1)$ , although the NoVs is nearly doubled. In the case of rapid delay $(\mu = 0.8)$ , the result of Theorem 3.1 also outperforms that of Theorem 3 in ^[38] by $27\%$ . When $r_1 = 1$ , it is observable that although the improvement effect starts to diminish as the lower bound of the delay increases, the improvement can still reach $20\%$ when $\mu = 0.5$ . Meanwhile, compared with Theorem 4 in ^[43], Theorem 3.1 has comparable NoVs, yet its results are evidently less conservative.

Table 3. The AUBs of

$r_2$ for various

$\mu$ and

$r_1$ .

	Methods	$\mu=0.1$	$\mu=0.2$	$\mu=0.5$	$\mu=0.8$	NoVs
$r_1=0$	^[43] Theorem 4	9.94	–	4.46	3.33	$521n^2+27n$
	^[38] Theorem 3	13.67( $\varrho=0.49)$	9.18( $\varrho=0.49)$	5.02( $\varrho=0.53)$	3.18 ( $\varrho=0.51)$	$259.5n^2+32.5n$
	Theorem 3.1	26.49( $\varrho=0.01$ )	13.16( $\varrho=0.04)$	5.93( $\varrho=0.27$ )	4.05( $\varrho=0.21$ )	$556n^2+70n$
	Remark 3.4	26.33( $\varrho=0.01$ )	13.14( $\varrho=0.03$ )	5.93( $\varrho=0.25$ )	4.05( $\varrho=0.21$ )	$466n^2+70n$
$r_1=1$	^[38] Theorem 3	13.48( $\varrho=0.43$ )	8.74( $\varrho=0.45$ )	4.18( $\varrho=0.57$ )	2.37( $\varrho=0.55)$	$259.5n^2+32.5n$
	Theorem 3.1	15.55( $\varrho=0.05$ )	9.05( $\varrho=0.13$ )	5.03( $\varrho=0.87$ )	2.81( $\varrho=0.85$ )	$556n^2+70n$
	Remark 3.4	15.55( $\varrho=0.05$ )	9.04( $\varrho=0.13$ )	5.03( $\varrho=0.88$ )	2.81( $\varrho=0.84$ )	$466n^2+70n$

| Show Table

DownLoad: CSV

Furthermore, the results presented in Remark 3.4 are tabulated in Table 3. Evidently, Remark 3.4 involves fewer NoVs compared to Theorem 3.1. Consequently, it exhibits a relatively lower computational complexity. Nevertheless, the results derived from Remark 3.4 are comparable to those obtained from Theorem 3.1.

Subsequently, we set the initial state as $x_0(t) = [1\ \ 0]^T$ and choose the delay function as $r(t) = 13.16/2+(13.16/2)cos(0.4t/13.16)$ . This function satisfies the conditions $r(t)\in[0, 13.16]$ and $\dot{r}(t)\in[-0.2, 0.2]$ . The state response is depicted in Figure 2. It is shown in the figure that the system is stable.

Figure 2. State responses of the single-area load frequency control system.

DownLoad: Full-Size Img PowerPoint

Example 4.3. Consider the single-area load frequency control system, which can be expressed as the system (2.1) with

$\begin{align*} A = \begin{bmatrix}-\frac{D}{M}&\frac{1}{M}&0&0\\0&-\frac{1}{T_{t}}&\frac{1}{T_{t}}&0\\-\frac{1}{T_{g}R}&0&-\frac{1}{T_{g}}&0\\\nu&0&0&0\end{bmatrix},\ \ \ \ A_{r} = \begin{bmatrix}0&0&0&0\\0&0&0&0\\-\frac{\nu K_{p}}{T_{g}}&0&0&-\frac{K_{i}}{T_{g}}\\0&0&0&0\end{bmatrix}, \end{align*}$

where $D$ stands for the generator's damping coefficient, $M$ is its moment of inertia, $T_t$ and $T_g$ represent the time constants corresponding to the turbine and governor, respectively, $R$ refers to the speed drop, and $v$ represents the frequency bias factor. $K_p$ and $K_i$ denote the proportional and integral gains of the $PI$ controller, respectively. Let $D = 1.0, M = 10, T_t = 0.3, T_g = 0.1, R = 0.05, v = 21, K_p = 0.8, K_i = \{0.1, 0.2, 0.3\}$ and $r_1 = 0, \mu = 0.1$ . The AUBs of $r_2$ under the appropriate $\varrho$ are shown in Table 4, demonstrating results that are evidently less conservative than those of Theorem 1 in ^[37] as well as Corollary 7 and Theorem 3 in ^[38]. Especially when integral gains $K_i = 0.1$ , the result shows a $30\%$ improvement in the AUBs compared to Theorem 3 in ^[38]. To further confirm the validity of our results, setting $K_i = 0.1$ and the initial state $x_0(t) = [1\ \ 0\ \ 0.5\ \ -1]^T$ , and choosing $r(t) = 12.74/2+(12.74/2)cos(0.2t/12.74)$ , which satisfies $r(t)\in[0, 12.74]$ and $\dot{r}(t)\in[-0.1, 0.1]$ , respectively, the state responses of the single-area load frequency control system are shown in Figure 3. Obviously, they tend to be stable.

