Microplastics in surface water and tissue of white leg shrimp,  <i>Litopenaeus vannamei</i>, in a cultured pond in Nakhon Pathom Province, Central Thailand

Akekawat Vitheepradit; Taeng-On Prommi; Akekawat Vitheepradit; Taeng-On Prommi

doi:10.3934/environsci.2023027

AIMS Environmental Science

2023, Volume 10, Issue 4: 478-503. doi: 10.3934/environsci.2023027

Previous Article Next Article

Research article Special Issues

Microplastics in surface water and tissue of white leg shrimp, Litopenaeus vannamei, in a cultured pond in Nakhon Pathom Province, Central Thailand

Akekawat Vitheepradit ¹,
Taeng-On Prommi ^{2
,
,}

1.
Department of Entomology, Kasetsart University, Bangkok 10900, Thailand
2.
Department of Science and Bioinnovation, Faculty of Liberal Arts and Science, Kasetsart University, Kamphaeng Saen Campus, Nakhon Pathom Province, 73140, Thailand

Received: 12 February 2023 Revised: 24 July 2023 Accepted: 02 August 2023 Published: 21 August 2023

The presence of microplastics in commercially important seafood species is a new issue of food safety concern. Although plastic debris has been found in the gastrointestinal tracts of several species, the prevalence of microplastics in edible shrimp tissues in Thailand has not yet been established. For the first time, the gastrointestinal tract (GT), heptapancreas (HEP), muscle (MU) and exoskeleton (EX) of farmed white leg shrimp (Litopenaeus vannamei) from commercial aquaculture facilities in Nakhon Pathom Province, Thailand, were analyzed for microplastics (MPs). The number of MP items per tissue was 27.36±2.28 in the GT, 17.42±0.90 in the HEP, 11.37±0.60 in the MU and 10.04±0.52 in the EX. MP concentrations were 137.78±16.48, 16.31±1.87, 1.69±0.13 and 4.37±0.27 items/gram (ww) in the GT, HEP, MU and EX, respectively. Microplastics ranged in size from < 100 to 200–250 μm, with fragment-shape (62.07%), fibers (37.31%) and blue (43.69%) was the most common. The most frequently found polymers in shrimp tissue organs and pond water were polyethylene terephthalate (PET), polyvinyl acetate (PVAc) and cellulose acetate butyrate (CAB). Shrimp consumption (excluding GT and EX) was calculated as 28.79 items/shrimp/person/day using Thailand's consumption of shrimp, MP abundance and shrimp consumption. The results of the study can be used as background data for future biomonitoring of microplastics in shrimp species that are significant from an ecological and commercial perspective. MP abundance in farmed L. vannamei may be related to feeding habits and the source of MPs could come from the aquaculture facilities operations.

Keywords:

Citation: Akekawat Vitheepradit, Taeng-On Prommi. Microplastics in surface water and tissue of white leg shrimp, Litopenaeus vannamei, in a cultured pond in Nakhon Pathom Province, Central Thailand[J]. AIMS Environmental Science, 2023, 10(4): 478-503. doi: 10.3934/environsci.2023027

Related Papers:

[1]	Xin Du, Quansheng Liu, Yuanhong Bi . Bifurcation analysis of a two–dimensional p53 gene regulatory network without and with time delay. Electronic Research Archive, 2024, 32(1): 293-316. doi: 10.3934/era.2024014
[2]	Fang Yan, Changyong Dai, Haihong Liu . Oscillatory dynamics of p53 pathway in etoposide sensitive and resistant cell lines. Electronic Research Archive, 2022, 30(6): 2075-2108. doi: 10.3934/era.2022105
[3]	Yongwei Yang, Yang Yu, Chunyun Xu, Chengye Zou . Passivity analysis of discrete-time genetic regulatory networks with reaction-diffusion coupling and delay-dependent stability criteria. Electronic Research Archive, 2025, 33(5): 3111-3134. doi: 10.3934/era.2025136
[4]	Weijie Ding, Xiaochen Mao, Lei Qiao, Mingjie Guan, Minqiang Shao . Delay-induced instability and oscillations in a multiplex neural system with Fitzhugh-Nagumo networks. Electronic Research Archive, 2022, 30(3): 1075-1086. doi: 10.3934/era.2022057
[5]	Yulin Zhao, Fengning Liang, Yaru Cao, Teng Zhao, Lin Wang, Jinhui Xu, Hong Zhu . MRI-based model for accurate prediction of P53 gene status in gliomas. Electronic Research Archive, 2024, 32(5): 3113-3129. doi: 10.3934/era.2024142
[6]	Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li . Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, 2021, 29(5): 2973-2985. doi: 10.3934/era.2021022
[7]	Rui Ma, Xin-You Meng . Dynamics of an eco-epidemiological model with toxicity, treatment, time-varying incubation. Electronic Research Archive, 2025, 33(5): 3074-3110. doi: 10.3934/era.2025135
[8]	Fengrong Zhang, Ruining Chen . Spatiotemporal patterns of a delayed diffusive prey-predator model with prey-taxis. Electronic Research Archive, 2024, 32(7): 4723-4740. doi: 10.3934/era.2024215
[9]	Xiaowen Zhang, Wufei Huang, Jiaxin Ma, Ruizhi Yang . Hopf bifurcation analysis in a delayed diffusive predator-prey system with nonlocal competition and schooling behavior. Electronic Research Archive, 2022, 30(7): 2510-2523. doi: 10.3934/era.2022128
[10]	Mengting Sui, Yanfei Du . Bifurcations, stability switches and chaos in a diffusive predator-prey model with fear response delay. Electronic Research Archive, 2023, 31(9): 5124-5150. doi: 10.3934/era.2023262

