Non-fragile sampled-data control for synchronizing Markov jump Lur'e systems with time-variant delay

1.   Introduction

Synchronization of chaotic systems stands as a prominent research area within nonlinear system science. Its potential utility spans diverse domains, including the transmission of digital signals, secure communication, and information processing ^[1,2,3]. Numerous chaotic systems, such as Chua's circuit and Hopfield network, can be effectively represented as Lur'e systems ^[4]. Consequently, the synchronization of Lur'e systems (LSs) has garnered substantial attention. Within the master-slave framework established by Pecora and Carroll ^[5], a wealth of research results addressing various synchronization issues, including quasi-synchronization ^[6], cluster synchronization ^[7], prespecified-time synchronization ^[8], and bipartite synchronization ^[9], have been reported.

In actual engineering systems, abrupt changes can occur in the structure or parameters due to component failures, sudden environmental disturbances, and changes in connections between subsystems. Markov chain often serves as a suitable candidate to model these abrupt change behaviors ^{[10,11,12,13]}. Synchronization of Markov jump Lur'e systems (MJLSs) was initially investigated by ^[14], where a delay feedback control scheme was introduced. Following that, reference ^[15] considered singular perturbations and developed a mode-dependent control strategy to ensure stochastic synchronization. Reference ^[16] took into account both time delays and external disturbances, presenting a design for state-feedback controllers to achieve finite-time $\mathcal{H}_\infty$ synchronization.

Recent years have witnessed the introduction of various networked control strategies, including quantized control ^[17], event-triggered control ^[18], and sampled-data control (SDC) ^[19], in the study of synchronization for MJLSs ^[20,21,22]. Under SDC strategies, the system's state or output is sampled at specific time instants, and the control action is updated and applied at those discrete time points, thereby effectively decreasing the amount of transmitted information and conserving communication bandwidth ^[23,24]. In the study of ^[21], MJLSs with a single time delay were examined. Building on a novel Lyapunov-Krasovskii functional (LKF), two SDC control approaches, formulated in terms of linear matrix inequalities (LMIs), were established to guarantee stochastic synchronization between the master and slave MJLSs. This research was extended to encompass multiple time delays in ^[22], where a design approach for mean-square exponential synchronization was developed.

Despite theoretical progress in the research of SDC for delayed Markov jump systems, there are still concerns that need to be addressed further. In particular, the time delays are assumed to be time-invariant in ^[22], whereas, in practical application circumstances, they often display time-variant behavior ^[25,26]. Incorporating time-variant delays is more challenging but may result in more generic solutions. Furthermore, the control designs in ^[22] do not account for the influence of gain fluctuations. In engineering implementations, controllers/filters frequently exhibit a degree of parameter inaccuracies due to digital rounding errors, memory constraints, and analog-digital conversion imprecision ^[27]. As indicated by ^[28], even a minor gain fluctuation can impair the effectiveness of control/filtering.

Motivated by the preceding discussion, this paper focuses on the issue of non-fragile sampled-data control for synchronizing delayed MJLSs. In contrast to the studies conducted in ^[21,22], the present research incorporates considerations for both time-variant delay and gain fluctuations. The main contributions of this paper can be summarized as follows: 1) The components of the nonlinear function within the MJLSs are assumed to be either Lipschitz continuous or monotonic nondecreasing. This enables the construction of two distinct two-sided loop LKFs; 2) Two sufficient conditions concerning the non-fragile sampled-data controller design are derived to ensure that the drive-response MJLSs are stochastically synchronized with a prescribed $\mathcal{L}_{2}-\mathcal{L}_{\infty }$ disturbance attenuation level (DAL). An exact mathematical expression of the desired controller gains can be obtained based on these conditions.