Table 4. The AUBs of

$r_2$ for various

$K_i$ .

Methods	$K_i=0.1$	$K_i=0.2$	$K_i=0.3$
^[37] Theorem 1	9.55	5.31	3.74
^[38] Corollary 7	9.04	5.12	3.39
^[38] Theorem 3	9.77	5.61	3.84
Corollary 3.1	11.01( $\varrho=0.07$ )	5.95( $\varrho=0.11$ )	3.85( $\varrho=0.13$ )
Theorem 3.1	12.74( $\varrho=0.17$ )	6.38( $\varrho=0.17$ )	4.05( $\varrho=0.15$ )

| Show Table

DownLoad: CSV

Figure 3. State responses of the single-area load frequency control system.

DownLoad: Full-Size Img PowerPoint

5. Conclusions

In this paper, an improved delay-segmentation-based non-continuous piecewise L–K functional is established. Then, a low-conservativeness stability criterion for linear systems with a periodical time-varying delay is presented based on the functional. Finally, we provide two simulation examples alongside an application to a single-area load frequency control system to demonstrate the effectiveness of the proposed approach.

Author contributions

Wei Wang: Writing-review & editing, formal analysis, validation, conceptualization, funding acquisition; Chang-Xin Li: Writing-original draft, software, methodology, investigation; Ao-Qian Luo: Writing-review & editing; Hui-Qin Xiao: Writing-review & editing, supervision. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare that they have not used Artificial Intelligence tools in the creation of this article.

Acknowledgments

This study is supported by the National Natural Science Foundation of China (No.62173136), the Natural Science Foundation of Hunan Province (Nos.2024JJ7130 and 2020JJ2013), and the Scientific Research and Innovation Foundation of Hunan University of Technology.