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 , 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]	Benson NU, Bassey DE, Palanisami T (2021) COVID pollution: impact of COVID-19 pandemic on global plastic waste footprint. Heliyon 7: e06343. https://doi:10.1016/j.heliyon.2021.e06343 doi: 10.1016/j.heliyon.2021.e06343
[2]	Jabeen K, Su L, Li J, et al. (2017) Microplastics and mesoplastics in fish from coastal and fresh waters of China. Environ Pollut 221: 141–149. https://doi.org/10.1016/j.envpol.2016.11.055 doi: 10.1016/j.envpol.2016.11.055
[3]	Frias J, Pagter E, Nash R, et al. (2018) Standardised protocol for monitoring microplastics in sediments. Deliverable D4.2. BASEMAN. Microplastic analyses in European waters: JPI Oceans. https://doi.org/10.13140/RG.2.2.36256.89601/1
[4]	Hanachi P, Karbalaei S, Walker TR, et al. (2019) Abundance and properties of microplastics found in commercial fish meal and cultured common carp (Cyprinus carpio). Environ Sci Pollut Res 26: 23777–23787. https://doi.org/10.1007/s11356-019-05637-6 doi: 10.1007/s11356-019-05637-6
[5]	Herrera A, Ŝtindlová A, Martínez I, et al. (2019) Microplastic ingestion by Atlantic chub mackerel (Scomber colias) in the Canary Islands coast. Mar Pollut Bull 139: 127–135. https://doi.org/10.1016/j.marpolbul.2018.12.022 doi: 10.1016/j.marpolbul.2018.12.022
[6]	Hossain MS, Sobhan F, Uddin MN, et al. (2019) Microplastics in fishes from the Northern Bay of Bengal. Sci Total Environ 690: 821–830. https://doi.org/10.1016/j.scitotenv.2019.07.065 doi: 10.1016/j.scitotenv.2019.07.065
[7]	Amin RM, Sohaimi ES, Anuar ST, et al. (2020) Microplastic ingestion by zooplankton in Terengganu coastal waters, southern South China Sea. Mar Pollut Bull 150: 110616. https://doi.org/10.1016/j.marpolbul.2019.110616 doi: 10.1016/j.marpolbul.2019.110616
[8]	Filgueiras AV, Preciado I, Cartón A, et al. (2020) Microplastic ingestion by pelagic and benthic fish and diet composition: a case study in the NW Iberian shelf. Mar Pollut Bull 160: 111623. https://doi.org/10.1016/j.marpolbul.2020.111623 doi: 10.1016/j.marpolbul.2020.111623
[9]	Kor K, Ghazilou A, Ershadifar H (2020) Microplastic pollution in the littoral sediments of the northern part of the Oman Sea. Mar Pollut Bull 155: 111166. https://doi.org/10.1016/j.marpolbul.2020.111166 doi: 10.1016/j.marpolbul.2020.111166
[10]	Barboza LGA, Lopes C, Oliveira P, et al. (2020) Microplastics in wild fish from North East Atlantic Ocean and its potential for causing neurotoxic effects, lipid oxidative damage, and human health risks associated with ingestion exposure. Sci Total Environ 717: 134625. https://doi.org/10.1016/j.scitotenv.2019.134625 doi: 10.1016/j.scitotenv.2019.134625
[11]	Zakeri M, Naji A, Akbarzadeh A, et al. (2020) Microplastic ingestion in important commercial fish in the southern Caspian Sea. Mar Pollut Bull 160: 111598. https://doi.org/10.1016/j.marpolbul.2020.111598. doi: 10.1016/j.marpolbul.2020.111598
[12]	Kolandhasamy P, Su L, Li J, et al. (2018) Adherence of microplastics to soft tissue of mussels: a novel way to uptake microplastics beyond ingestion. Sci Total Environ 610–611: 635–640. https://doi.org/10.1016/j.scitotenv.2017.08.053 doi: 10.1016/j.scitotenv.2017.08.053
[13]	Baalkhuyur FM, Qurban MA, Panickan P, et al. (2020) Microplastics in fishes of commercial and ecological importance from the Western Arabian Gulf. Mar Pollut Bull 152: 110920. https://doi.org/10.1016/j.marpolbul.2020.110920 doi: 10.1016/j.marpolbul.2020.110920