Notations: Throughout, the notation $\mathcal{E}\{\cdot \}$ represents the mathematical expectation. $col\{\cdots \}$ and $diag\{\cdots \}$ stand for a column vector and a block-diagonal matrix, respectively. Represent by $\mathbb{R}^{n}$ the $n$-dimensional Euclidean space, by $\mathbb{R}^{l\times n}$ the set of all $l\times n$ real matrices, by $\mathbb{S}^{n}$ the $n\times n$ symmetric matrices, and by $\mathbb{S} _{+}^{n}$ the $n\times n$ symmetric and positive-define matrices. The superscripts "$-1$" and "$T$" stand for the inverse and transpose of a matrix, respectively. $\mathscr{S}(P)$ is the sum of matrix $P$ and its transpose (i.e., $\mathscr{S}(P) = P+P^{T}$). $\lambda _{\max }(P)$ is used to denote the largest eigenvalue of $P$. $I_{n}$ and $0_{l\times n}$ represent the $n\times n$ identity matrix and $l\times n$ zero matrix, respectively. The symbol "$\ast$" denotes a symmetric block.

2.   Preliminaries

Consider the following MJLSs with time-variant delay:

where $x(t) = col\{x_{1}(t), \, x_{2}(t), \, \ldots, \, x_{n}(t)\}$ and $y(t) = col\{y_{1}(t), \, y_{2}(t), \, \ldots, \, y_{n}(t)\}$ denote the state vectors of the drive and response system, respectively; $\hat{z}(t) = col\{\hat{z}_{1}(t), \, \hat{z}_{2}(t), \, \ldots, \, \hat{z}_{m}(t)\}$ and $\check{z}(t) = col\{\check{z} _{1}(t), \, \check{z}_{2}(t), \, \ldots, \, \check{z}_{m}(t)\}$ are the measurement output of the drive and response system, respectively; $u(t)\in \mathbb{R} ^{n}$ is the control input; $f(Dx(t)) = \left[f_{1}(d_{1}^{T}x_{1}(t))\text{, }\ldots \text{, }f_{n}(d_{n}^{T}x_{n}(t)) \right]$ and $f(Dy(t)) =$ $\left[f_{1}(d_{1}^{T}y_{1}(t))\text{, } \ldots \text{, }f_{n}(d_{n}^{T}y_{n}(t)) \right]$ are nonlinear function vectors, where $d_{i}^{T}\,$is the $i$th row of matrix D; $\omega (t)$ denotes the exterior disturbance; and $\sigma (t)$ is the time delay, which, as in ^[29,30,31], is considered to be continuous and satisfies the following constraints:

where $\sigma _{1}$ and $\sigma _{2}$ are the lower and upper bounds of the variable time delay, respectively. System matrices $A(\delta (t))$, $W(\delta (t))$, $H(\delta (t))$, $G(\delta (t))$, and $C(\delta (t))$ are known matrices with appropriate dimensions. $x(s) = \hat{\rho}(s)$ and $y(s) = \check{\rho}(s)$, $s\in \lbrack -\sigma _{2}, \, 0]$ denote the initial condition with $\sigma _{2}$ being the upper bounds of delay function $\sigma (t)$. The time-homogenous Markov jump process with right continuous trajectories is represented by $\{\delta (t)\}$ that takes values in $\Gamma = \{1, \, 2, \, \ldots, \, \tilde{\Gamma}\}$. The transition probability matrix (TPM) is characterized by $\tilde{\pi} = {\pi_{mn}}$, which is defined as follows:

where $\varphi > 0$, $\lim_{\varphi \rightarrow 0}(\frac{o(\varphi)}{\varphi }) = 0$, and $\pi _{mn}$, for all $m, n \in \Gamma$, denotes the switching rate from $m$ to $n$ with $\pi _{mn}\geq 0$, $m\neq n$, and $\pi _{mm} = -\sum_{n = 1, \, n\neq m}^{\tilde{\Gamma}}\pi _{mn} < 0$ ^[32,33]. The sampled-data controller considered in this study differs from the state-feedback controllers discussed in ^[34,35,36]. It is based on output feedback and takes into account gain fluctuations. The controller structure is given by:

where$\ K(\delta (t))$ denotes the controller gain matrix to be determined and $t_{k}$ $(k = 0, \, 1, \, 2, \, \ldots)$ is the updated instant time of the zero-order-hold ^[37]. Throughout this work, the sampling periods are considered to be bounded and time-variant, satisfying

where $h_{k}$ denotes the variable sampling interval, and scalars $h_{1}$ and $h_{2}$ denote the lower and upper bounds of $h_{k}$, respectively. $\Delta K(\delta (t))$ stands for the gain uncertainty with the following form:

where $E(\delta (t))$ and $N(\delta (t))$ are certain real constant matrices, and $\Theta$ stands for an uncertain matrix meeting $\Theta ^{T}\Theta \leq I$ ^[38,39,40]. Thus, by (2.4), (2.3) can be restructured as