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	A. Seuret, F. Gouaisbaut, Allowable delay sets for the stability analysis of linear time-varying delay systems using a delay-dependent reciprocally convex lemma, IFAC-PapersOnLine, 50 (2017), 1275–1280. https://doi.org/10.1016/j.ifacol.2017.08.131 doi: 10.1016/j.ifacol.2017.08.131
[2]	Y. L. Fan, C. K. Zhang, Y. F. Liu, Y. He, Q. G. Wang, Stability analysis of systems with time-varying delays for conservatism and complexity reduction, Syst. Control Lett., 193 (2024), 105948. https://doi.org/10.1016/j.sysconle.2024.105948 doi: 10.1016/j.sysconle.2024.105948
[3]	Y. He, C. K. Zhang, H. B. Zeng, M. Wu, Additional functions of variable-augmented-based free-weighting matrices and application to systems with time-varying delay, Internat. J. Systems Sci., 54 (2023), 991–1003. https://doi.org/10.1080/00207721.2022.2157198 doi: 10.1080/00207721.2022.2157198
[4]	P. Pepe, G. Pola, M. D. Di Benedetto, On Lyapunov–Krasovskii characterizations of stability notions for discrete-time systems with uncertain time-varying time delays, IEEE Trans. Automat. Control, 63 (2018), 1603–1617. https://doi.org/10.1109/TAC.2017.2749526 doi: 10.1109/TAC.2017.2749526
[5]	C. K. Zhang, W. H. Chen, C. Zhu, Y. He, M. Wu, Stability analysis of discrete-time systems with time-varying delay via a delay-dependent matrix-separation-based inequality, Automatica, 156 (2023), 111192. https://doi.org/10.1016/j.automatica.2023.111192 doi: 10.1016/j.automatica.2023.111192
[6]	Y. Mo, W. Yu, H. Hou, S. Dasgupta, Razumikhin-type ISS Lyapunov function and small gain theorem for discrete time time-delay systems with application to a biased min-consensus protocol, Automatica, 174 (2025), 112111. https://doi.org/10.1016/j.automatica.2025.112111 doi: 10.1016/j.automatica.2025.112111
[7]	T. S. Peng, H. B. Zeng, W. Wang, X. M. Zhang, X.-G. Liu, General and less conservative criteria on stability and stabilization of T–S fuzzy systems with time-varying delay, IEEE Trans. Fuzzy Syst., 31 (2023), 1531–1541. https://doi.org/10.1109/TFUZZ.2022.3204899 doi: 10.1109/TFUZZ.2022.3204899
[8]	Y. Li, Y. He, Y. Yang, Q. Liu, Membership-functions-derivative-dependent stability criteria for delayed {T}akagi{-S}ugeno fuzzy systems, IEEE Trans. Fuzzy Syst., 32 (2024), 4685–4698. https://doi.org/10.1109/TFUZZ.2024.3409782 doi: 10.1109/TFUZZ.2024.3409782
[9]	Y. Chen, G. Chen, Hierarchical admissibility criteria for T-S fuzzy singular systems with time-varying delay, Fuzzy Sets Syst., 492 (2024), 109065. https://doi.org/10.1016/j.fss.2024.109065 doi: 10.1016/j.fss.2024.109065
[10]	Z. Sheng, C. Lin, B. Chen, Q. G. Wang, Stability and stabilization of T–S fuzzy time-delay systems under sampled-data control via new asymmetric functional method, IEEE Trans. Fuzzy Syst., 31 (2023), 3197–3209. https://doi.org/10.1109/TFUZZ.2023.3247030 doi: 10.1109/TFUZZ.2023.3247030
[11]	H. B. Zeng, Z. J. Zhu, T. S. Peng, W. Wang, X. M. Zhang, Robust tracking control design for a class of nonlinear networked control systems considering bounded package dropouts and external disturbance, IEEE Trans. Fuzzy Syst., 32 (2024), 3608–3617. https://doi.org/10.1109/TFUZZ.2024.3377799 doi: 10.1109/TFUZZ.2024.3377799
[12]	L. Jin, Y. He, C. K. Zhang, L. Jiang, W. Yao, M. Wu, Delay-dependent stability of load frequency control with adjustable computation accuracy and complexity, Control Eng. Pract., 135 (2023), 105518. https://doi.org/10.1016/j.conengprac.2023.105518 doi: 10.1016/j.conengprac.2023.105518
[13]	C. G. Wei, X. C. Shangguan, Y. He, C. K. Zhang, D. Xu, Delay-dependent stability evaluation for temperature control load participating in load frequency control of microgrid, IEEE Trans. Ind. Electron., 72 (2025), 449–459. https://doi.org/10.1109/TIE.2024.3404126 doi: 10.1109/TIE.2024.3404126
[14]	W. M. Wang, H. B. Zeng, J. M. Liang, S. P. Xiao, Sampled-data-based load frequency control for power systems considering time delays, J. Franklin Inst., 362 (2025), 107477. https://doi.org/10.1016/j.jfranklin.2024.107477 doi: 10.1016/j.jfranklin.2024.107477
[15]	W. Wang, R. K. Xie, L. Ding, Stability analysis of load frequency control systems with electric vehicle considering time-varying delay, IEEE Access, 13 (2025), 3562–3571. https://doi.org/10.1109/ACCESS.2024.3519343 doi: 10.1109/ACCESS.2024.3519343
[16]	G. Zhuang, Q. Ma, B. Zhang, S. Xu, J. Xia, Admissibility and stabilization of stochastic singular {M}arkovian jump systems with time delays, Syst. Control Lett., 114 (2018), 1–10. https://doi.org/10.1016/j.sysconle.2018.02.004 doi: 10.1016/j.sysconle.2018.02.004
[17]	W. Le, Y. Ding, W. Wu, H. Liu, New stability criteria for semi-{M}arkov jump linear systems with time-varying delays, AIMS Mathematics, 6 (2021), 4447–4462. https://doi.org/10.3934/math.2021263 doi: 10.3934/math.2021263
[18]	W. Wang, J. Liang, M. Liu, L. Ding, H. Zeng, Novel robust stability criteria for Lur'e systems with time-varying delay, Mathematics, 12 (2024), 583. https://doi.org/10.3390/math12040583 doi: 10.3390/math12040583
[19]	W. Wang, J. M. Liang, H. B. Zeng, Sampled-data-based stability and stabilization of Lurie systems, Appl. Math. Comput., 501 (2025), 129455. https://doi.org/10.1016/j.amc.2025.129455 doi: 10.1016/j.amc.2025.129455
[20]	X. M. Zhang, Q. L. Han, Robust ${H}_{\infty}$ filtering for a class of uncertain linear systems with time-varying delay, Automatica, 44 (2008), 157–166. https://doi.org/10.1016/j.automatica.2007.04.024 doi: 10.1016/j.automatica.2007.04.024
[21]	X. M. Zhang, Q. L. Han, X. Ge, Sufficient conditions for a class of matrix-valued polynomial inequalities on closed intervals and application to ${H}_{\infty}$ filtering for linear systems with time-varying delays, Automatica, 125 (2021), 109390. https://doi.org/10.1016/j.automatica.2020.109390 doi: 10.1016/j.automatica.2020.109390