[14]	Li B, Su L, Zhang H, et al. (2020) Microplastics in fishes and their living environments surrounding a plastic production area. Sci Total Environ 727: 138662. https://doi.org/10.1016/j.scitotenv.2020.138662. doi: 10.1016/j.scitotenv.2020.138662
[15]	Claessens M, Van Cauwenberghe L, Vandegehuchte MB, et al. (2013) New techniques for the detection of microplastics in sediments and field collected organisms. Mar Pollut Bull 70: 227–233. https://doi.org/10.1016/j.marpolbul.2013.03.009 doi: 10.1016/j.marpolbul.2013.03.009
[16]	Carbery M, O'Connor W, Palanisami T (2018) Trophic transfer of microplastics and mixed contaminants in the marine food web and implications for human health. Environ Int 115: 400–409. https://doi.org/10.1016/j.envint.2018.03.007 doi: 10.1016/j.envint.2018.03.007
[17]	Wang Q, Zhu X, Hou C, et al. (2021) Microplastic uptake in commercial fishes from the Bohai Sea, China. Chemosphere 263: 127962. https://doi.org/10.1016/j.Chemosphere.2020.127962 doi: 10.1016/j.Chemosphere.2020.127962
[18]	Lindeque PK, Cole M, Coppock RL, et al. (2020) Are we underestimating microplastic abundance in the marine environment? A comparison of microplastic capture with nets of different mesh-size. Environ Pollut 265: 114721. https://doi.org/10.1016/j.envpol.2020.114721 doi: 10.1016/j.envpol.2020.114721
[19]	Sfriso AA, Tomio Y, Rosso B, et al. (2020) Microplastic accumulation in benthic invertebrates in Terra Nova Bay (Ross Sea, Antarctica). Environ Int 137: 10558. https://doi.org/10.1016/j.envint.2020.105587 doi: 10.1016/j.envint.2020.105587
[20]	Daniel DB, Ashraf PM, Thomas SN (2020) Abundance, characteristics and seasonal variation of microplastics in Indian white shrimps (Fenneropenaeus Indicus) from coastal waters off Cochin, Kerala, India. Sci Total Environ 737: 139839. https://doi.org/10.1016/j.scitotenv.2020.139839 doi: 10.1016/j.scitotenv.2020.139839
[21]	Devriese LI, van der Meulen MD, Maes T, et al. (2015) Microplastic contamination in brown shrimp (Crangon crangon, Linnaeus 1758) from coastal waters of the southern North Sea and channel area. Mar Pollut Bull 98: 179–187. https://doi.org/10.1016/j.marpolbul.2015.06.051 doi: 10.1016/j.marpolbul.2015.06.051
[22]	Hossain MS, Rahman MS, Uddin MN, et al. (2020) Microplastic contamination in Penaeid shrimp from the Northern Bay of Bengal. Chemosphere 238: 124688. https://doi.org/10.1016/j.chemosphere.2019.124688 doi: 10.1016/j.chemosphere.2019.124688
[23]	Gray AD, Weinstein JE (2017) Size- and shape-dependent effects of microplastic particles on adult daggerblade grass shrimp (Palaemonetes pugio). Environ Toxicol 36: 3074–3080. https://doi.org/10.1002/etc.3881 doi: 10.1002/etc.3881
[24]	Qu X, Su L, Li H, et al. (2018) Assessing the relationship between the abundance and properties of microplastics in water and in mussels. Sci Total Environ 621: 679–686. https://doi.org/10.1016/j.scitotenv.2017.11.284 doi: 10.1016/j.scitotenv.2017.11.284
[25]	Phuong NN, Duong TT, Le TPQ, et al (2022) Microplastics in Asian freshwater ecosystems: current knowledge and perspectives. Sci Total Environ 808: 151989. https://doi:10.1016/j.scitotenv.2021.151989 doi: 10.1016/j.scitotenv.2021.151989
[26]	Masura J, Baker J, Foster G, et al. (2015) Laboratory Methods for the Analysis of Microplastics in the Marine Environment: Recommendations for Quantifying Synthetic Particles in Waters and Sediments. Maryland, USA: NOAA Marine Debris Division.