For $\delta(t) = m$, $m \in \Gamma$, from (2.1), (2.2), and (2.5), we can obtain the following synchronization error system:

where $f(D\eta (t)) = f(D\eta (t)+Dy(t))-f(Dy(t))$. The nonlinear functions are supposed to adhere to one of the two distinct assumptions outlined below:

Assumption 1. There exists a matrix $L = diag\{L_{i}, \, \ldots, \, L_{n}\} > 0$ that ensures that

for any two different scalars $\eta_1, \, \eta_2 \in \mathbb{R}$.

Assumption 2. There exists a matrix $L = diag\{L_{i}, \, \ldots, \, L_{n}\} > 0$ that ensures that

for any two different scalars $\eta_1, \, \eta_2 \in \mathbb{R}$.

Remark 1. In most extant works discussing systems with nonlinear functions, Assumptions 1 or 2 are two extensively employed hypotheses (see, e.g., ^[41,42,43]). Assumption 1 imposes a condition on the nonlinear function vector, requiring only Lipschitz continuity of its components. Assumption 2 strengthens this requirement by demanding not only Lipschitz continuity but also nondecreasing monotonicity. Examples of functions that satisfy Assumption 1 but violate Assumption 2 include the cosine function $cos(t)$ and the exponential function $exp(-t^2)$.

Definition 1. ^[44] Error system (2.6) is said to be stochastically stable if, when $\omega (t)\equiv 0$,

holds true.

Definition 2. Given a scalar $\gamma > 0$, drive-response MJLSs (2.1) and (2.2) are said to be stochastically synchronized with a prescribed $\mathcal{L}_{2}-\mathcal{L}_{\infty }$ DAL $\gamma$ if error system (2.6) is stochastically stable and

holds for all nonzero $\omega (t)\in \mathcal{L}_{2}[0, \infty]$ and the zero initial condition.

The aim of this paper is to figure out a non-fragile sampled-data feedback controller in the compact form of (2.5) to ensure that drive-response MJLSs (2.1) and (2.2) are stochastically synchronized with a prescribed $\mathcal{L}_{2}-\mathcal{L}_{\infty }$ DAL $\gamma$.

The $\mathcal{L}_{2}-\mathcal{L}_{\infty }$ DAL, also called the energy-to-peak DAL, was proposed by Wilson in ^[45]. The level is a metric that quantifies the controller's performance to limit the impact of energy-bounded disturbance on the peak of the system's output.

In order to address such an issue, the following four lemmas should be employed:

Lemma 1. ^[46,47] For a given matrix $R\in \mathbb{S}_{+}^{n}$ and any differentiable function $\mu$ in $[\lambda _{1}, \, \lambda _{2}]\rightarrow \mathbb{R}^{n}$, we have

where

Lemma 2. ^[48] For a scalar $\beta \in (0, \, 1)$, matrices $\theta _{1}$ and $\theta _{2}\in \mathbb{S}_{+}^{n}$, and $\theta _{3}$ and $\theta _{4}\in \mathbb{R}^{n\times n}$, the following inequality holds true:

where $\tilde{\theta}_{1} = (\theta _{1}-\theta _{4}\theta _{2}^{-1}\theta _{4}^{T})$ and $\tilde{\theta}_{2} = (\theta _{2}-\theta _{3}^{T}\theta _{1}^{-1}\theta _{3})$.