[22]	G. M. Zhuang, Z. K. Wang, X. P. Xie, J. W. Xia, Embedded-adaptive-observer-based ${H}_\infty$ integrated fault estimation and memory fault-tolerant control for nonlinear delayed implicit robotic arm, IEEE Trans. Ind. Inform., 20 (2024), 8317–8327. https://doi.org/10.1109/TII.2024.3372009 doi: 10.1109/TII.2024.3372009
[23]	R. H. Gielen, M. Lazar, S. V. Raković, Necessary and sufficient Razumikhin-type conditions for stability of delay difference equations, IEEE Trans. Automat. Control, 58 (2013), 2637–2642. https://doi.org/10.1109/TAC.2013.2255951 doi: 10.1109/TAC.2013.2255951
[24]	H. C. Lin, H. B. Zeng, X. M. Zhang, W. Wang, Stability analysis for delayed neural networks via a generalized reciprocally convex inequality, IEEE Trans. Neural Netw. Learn. Syst., 34 (2023), 7491–7499. https://doi.org/10.1109/TNNLS.2022.3144032 doi: 10.1109/TNNLS.2022.3144032
[25]	H. B. Zeng, Z. J. Zhu, W. Wang, X. M. Zhang, Relaxed stability criteria of delayed neural networks using delay-parameters-dependent slack matrices, Neural Netw., 180 (2024), 106676. https://doi.org/10.1016/j.neunet.2024.106676 doi: 10.1016/j.neunet.2024.106676
[26]	F. S. S. de Oliveira, F. O. Souza, Further refinements in stability conditions for time-varying delay systems, Appl. Math. Comput., 369 (2020), 124866. https://doi.org/10.1016/j.amc.2019.124866 doi: 10.1016/j.amc.2019.124866
[27]	W. Wang, H. B. Zeng, K. L. Teo, Y. J. Chen, Relaxed stability criteria of time-varying delay systems via delay-derivative-dependent slack matrices, J. Franklin Inst., 360 (2023), 6099–6109. https://doi.org/10.1016/j.jfranklin.2023.04.019 doi: 10.1016/j.jfranklin.2023.04.019
[28]	P. Park, J. W. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47 (2011), 235–238. https://doi.org/10.1016/j.automatica.2010.10.014 doi: 10.1016/j.automatica.2010.10.014
[29]	H. B. Zeng, Y. He, M. Wu, J. She, Free-matrix-based integral inequality for stability analysis of systems with time-varying delay, IEEE Trans. Automat. Control, 60 (2015), 2768–2772. https://doi.org/10.1109/TAC.2015.2404271 doi: 10.1109/TAC.2015.2404271
[30]	C. K. Zhang, Y. He, L. Jiang, M. Wu, H. B. Zeng, Stability analysis of systems with time-varying delay via relaxed integral inequalities, Syst. Control Lett., 92 (2016), 52–61. https://doi.org/10.1016/j.sysconle.2016.03.002 doi: 10.1016/j.sysconle.2016.03.002
[31]	A. Seuret, F. Gouaisbaut, Stability of linear systems with time-varying delays using {B}essel–{L}egendre inequalities, IEEE Trans. Automat. Control, 63 (2018), 225–232. https://doi.org/10.1109/TAC.2017.2730485 doi: 10.1109/TAC.2017.2730485
[32]	H. B. Zeng, X. G. Liu, W. Wang, A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems, Appl. Math.Comput., 354 (2019), 1–8. https://doi.org/10.1016/j.amc.2019.02.009 doi: 10.1016/j.amc.2019.02.009
[33]	W. M. Wang, Y. W. Wang, H. B. Zeng, J. Huang, Further results on stability analysis of {T–S} fuzzy systems with time-varying delay, IEEE Trans. Fuzzy Syst., 32 (2024), 3529–3541. https://doi.org/10.1109/TFUZZ.2024.3375449 doi: 10.1109/TFUZZ.2024.3375449
[34]	F. Gouaisbaut, D. Peaucelle, Delay-dependent stability analysis of linear time delay systems, IFAC Proc. Vol., 39 (2006), 54–59. https://doi.org/10.3182/20060710-3-IT-4901.00010 doi: 10.3182/20060710-3-IT-4901.00010
[35]	Q. L. Han, A new delay-dependent absolute stability criterion for a class of nonlinear neutral systems, Automatica, 44 (2008), 272–277. https://doi.org/10.1016/j.automatica.2007.04.009 doi: 10.1016/j.automatica.2007.04.009
[36]	X. M. Zhang, Q. L. Han, A. Seuret, F. Gouaisbaut, Y. He, Overview of recent advances in stability of linear systems with time-varying delays, IET Control Theory Appl., 13 (2019), 1–16. https://doi.org/10.1049/iet-cta.2018.5188 doi: 10.1049/iet-cta.2018.5188
[37]	H. B. Zeng, Y. He, K. L. Teo, Monotone-delay-interval-based Lyapunov functionals for stability analysis of systems with a periodically varying delay, Automatica, 138 (2022), 110030. https://doi.org/10.1016/j.automatica.2021.110030 doi: 10.1016/j.automatica.2021.110030
[38]	Y. Chen, H. B. Zeng, Y. Li, Stability analysis of linear delayed systems based on an allowable delay set partitioning approach, Automatica, 163 (2024), 111603. https://doi.org/10.1016/j.automatica.2024.111603 doi: 10.1016/j.automatica.2024.111603
[39]	J. de Canniere, H. van Brussel, J. van Bogaert, A contribution to the mathematical analysis of variable spindle speed machining, Appl. Math. Model., 5 (1981), 158–164. https://doi.org/10.1016/0307-904X(81)90038-X doi: 10.1016/0307-904X(81)90038-X
[40]	S. Jayaram, S. G. Kapoor, R. E. DeVor, Analytical stability analysis of variable spindle speed machining, J. Manuf. Sci. Eng., 122 (2000), 391–397. https://doi.org/10.1115/1.1285890 doi: 10.1115/1.1285890
[41]	W. Michiels, V. van Assche, S. I. Niculescu, Stabilization of time-delay systems with a controlled time-varying delay and applications, IEEE Trans. Automat. Control, 50 (2005), 493–504. https://doi.org/10.1109/TAC.2005.844723 doi: 10.1109/TAC.2005.844723
[42]	W. Wang, W. M. Wang, H. B. Zeng, Stability analysis of systems with cyclical delay via an improved delay-monotonicity-dependent Lyapunov functional, J. Franklin Inst., 360 (2023), 99–108. https://doi.org/10.1016/j.jfranklin.2022.11.032 doi: 10.1016/j.jfranklin.2022.11.032
[43]	H. B. Zeng, Y. J. Chen, Y. He, X. M. Zhang, A delay-derivative-dependent switched system model method for stability analysis of linear systems with time-varying delay, Automatica, 175 (2025), 112183. https://doi.org/10.1016/j.automatica.2025.112183 doi: 10.1016/j.automatica.2025.112183