[27]	Avio CG, Gorbi S, Regoli F (2015) Experimental development of a new protocol for extraction and characterization of microplastics in fish tissues: first observations in commercial species from Adriatic Sea. Mar Environl Res 111: 18–26. https://doi.org/10.1016/j.marenvres.2015.06.014. doi: 10.1016/j.marenvres.2015.06.014
[28]	Zhang K, Su J, Xiong X, et al. (2016) Microplastic pollution of lakeshore sediments from remote lakes in Tibet plateau, China. Environ Pollut 219: 450–455. https://doi.org/10.1016/j.envpol.2016.05.048 doi: 10.1016/j.envpol.2016.05.048
[29]	Hidalgo-Ruz V, Gutow L, Thompson RC, et al. (2012) Microplastics in the marine environment: a review of the methods used for identification and quantification. Environ Sci Technol 46: 3060–3075. https://doi.org/10.1021/es2031505 doi: 10.1021/es2031505
[30]	Su L, Cai H, Kolandhasamy P, et al. (2018) Using the Asian clam as an indicator of microplastic pollution in freshwater ecosystems. Environ Pollut 234: 347−355. https://doi.org/10.1016/j.envpol.2017.11.075 doi: 10.1016/j.envpol.2017.11.075
[31]	Bergmann M, Wirzberger V, Krumpen T, et al. (2017) High quantities of microplastic in Arctic deep-sea sediments from the HAUSGARTEN Observatory. Environ Sci Technol 51: 11000–11010. https://doi.org/10.1021/acs.est.7b03331 doi: 10.1021/acs.est.7b03331
[32]	Smith M, Love DC, Rochman CM, et al. (2018) Microplastics in seafood and the implications for human health. Curr Environ Health Rep 5: 375–386. https://doi.org/10.1007/s40572-018-0206-z doi: 10.1007/s40572-018-0206-z
[33]	Hossain MS, Rahman MS, Uddin MN, et al. (2020) Microplastic contamination in Penaeid shrimp from the Northern Bay of Bengal. Chemosphere 238: 124688. https://doi.org/10.1016/j.chemosphere.2019.124688 doi: 10.1016/j.chemosphere.2019.124688
[34]	Gurjar UR, Xavier M, Nayak BB, et al. (2021) Microplastics in shrimps: a study from the trawling grounds of north eastern part of Arabian Sea. Environ Sci Pollut Res 28: 48494–48504. https://doi.org/10.1007/s11356-021-14121-z doi: 10.1007/s11356-021-14121-z
[35]	Wang S, Zhang C, Pan Z, et al. (2020) Microplastics in wild freshwater fish of different feeding habits from Beijiang and Pearl River Delta regions, south China. Chemosphere 258: 127345. https://doi.org/10.1016/j.chemosphere.2020.127345 doi: 10.1016/j.chemosphere.2020.127345
[36]	Curren E, Leaw CP, Lim PT, et al. (2020) Evidence of marine microplastics in commercially harvested seafood. Front Bioeng Biotechnol 8: 562760. https://doi.org/10.3389/fbioe.2020.562760 doi: 10.3389/fbioe.2020.562760
[37]	Reunura T, Prommi TO (2022) Detection of microplastics in Litopenaeus vannamei (Penaeidae) and Macrobrachium rosenbergii (Palaemonidae) in cultured pond. PeerJ 10: e12916. https://doi.org/10.7717/peerj.12916 doi: 10.7717/peerj.12916
[38]	Yoon H, Park B, Rim J, et al. (2022) Detection of microplastics by various types of whiteleg shrimp (Litopenaeus vannamei) in the Korean Sea. Separations 9: 332. https://doi.org/10.3390/separations9110332 doi: 10.3390/separations9110332
[39]	Barrows A, Cathey SE, Petersen CW (2018) Marine environment microfiber contamination: global patterns and the diversity of microparticle origins. Environ Pollut 237: 275–284. https://doi.org/10.1016/j.envpol.2018.02.062 doi: 10.1016/j.envpol.2018.02.062
[40]	Capparelli MV, Molinero J, Moulatlet GM, et al. (2021) Microplastics in rivers and coastal waters of the province of Esmeraldas, Ecuador. Mar Pollut Bull 173: 113067. https://doi.org/10.1016/j.marpolbul.2021.113067 doi: 10.1016/j.marpolbul.2021.113067
[41]	Valencia-Castañeda G, Ruiz-Fernández AC, Frías-Espericueta MG, et al. (2022) Microplastics in the tissues of commercial semi-intensive shrimp pond-farmed Litopenaeus vannamei from the Gulf of California ecoregion. Chemosphere 297: 134194. https://doi.org/10.1016/j.chemosphere.2022.134194 doi: 10.1016/j.chemosphere.2022.134194
[42]	Wu F, Wang Y, Leung JYS, et al. (2020) Accumulation of microplastics in typical commercial aquatic species: a case study at a productive aquaculture site in China. Sci Total Environ 708: 135432. https://doi.org/10.1016/j.scitotenv.2019.135432 doi: 10.1016/j.scitotenv.2019.135432
[43]	Keshavarzifard M, Vazirzadeh A, Sharifnia M (2021) Occurrence and characterization of microplastics in white shrimp, Metapenaeus affinis, living in a habitat highly affected by anthropogenic pressures, northwest Persian Gulf. Mar Pollut Bull 169: 112581. https://doi.org/10.1016/j.marpolbul.2021.112581 doi: 10.1016/j.marpolbul.2021.112581
[44]	Valencia-Castañeda G, Ruiz-Fernández AC, Frías-Espericueta MG, et al. (2022) Microplastics in the tissues of commercial semi-intensive shrimp pond-farmed Litopenaeus vannamei from the Gulf of California ecoregion. Chemosphere 297: 134194. https://doi.org/10.1016/j.chemosphere.2022.134194 doi: 10.1016/j.chemosphere.2022.134194
[45]	Weinstein JE, Crocker BK, Gray AD (2016) From macroplastic to microplastic: Degradation of high-density polyethylene, polypropylene, and polystyrene in a salt marsh habitat. Environ Toxicol Chem 35: 1632–1640. https://doi.org/10.1002/etc.3432 doi: 10.1002/etc.3432
[46]	Dall W, Hill BJ, Rothlisberg PC, et al. (1990) The Biology of the Penaeidae. Academic Press, San Diego, USA.
[47]	Yan MT, Li WX, Chen XF, He YH, Zhang XY, Gong H (2021) A preliminary study of the association between colonization of microorganism on microplastics and intestinal microbiota in shrimp under natural conditions. J Hazard Mater 408: 124882. https://doi.org/10.1016/j.jhazmat.2020.124882 doi: 10.1016/j.jhazmat.2020.124882
[48]	Keteles KA, Fleeger JW (2001) The contribution of ecdysis to the fate of copper, zinc and cadmium in grass shrimp, Palaemonetes pugio Holthius. Mar Pollut Bull 42: 1397–1402. https://doi.org/10.1016/S0025-326X(01)00172-2 doi: 10.1016/S0025-326X(01)00172-2
[49]	Duan Y, Xiong D, Wang Y, et al. (2021) Toxicological effects of microplastics in Litopenaeus vannamei as indicated by an integrated microbiome, proteomic and metabolomic approach. Sci Total Environ 761: 43311. https://doi.org/10.1016/j.scitotenv.2020.143311 doi: 10.1016/j.scitotenv.2020.143311
[50]	Hsieh SL, Wu YC, Xu RQ, et al. (2021) Effect of polyethylene microplastics on oxidative stress and histopathology damages in Litopenaeus vannamei. Environ Pollut 288: 117880. https://doi.org/10.1016/j.envpol.2021.117800 doi: 10.1016/j.envpol.2021.117800
[51]	Barboza LGA, Frias JPGL, Booth AM, et al. (2019) Microplastics Pollution in the Marine Environment v. 3. In: Sheppard C (Ed.), World Seas: An Environmental Evaluation. Volume Ⅲ: Ecological Issues and Environmental Impacts. 2ed. Academic Press (Elsevier), London, pp. 329–351.
[52]	Karami A, Golieskardi A, Choo CK, et al. (2018) Microplastic and mesoplastic contamination in canned sardines and sprats. Sci Total Environ 612: 1380–1386. https://doi.org/10.1016/j.scitotenv.2017.09.005 doi: 10.1016/j.scitotenv.2017.09.005
[53]	Wright SL, Kelly FJ (2017) Plastic and human health: a micro issue? Environ. Sci. Technol. 51: 6634–6647. https://doi.org/10.1021/acs.est.7b00423 doi: 10.1021/acs.est.7b00423
[54]	Volkheimer G (1977) Persorption of particles: physiology and pharmacology. Adv Pharmacol Chemother 14: 163–187. https://doi.org/10.1016/s1054-3589(08)60188-x doi: 10.1016/s1054-3589(08)60188-x
[55]	Collard F, Gilbert B, Compère P, et al. (2017) Microplastics in livers of European anchovies (Engraulis encrasicolus, L.). Environ Pollut 229: 1000–1005. https://doi.org/10.1016/j.envpol.2017.07.089 doi: 10.1016/j.envpol.2017.07.089
[56]	Jovanović B, Gökdağ K, Güven O, et al. (2018) Virgin microplastics are not causing imminent harm to fish after dietary exposure. Mar Pollut Bull 130: 123–131. https://doi.org/10.1016/j.marpolbul.2018.03.016 doi: 10.1016/j.marpolbul.2018.03.016
[57]	FAO. The State of World Fisheries and Aquaculture (2020) Sustainability in Action; FAO: Rome, Italy.
[58]	Barboza LGA, Vethaak AD, Lavorante BR, et al. (2018) Marine microplastic debris: An emerging issue for food security, food safety and human health. Mar Pollut Bull 133: 336–348. https://doi.org/10.1016/j.marpolbul.2018.05.047 doi: 10.1016/j.marpolbul.2018.05.047
[59]	Kim IS, Chae DH, Kim SK, et al. (2015) Factors influencing the spatial variation of microplastics on high-tidal coastal beaches in Korea. Arch Environ Contam Toxicol 69: 299–309. https://doi.org/10.1007/s00244-015-0155-6 doi: 10.1007/s00244-015-0155-6
[60]	Beer S, Garm A, Huwer B, et al. (2018) No increase in marine microplastic concentration over the last three decades - a case study from the Baltic Sea. Sci Total Environ 621: 1272–1279. https://doi.org/10.1016/j.scitotenv.2017.10.101 doi: 10.1016/j.scitotenv.2017.10.101
[61]	Zhu L, Bai H, Chen B, et al. (2018) Microplastic pollution in North Yellow Sea, China: observations on occurrence, distribution and identification. Sci Total Environ 636: 20–29. https://doi.org/10.1016/j.scitotenv.2018.04.182 doi: 10.1016/j.scitotenv.2018.04.182
[62]	Cheung PK, Fok L (2016) Evidence of microbeads from personal care product contaminating the sea. Mar Pollut Bull 109: 582–585. https://doi.org/10.1016/j.marpolbul.2016.05.046 doi: 10.1016/j.marpolbul.2016.05.046
[63]	Browne MA, Crump P, Niven SJ, et al. (2011) Accumulation of microplastic on shorelines worldwide: sources and sinks. Environ Sci Technol 45: 9175–9179. https://doi.org/10.1021/es201811s doi: 10.1021/es201811s
[64]	Qiu Q, Peng J, Yu X, et al. (2015) Occurrence of microplastics in the coastal marine environment: first observation on sediment of China. Mar Pollut Bull 98: 274–280. https://doi.org/10.1016/j.marpolbul.2015.07.028 doi: 10.1016/j.marpolbul.2015.07.028
[65]	Liu T, Zhao Y, Zhu M, et al. (2020) Seasonal variation of micro and meso-plastics in the seawater of Jiaozhou Bay, the Yellow Sea. Mar Pollut Bull 152: 110922. https://doi.org/10.1016/j.marpolbul.2020.110922 doi: 10.1016/j.marpolbul.2020.110922

This article has been cited by:

Xin Du, Quansheng Liu, Yuanhong Bi, Bifurcation analysis of a two–dimensional p53 gene regulatory network without and with time delay, 2023, 32, 2688-1594, 293, 10.3934/era.2024014

Reader Comments

Your name:*

Email:*
© 2023 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)