Lemma 3. ^[49] For real matrices $R$ and $S$ of suitable dimensions and a scalar $\alpha > 0$, one has

Lemma 4. ^[50] The inequality

is equivalent to

3.   Main results

This section focuses on the non-fragile sampled-data synchronization problem for MJLSs with time-variant delay. Sufficient conditions are provided to ensure stochastic synchronization with a predefined $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ DAL for the drive-response MJLSs and the corresponding desired controller gains will be given.

To go further, we need to introduce some notations as follows:

where $\eta _{t}^{T}(s) = \eta ^{T}(t+s)$ and $e_j(s)\, (j = 1, \, \cdots, \, 4)$ are given by

For the nonlinear function of error system (2.6), we consider the following two different assumptions.

Under Assumption 1, the following inequality holds true

In this case, we can propose the following condition:

Theorem 1. Under Assumption 1, for given scalars $\gamma > 0$, $h_{2}\geq h_{1} > 0$, $u > 0$, suppose that there exist scalars $\epsilon _{m} > 0$, matrices $\tilde{P}$ in $\mathbb{S}_{+}^{5n}$, $P_{m}$, $S_{1}$, $S_{2}$, $R_{1}$, $R_{2}$, $S_{5}$, $S_{6}$ in $\mathbb{S}_{+}^{n}$, $S_{3}$ in $\mathbb{S}^{2n}$, $S_{4}$ in $\mathbb{S}^{n}$, diagonal matrix $T_{5}$ in $\mathbb{S}_{+}^{n}$, arbitrary matrices $M_{1}$, $M_{2}$ in $\mathbb{R} ^{3n\times 3n}$, $M_{3}$, $M_{4}$, $T_{1}$, $S_{7}$, $S_{8}$, $X_{m}$ in $\mathbb{R}^{n\times n}$, and $T_{3}$, $T_{4}$ in $\mathbb{R}^{19n\times 2n}$, such that

hold for $m\in \Gamma$, $h_{k}\in \{h_{1}, h_{2}\}$, and $\chi \in \{\sigma _{1}, \sigma _{2}\}$, where

Then, drive-response MJLSs (2.1) and (2.2) are stochastically synchronized with a predefined $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ DAL if the SDC gains in (2.3) are given by

Proof. For $\delta (t) = m\in \Gamma$, choose the following LKF:

with

where

In consideration of

one has

In light of

and (3.9), we can write

Define $\mathfrak{L}$ as the infinitesimal generator of random process $\{\eta (t), \, \delta (t)\}$. For each $\delta (t) = m$,

Subsequently, we can deduce that

It follows that

For the integral in (3.13), by Lemma 1, one has

Employing Jensen's inequality, for the integral items in (3.14), we have

For any matrices $M_{1}$, $M_{2}$ in $\mathbb{R}^{3n\times 3n}$ and $M_{3}$, $M_{4}$ in $\mathbb{R}^{n\times n}$, from Lemma 2 we can obtain the following inequalities:

with

Furthermore, along (2.6), using the free-weighting-matrix approach ^[51], for any matrices $T_{1}$, $T_{2}$ in $\mathbb{R}^{n\times n}$, the following is satisfied:

On the other hand, by the definitions of $\alpha _{1}(\sigma (t))$ and $\alpha _{2}(\sigma (t))$, for any matrices $T_{3}$, $T_{4}$ in $\mathbb{R} ^{19n\times 2n}$, one has

For any diagonal matrix $T_{5}$ in $\mathbb{S}_{+}^{n}$, from (3.1), the following holds:

For $t\in \left[t_{k}, \, t_{k+1}\right)$, combining the above inequalities, one obtains

where

By setting $T_{2} = uT_{1}$ and using (3.7), $\Pi _{1}^{h_{k}}\big{(}\sigma (t)\big{)}$ can be rearranged as $\tilde{\Pi} _{1}^{h_{k}}\big{(}\sigma (t)\big{)}+\mathscr{S}(\bar{E}_{m}\Theta \bar{N}_{m}^{T})$. Using Lemma 3, one has

and, thus, it can be concluded that $\tilde{\Pi}_{1}^{h_{k}}\big{(}\sigma (t)\big{)}+\mathscr{S}(\bar{E}_{m}\Theta \bar{N}_{m}^{T})$ can be guaranteed by

which is equivalent to (3.2) by Lemma 4. Since $\Pi _{1}^{h_{k}}(\cdot)$ and $\Pi _{2}^{h_{k}}(\cdot)$ are convex functions, it is easy to get $\xi ^{T}(t)\Pi _{1}^{h_{k}}\big{(}\sigma (t)\big{)}\xi (t) < 0$ from (3.2) and (3.3). Similarly, we can show that $\xi ^{T}(t)\Pi _{2}^{h_{k}}\big{(}\sigma (t)\big{)}\xi (t) < 0$ can be assured by (3.4) and (3.5). Thus, for $t\in \lbrack t_{k}, t_{k+1})$, one has

When $\omega (t)\equiv 0$, from (3.24), there exists a scalar $\alpha > 0$ such that

Using Dynkin's formula, one has

Since $\mathscr{V}_{5}(t_k) = 0$ and $\lim_{t\rightarrow t_{k}}\mathscr{V}\big{(}\eta (t), \, \delta (t), \, t\big{)} = \mathscr{V}\big{(}\eta (t_{k}), \, \delta (t_{k}), \, t_{k}\big{)}$, $\mathscr{V}\big{(}\eta (t), \, \delta (t), \, t\big{)}$ is continuous. Thus, from (3.26), one obtains

Therefore, from Definition 1, error system (2.6) is stochastically stable.

Next, we consider the case that $\omega (t)\neq 0$ and introduce a $\mathcal{L} _{2}-\mathcal{L}_{\infty }$ performance index function of error system (2.6) as follows:

Then, under the zero initial condition, using Dynkin's formula for (3.24) gives

From (3.24), one gets $I(t)\leq 0$. Thus,

Applying Lemma 4, (3.6) is equivalent to

In virtue of (3.8), (3.27), and (3.28), one has

Hence, system (2.6) has a $\mathcal{L}_{2}-\mathcal{L} _{\infty }$ performance. In this way, the proof is completed. □

On the basis of Assumption 2, the following inequalities

hold true for $i = 1, \, \ldots, \, n$.

Based on LKF:

and we can easily derive the following theorem:

Theorem 2. Under Assumption 2, for given scalars $\gamma > 0$, $h_{2}\geq h_{1} > 0$, $u > 0$, suppose that there are scalar $\epsilon _{m} > 0$, matrices $\tilde{P}$ in $\mathbb{S}_{+}^{5n}$, $P_{m}$, $S_{1}$, $S_{2}$, $R_{1}$, $R_{2}$, $S_{5}$, $S_{6}$ in $\mathbb{S}_{+}^{n}$, $S_{3}$ in $\mathbb{S}^{2n}$, $S_{4}$ in $\mathbb{S}^{n}$, diagonal matrices $T_{5}$, $T_{6}$, $\Lambda = diag\{r_{1}, \ldots, r_{n}\}$ in $\mathbb{S}_{+}^{n}$, arbitrary matrices $M_{1}$, $M_{2}$ in $\mathbb{R}^{3n\times 3n}$, $M_{3}$, $M_{4}$, $T_{1}$, $S_{7}$, $S_{8}$, $X_{m}$ in $\mathbb{R}^{n\times n}$, and $T_{3}$, $T_{4}$ in $\mathbb{R}^{19n\times 2n}$, such that the LMIs in (3.6) and

hold for $m\in \Gamma$, $h_{k}\in \{h_{1}, \, h_{2}\}$, and $\chi \in \{\sigma _{1}, \, \sigma _{2}\}$, where

Then, drive-response MJLSs (2.1) and (2.2) are stochastically synchronized with a predefined $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ DAL if the SDC gains in (2.3) are given by (3.7).

Proof. For any $t\in \left[t_{k}, \, t_{k+1}\right)$, calculate that

Further, by (3.29) and (3.30), for any diagonal matrices $T_{5}$, $T_{6}$ in $\mathbb{S}_{+}^{n}$, the following hold true:

The remainder of the proof is consistent with Theorem 1, which is omitted here.

Remark 2. On the basis of the LMI feasible solutions, Theorems 1 and 2 establish two different conditions for the desired SDC gains, which are capable of being easily verified by publicly accessible MATLAB toolboxes. For a clearer understanding of the proposed controller design, a flowchart is presented in Figure 1. The LKF term $2\sum_{i = 1}^{n}\int_{0}^{d_{i}^{T}\eta _{i}(t)}r_{i}f_{i}(s)ds$ is introduced based on Assumption 2, which has the potential to effectively mitigate conservatism. This will be confirmed by comparing the maximum allowable upper bound (MAUB) $h_{2}$ in the following section.

When there is no gain perturbation (i.e., $\Delta K(\delta (t)) = 0)$ and no mode switching (i.e., $m = 1$), (2.3) can be rewritten as $u(t) = KC\eta (t_{k})$. Correspondingly, error system (2.6) is simplified to

The following criterion can be obtained.

Corollary 1. Under Assumption 1, for given scalars $\gamma > 0$, $h_{2}\geq h_{1} > 0$, $u > 0$, suppose that there are matrices $\tilde{P}$ in $\mathbb{S}_{+}^{5n}$, $P$, $S_{1}$, $S_{2}$, $R_{1}$, $R_{2}$, $S_{5}$, $S_{6}$ in $\mathbb{S} _{+}^{n}$, $S_{3}$ in $\mathbb{S}^{2n}$, $S_{4}$ in $\mathbb{S}^{n}$, diagonal matrix $T_{5}$ in $\mathbb{S}_{+}^{n}$, arbitrary matrices $M_{1}$, $M_{2}$ in $\mathbb{R}^{3n\times 3n}$, $M_{3}$, $M_{4}$, $T_{1}$, $S_{7}$, $S_{8}$, $X$ in $\mathbb{R}^{n\times n}$, and $T_{3}$, $T_{4}$ in $\mathbb{R} ^{19n\times 2n}$, such that

hold for $h_{k}\in \{h_{1}, \, h_{2}\}$ and $\chi \in \{\sigma _{1}, \, \sigma _{2}\}$, where

Then, error system (3.36) is stochastically stable and has a predefined $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ DAL if the SDC gain is determined by $K = T_{1}^{-1}X$.

Corollary 2. Under Assumption 2, for given scalars $\gamma > 0$, $h_{2}\geq h_{1} > 0$, $u > 0$, suppose that there are matrices $\tilde{P}$ in $\mathbb{S}_{+}^{5n}$, $P$, $S_{1}$, $S_{2}$, $R_{1}$, $R_{2}$, $S_{5}$, $S_{6}$ in $\mathbb{S}_{+}^{n}$, $S_{3}$ in $\mathbb{S}^{2n}$, $S_{4}$ in $\mathbb{S}^{n}$, diagonal matrix $T_{5}$ in $\mathbb{S}_{+}^{n}$, arbitrary matrices $M_{1}$, $M_{2}$ in $\mathbb{R} ^{3n\times 3n}$, $M_{3}$, $M_{4}$, $T_{1}$, $S_{7}$, $S_{8}$, $X$, in $\mathbb{R}^{n\times n}$, and $T_{3}$, $T_{4}$ in $\mathbb{R}^{19n\times 2n}$, such that the LMIs in (3.41) and

hold for $h_{k}\in \{h_{1}, \, h_{2}\}$ and $\chi \in \{\sigma _{1}, \, \sigma _{2}\}$, where

Then, error system (3.36) is stochastically stable and has a predefined $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ DAL if the SDC gain is determined by $K = T_{1}^{-1}X$.

4.   Numerical example

Consider two-mode ($m = 1, \, 2$) drive-response MJLSs modeled in (2.1) and (2.2) with parameters (refer to ^[20]):

Consider the controller gain fluctuation (2.4) with the following parameters:

The time delay is chosen as $\sigma (t) = 0.4+\beta \left\vert \sin t\right\vert$, which implies that $\sigma _{1}$ and $\sigma _{2}$ are $0.4$ and $0.4+\beta$, respectively. In addition, the TPM here is assumed to be

In the following, we set $u = 0.5$, $h_1 = 0.001$, and consider two cases of nonlinear functions:

Case 1: $f(Dx(t)) = col\{\frac{1}{2}(\left\vert x_{1}(t)+1\right\vert -\left\vert x_{1}(t)-1\right\vert), \, 0, \, 0\}$.

In this case, it's easy to see that $f(\cdot)$ satisfies both Assumptions 1 and 2 with $L = diag\{1, \, 0, \, 0\}$. The prescribed $\mathcal{L}_{2}- \mathcal{L}_{\infty }$ performance $\gamma$ is chosen as $0.25$. In view of Theorems 1 and 2, Table 1 details MAUB $h_{2}$ when $\sigma _{1}$ is fixed and $\beta$ increases from $0.2$ to $1.8$. As can be seen from Table 1, the MAUB $h_{2}$ depends on the value of $\beta$, and $h_{2}$ in Theorem 1 is always less than $h_{2}$ in Theorem 2. Thus, in comparison to Theorem 1, Theorem 2 yields less conservative results. Furthermore, it demonstrates the effectiveness of the intentionally introduced LKF $2\sum_{i = 1}^{n}\int_{0}^{d_{i}^{T}\eta _{i}(t)}r_{i}f_{i}(s)ds$.

Case 2: $f(Dx(t)) = col\{\sin (x_{1}), \, 0, \, 0\}$.

Obviously, the nonlinear function $f(\cdot)$ satisfies Assumption 1 but does not satisfy Assumption 2 with $L = diag\{1, \, 0, \, 0\}$. Therefore, the proposed condition in Theorem 2 is no longer valid, while the one in Theorem 1 remains applicable. We set $h_{2} = 0.121$ and $\gamma = 0.25$. By solving the LMIs in Theorem 1, One has

Then, the control gain matrices can be obtained as

For the simulation, we set the initial conditions of the drive-response MJLSs to be $x(s) = col\{0.2, \, 0.3, \, 0.2\}$, $y(s) = col\{-0.3, \, -0.1, \, 0.4\}$, $s\in \lbrack -1.8, \, 0]$, the external disturbance $\omega (t) = col\{\exp (-0.5t)$, $\exp (-0.5t)$, $\exp (-0.5t)\}$, and the above parameters. The chaotic behavior of drive system (2.1) with $u(t) = 0$ is shown in Figure 2. Figures 3 and 4 depict the Markov-jump signal and sampling intervals, respectively. Under the designed SDC method, the synchronization between drive-response MJLSs (2.1) and (2.2) is achieved in Figure 5. Figure 6 depicts the trajectories of the error system (2.6). Define a new function

Its evolution versus time is shown in Figure 7. From the figure, one can observe that $\sup_{t}\mathcal{L}(t) =$ $0.0192 <$ $\gamma = 0.21$, which implies that the prescribed $\mathcal{L}_{2}- \mathcal{L}_{\infty }$ performance $\gamma$ is validity guaranteed.

5.   Conclusions

The non-fragile sampled-data synchronization control issue has been studied for MJLSs with time-variant delay. On the foundation of two different assumptions of the nonlinear function vector, two time-dependent two-sided loop LKFs (see (3.8) and (3.27)) have been constructed. By employing these two LKFs and several inequalities, numerically tractable conditions for the design of a non-fragile sampled-data controller (2.5) have been provided in Theorems 1 and 2 to guarantee the drive and response MJLSs realize stochastic synchronization with a prescribed $\mathcal{L}_{2}-\mathcal{L}_{\infty }$ DAL. Finally, an example has been given to verify the validity of the non-fragile sampled-data controller approaches. In this paper, the transition probabilities of the Markov jump process are assumed to be completely known. In practical applications, however, it is often costly or difficult to obtain all the elements in the TPM. As an extension of the present work, we will further investigate the non-fragile sampled-data synchronization control for Markov jump systems with partially unknown transition probabilities.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of the Anhui Higher Education Institutions (Grant No. 2023AH052495).

Conflict of interest

All authors declare no conflicts of interest in this paper.