This article has been cited by:

1.	Chunwei Zhang, Shuhui Pei, Ardashir Mohammadzadeh, An Adaptive-switching Fuzzy Structural Vibration Control System: Theory and Experiment, 2025, 13, 2523-3920, 10.1007/s42417-025-01881-y
2.	Zheshu Jia, A Novel Bio-inspired Optimised Algorithm with Deep Learning in the Pursuit-Evasion Framework for Task Assessment, 2025, 2153-0785, 10.1007/s13235-025-00654-2

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.1

Metrics

Article views(366) PDF downloads(38) Cited by(2)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(3) / Tables(4)

AIMS Mathematics

Stability analysis of linear systems with a periodical time-varying delay based on an improved non-continuous piecewise Lyapunov functional

Related Papers:

Abstract

1. Introduction

2. Preliminary

2.1. Notations

2.2. System description

2.3. Lemma

3. Main results

4. Numerical examples

5. Conclusions

Author contributions

Use of Generative-AI tools declaration

Acknowledgments

Data availability statement

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

Stability analysis of linear systems with a periodical time-varying delay based on an improved non-continuous piecewise Lyapunov functional

Related Papers:

Abstract

1. Introduction

2. Preliminary

2.1. Notations

2.2. System description

2.3. Lemma

3. Main results

4. Numerical examples

5. Conclusions

Author contributions

Use of Generative-AI tools declaration

Acknowledgments

Data availability statement

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog