Synchronization of time-delay systems with impulsive delay via an average impulsive estimation approach

1.   Introduction

When control systems produce a state change at one discrete instant suddenly, system evolutions exhibit impulsive behavior, and such systems can also be called impulsive dynamical systems, which belong to a special class of hybrid systems. Impulsive systems have attracted the attention of many researchers due to their important applications in various fields such as network control systems ^[1], financial interest rate adjustment ^[2], and pharmaceutical management ^[3]. It is well known that delays are inevitable in signal transmission and impulsive input, so the study of impulsive delay is very necessary. Furthermore, time delay systems have been widely studied in many fields, such as neural networks ^[4]. Recently, impulsive time-delay systems have been studied in some studies ^[5,6,7,8,9]. Specially, reference ^[9] obtained an implicit function related time delay and synchronization rate of impulsive systems. This function was used to reveal the potential influence of delay in continuous dynamical systems on the synchronization of the systems. However, there are few studies on synchronization of hybrid time-delay systems, which is a topic worth investigating.

In addition, impulsive control is an effective method of regulating the synchronization of time-delay systems, as it has a simple structure and makes it possible to change the state of the systems in a fraction of a second and transmit information at discrete times. This feature of impulsive control reduces communication costs to a certain extent and has been used effectively in secure communication, epidemiological model control, and biomedical systems ^{[10,11,12,13]}. Unstable or asynchronous systems tend to be stable or synchronous under impulsive control. For instance, reference ^[14] proposed a class of event-triggered impulsive controllers with time delay for system stability. The synchronization of linear dynamical systems with impulsive delay was discussed in ^[15] by pinning control. However, the interaction between the system time delay and the impulsive delay can increase the difficulty of analyzing stability and synchronization, and there are fewer relevant studies. The literature ^[16] concluded that the impulsive delay had a beneficial effect on the stability of time-delay systems through a Razumikhin-type inequality with impulsive delay. Based on the idea of average impulsive delay (AID), reference ^[17] addressed the influence of impulsive delay on time-delay systems. However, the impulsive control in most researches usually does not take into account delay, i.e., $\Delta z(t_ u) = z(t_ u)- z(t^-_ u) = A_ u z(t^-_ u)$ ^[18,19], or just consider "pure" delay, that is, $\Delta z(t_ u) = f(z(t_ u-\rho_ u)^-)$ ^[16,17,20]. We consider a more universal impulsive delay: $z(t_ u) = D_ u z (t^{-} _ u)+E_ u z ((t_ u- \rho _ u)^{-})$. Furthermore, many related works imposed strict limitations on impulsive size or impulsive interval length ^[17,19,21]. For instance, reference ^[21] required that input delays must be small enough, whereas time delays may even be longer than impulsive intervals. Reference ^[19] considered more general input delays, but required that the length of impulsive interval is subject to a common upper or lower bound. Therefore, obtaining relaxing criterion for the synchronization of impulsive time-delay systems is crucial.

On the other hand, if the time-delay system is stable or synchronous but the impulses are unstable, we refer to this situation as an impulsive perturbation ^[22,23]. However, in the vast majority of current studies on impulsive perturbation problems usually in the Lyapunov sense, requires that $V(t_ u)\leq \exp\left\lbrace \sigma\right\rbrace V((t_ u-\rho_ u)^-)$ with $\sigma > 0$ ^[17,20]. However, due to the fact that impulsive delays may not be invariable forever, it is more practical to set a flexible $\sigma_ u$ rather than setting the same threshold at every impulsive moment. Additionally impulsive delays may affect the characteristics of original system ^[20,24]. It is natural to consider the question: Can we extract the delay information of the impulse term and then integrate it into the rate coefficient $\sigma_ u$ so to ensure the influence of impulsive perturbation even though a number of $\sigma_ u < 0$?

Based on the above proposed problems, we employ the Razumikhin-type Lyapunov function to give sufficient conditions for synchronization of the impulsive system with mixed time delays, and the key contributions of this paper are as follows:

1). Differ from ^[9] proposed an implicit function, we consider the relationship between the system time delay and the rate of synchronization through a display inequality $\epsilon = q_1exp\left \{\rho \frac{\lambda \alpha \bar t-p\rho^*+\hat{\sigma }_+^* }{\bar t} \right \}+ q_2exp\left \{\mu \frac{\lambda \alpha \bar t-p\rho^*+\hat{\sigma }_+^* }{\bar t} \right \}$ which shows that the system time delay has a positive or negative effect on synchronization in this paper.

2). The size relationship between impulsive delay and impulsive interval length has no strict limitation, which was restricted in ^[16,20]. In addition, the time delay in continuous dynamics is smaller or greater than impulsive delay.

3). Through the presented concept of average impulsive estimation (AIE), the effect of impulsive perturbation is ensured by integrating the delay information obtained in impulse into impulsive estimation $\sigma_ u$, even if some $\sigma_ u$ are negative. Furthermore, the limitation of having an universal threshold for impulsive estimates at every impulsive point is eliminated. Compared with recent relevant studies ^{[16,17,20,25]}, the results in this paper are less conservative.

2.   Model description and preliminaries

2.1.   Notations

Let $\mathcal{R}$ and $\mathcal{R_+}$ denote the set of real numbers and positive real numbers, respectively. Denote the set of positive integers by $\mathcal{Z_+}$, the set of nonnegative integers by $\mathcal{Z}^0_+$, the set of $n$-dimensional real-valued vectors by $\mathcal{R}^n$, and the set of $n\times m$-dimensional real matrices by $\mathcal{R}^{n\times m}$. $\left \| \cdot \right \|$ denotes the vector Euclidean norm. Let $PC_v: = PC([-v, 0], \mathcal{R}^n)$ is the set of piece-wise right-continuous function $\phi:[-v, 0]\rightarrow \mathcal{R}_+$ with the norm $\left \| \phi \right \|_v: = \sup_{-v\le \theta \le 0} \left \| \phi (\theta) \right \|$. $D^+\hbar(\cdot)$ denotes the upper-right Dini-derivative of $\hbar(\cdot)$.

2.2.   Model

Consider a class of delayed system involved finite distributed delay:

where $\xi(t) = (\xi_1(t), \cdots, \xi_n(t))^T\in\mathcal{R}^n$ is the state vector, $C$, $A$, $B$, $\hat{B} \in\mathcal{R}^{n\times n}$ are constant system matrices, $f(\cdot)$, $\bar{f}(\cdot)\in\mathcal{R}^n$ represent the neuron activation functions and $f(0)\equiv0$ and $\bar{f}(0)\equiv0$. $\rho$ and $\mu$ denote time delay and distributed delay, respectively. $k(\cdot):[0, \mu]\rightarrow \mathcal{R}_+$ is the delay kernel satisfying $\int_{0}^{\mu }k(s)ds = 1$, $\int_{0}^{\mu }sk(s)ds < +\infty$. $v = max\{\rho, \mu\}$. Let the initial condition is $\psi\in PC_v$. Denote external input by $J$.

Consider Eq (2.1) as drive system, then corresponding response system with delayed impulsive control can be

in which $\left \{ t_ u \right \}$ is a strictly increasing sequence such that $\lim_{ u \to +\infty} t_{ u} = +\infty$ and $t_0$ is the initial time. $\zeta (t^{-})$ and $\zeta (t^{+})$ denote the left limit and right limit at time t, respectively. For this paper, let $\zeta(t)$ is right-continuous at every $t_ u$, that is, $\zeta (t^+_ u) = \zeta (t_ u)$. $\rho_ u$, $u\in \mathcal{Z}_+$, is impulse input delays satisfying $0\le \rho _ u\le t_ u-t_{ u-1}$, $\rho_0 = 0$ and $\rho_ u\le v$. Let synchronization error is $z (t) = \zeta (t)-\xi (t)$ and impulsive control input is $H(t_ u) = D_ u z (t_ u^{-})+E_ u z ((t_ u- \rho _ u)^{-})- z (t_ u^{-})$, such that $z (t_ u)$ can be expressed by

Then it is easy to obtain error system as follows:

where $g(z (t)) = f(\zeta (t))-f(\xi (t))$, $\bar g(z (t-\rho)) = \bar{f}(\zeta (t-\rho))-\bar{f}(\xi (t-\rho))$.

2.3.   Properties and definitions

Definition 1. ^[26] Suppose that there are positive numbers $N_0$ and $\bar t$, such that

where $N(\hat{t}, \check{t})$ is the number of impulsive times $\left\lbrace t_ u\right\rbrace$ occurring in $(\hat{t}, \check{t}]$, $\check{t} > \hat{t} > t_0$. Then $\bar t$ is the average impulsive interval (AII) of impulsive instant sequence $\left\lbrace t_ u\right\rbrace$ and $N_0$ is the chatter bound.

Definition 2. ^[27] Suppose that there exist positive numbers $\hat\rho _0$ and $\rho_*$ such that

where $N(t, t_0)$ is the number of impulses on the interval $(t_0, t]$, then $\rho^{*}$ is the AID of impulsive delay sequence $\left\lbrace \rho_ u\right\rbrace$.

Let $\mathscr{H}\left [\left \{ t_ u, \rho_ u \right \} \right]$ is the class consisting of impulse time sequence $\left\lbrace t_ u\right\rbrace$ satisfying AII condition Eq (2.4) and impulsive delay sequence $\left\lbrace \rho_ u\right\rbrace$ satisfying AID condition Eq (2.5).

Definition 3. ^[28] Response system (2.2) can achieve exponential synchronization with drive system (2.1) if there exist positive scalars $\chi$, $\lambda$ satisfy

Definition 4. ^[29] Function $V:\left [t_0-v, +\infty \right) \times \mathcal{R} ^n\longrightarrow \mathcal{R} _+$ belong to the class $\mathcal{V} ^*$ when following conditions are met:

(1). $V$ is continuous on each set $\left [t_{ u-1}, t_{ u} \right) \times \mathcal{R} ^{n_ z}$ and $\lim_{(t, z) \to (t_{ u}^{-}, z)} V(t, z) = V(t_{ u}^{-}, z)$ exists;

(2). $V(t, z)$ is locally Lipschitz in $z$ and $V(t, 0)\equiv 0$, $\forall t\in \mathcal{R} _+$;

(3). $V(t, z)$ satisfies $l_1\left \| z \right \|^\alpha \le V(t, z)\le l_2\left \| z \right \|^\alpha$, where $l_1, l_2, \alpha$ are positive scalars.

If $V\in \mathcal{V} ^*$ is a locally Lipschitz function, then $D^+V(t, z (0))$ along with the state trajectory of system (2.3) is defined by

in which $(t, z)\in \left [t_0, +\infty \right)\times PC_v.$

Assumption 1: For any $s\in \mathcal{R}$, $z\in \mathcal{R}$, there exist Lipschitz constants $r_i > 0$ and $\bar r_i > 0$, such that

where $i = 1, 2, \cdot\cdot\cdot, n$ and $f_{i}(0) = \bar f_{i}(0) = 0$.

3.   Main results

In this section, we obtain some sufficient conditions for exponential synchronization of systems (2.1) and (2.2).

3.1.   The case of impulsive perturbation

{From impulsive perturbations point of view, we establish some criteria for exponential synchronization.}

Theorem 1: Considering system (2.3) under Assumption 1. Suppose that there exists a function $V\in \mathcal{V} ^*$, scalars $p > 0$, $c$ with $p > c > 0,$ $q_1 = \sum_{i = 1}^{n} r_i\max_{j} \left | g_{ij} \right |$, $q_2 = \sum_{i = 1}^{n} \bar{r}_i\max_{j} \left | \hat g_{ij} \right |$, $\epsilon = q_1\exp\left\lbrace c\rho \right\rbrace +q_2\exp\left\lbrace c\mu \right\rbrace$, $\sigma _ u = \ln\left({\sum_{i = 1}^{n}\max_j\left | d_{ij}^{(u)} \right | +\sum_{i = 1}^{n}\max_j\left | e_{ij}^{(u)} \right | }\right)$, $\Gamma _ u = \frac{\sum_{i = 1}^{n}\max_j\left | d_{ij}^{(u)} \right | }{\exp\left\lbrace \sigma_ u\right\rbrace }$, $\hat\Gamma _ u = \frac{\sum_{i = 1}^{n}\max_j\left | e_{ij}^{(u)} \right | }{\exp\left\lbrace \sigma_ u\right\rbrace }$, $\hat\sigma_{+}^{0} > 0$, and $\sigma_{+}^{*} > 0$ such that for every $t\in\mathcal{R} _+$, we have

where $N(t, t_0)$ is the same as in Definition 1, and

Then drive system (2.1) can achieve exponential synchronization with response system (2.2) over the class $\mathscr{H}\left [\left \{ t_ u, \rho_ u \right \} \right]$.

Proof: Define $V(t) = V(t, z (t)) = \left \| z (t) \right \| = \sum_{i = 1}^{n}\left | z_i (t) \right |$, and $V_0 = \sup_{u\in[t_0-v, t_0]} {V(u)}$.

The proof is divided into the next three steps.

Step 1: We firstly need to prove that for some $t\in \left [t_0, t_ u \right)$ one has

where $\Omega _ u = \exp\left \{ \sum_{j = 0, p\rho_j+\sigma _j > 0}^{ u-1} (p\rho_j+\sigma _j) \right \}$.

In order to prove Eq (3.5), we construct the function

From Eq (3.5), it yields that

We will show that Eq (3.6) holds for $u = 1$, i.e.,

It is easy for us to get $\Lambda (t)\leq V_0$ for $t\in[t_0-v, t_0]$, which indicates that $\Lambda (t_0)\leq V_0$. If (3.7) is not true, then there exists $t^*\in \left (t_0, t_1 \right)$ such that $\Lambda (t^*) > V_0$, $\Lambda (t)\leq V_0$ for $t\in(t_0-v, t^*)$ and $D^+\Lambda (t)|_{t = t^*}\ge 0$. Obviously, $\Lambda(t^*) > \Lambda(t)$ for $t\in(t^*-\mu, t^*)$, then we have

From Eq (3.1) we can obtain

Hence, there is

which contradicts $D^+\Lambda (t)|_{t = t^*}\ge 0$. Then, provided that (3.6) is true when $t\in[t_0, t_ m)$, $1\leq m\leq u-1$, that is $\Lambda(t)\leq\Omega_ m V_0$, $t\in[t_0, t_ m)$. Next, we will show that (3.6) is true when $t\in[t_0, t_{ m+1})$, i.e., we need to prove that $\Lambda(t)\leq\Omega_{ m+1}V_0$, for $t\in[t_ m, t_{ m+1})$.

When $t = t_ m$, from (3.2) we can get

Therefore, Eq (3.6) holds for $t = t_ m$. Provided that Eq (3.6) is not true for $t\in(t_ m, t_{ m+1})$, then there exists $t_*\in(t_ m, t_{ m+1})$ has $\Lambda (t_*) > \Omega _{ m+1}V_0$, $\Lambda (t)\leq\Omega _{ m+1}V_0$ for $t\in(t_0-v, t_*)$ and $D^+\Lambda (t)|_{t = t_*}\ge 0$. Similar with the argument used in Eq (3.7), we can obtain $D^+\Lambda (t)|_{t = t_*} < 0$, it contradicts $D^+\Lambda (t)|_{t = t_*}\ge 0$. Hence, we can found Eq (3.6) holds through using mathematical induction, in which $t\in[t_0, t_ u)$, which implies Eq (3.5) holds for $t\in[t_0, t_ u)$, $u\in \mathcal{Z} _+$.

Step 2: According to Eq (3.5), condition Eqs (2.4), (2.5) and (3.3), we have

where $t\geq t_0$.

Step 3: Based on condition Eqs (3.4), (3.9) and Assumption $l_1\left \| z \right \|^\alpha \le V(t, z)\le l_2\left \| z \right \|^\alpha$, which can derive that

where $\chi = \left(\frac{l_2}{l_1}\exp\left\lbrace pN_0\rho^*+p\hat\rho_0+N_0\sigma ^*_++\hat\sigma ^0_+\right\rbrace \right) ^{\frac{1}{\alpha } }$, $\lambda = \frac{c\bar t-p\rho^*-\hat\sigma ^*_+}{\alpha \bar t}$, which implies the system (2.2) can be exponentially synchronized with System (2.1) over the class $\mathscr{H}\left [\left \{ t_ u, \rho_ u \right \} \right]$.

Remark 1: System under consideration with hybrid delayed impulses is discussed in Theorem 1 using the Lyapunov–Razumikhin method. Condition Eq (3.1) describes the continuous evolution of the considered system, and it follows from $a > 0$ that the continuous dynamics are stabilizing. Condition Eq (3.2) overviews the impulsive effect. In the case where $\Gamma_ u = 0$, $\hat\Gamma_ u = 1$, Eq (3.2) simplifies to $U(t_ u, h(z))\leq \exp\left\lbrace d\right\rbrace U(\pi, z)$, where $\pi = t_ u-\rho_ u$ ^[20]. In the case where $\Gamma_ u = 1$, $\hat\Gamma_ u = 0$, Eq (3.2) simplifies to $U(t_ u, h_ u(z(0)))\leq \exp\left\lbrace d\right\rbrace U(t^-_ u, z(0))$ ^[7].

Remark 2: According to the derivation condition $\epsilon = q_1\exp\left\lbrace c\rho \right\rbrace +q_2\exp\left\lbrace c\mu \right\rbrace$ of Theorem 1, it can be learnt that the parameter $c$ will increase with the decrease of $\rho$ and $\mu$. Furthermore, the synchronization rate $\lambda = \frac{c\bar t-p\rho^*-\hat\sigma ^*_+}{\alpha \bar t}$ will increase. Therefore we have $\epsilon = q_1exp\left \{\rho \frac{\lambda \alpha \bar t-p\rho^*+\hat{\sigma }_+^* }{\bar t} \right \}+ q_2exp\left \{\mu \frac{\lambda \alpha \bar t-p\rho^*+\hat{\sigma }_+^* }{\bar t} \right \}$. This means that, in some cases, system delays $\rho$ might have potentially negative impact on the synchronization between the systems (2.1) and (2.2).

Remark 3: According to condition Eq (3.2), $\sigma_ u$ is called as impulsive estimate. In order to analyze the function of the impulsive estimation sequence $\left\lbrace \sigma_ u\right\rbrace$, the concepts of average positive impulsive estimation (APIE) and AIE are introduced in this paper.

Assuming the existence of scalars $\hat\sigma^0_+ > 0$ and $\sigma^*_+ > 0$ satisfying condition Eq (3.3), $\sigma^*_+$ is called as APIE.

Supposing that there are some $\hat\sigma^0 > 0$ and $\sigma^* > 0$ such that

where $N(t, t_0)$ is given in Definition 2, thus $\sigma ^{*}$ is referred to as AIE of the impulsive estimation sequence $\left\lbrace \sigma_ u\right\rbrace$. When $\sigma_ u > 0$, there is $\sigma^* = \sigma^*_+$. Since $t_0$ does not act as an impulse point, we assume that $\sigma_0 = 0$.

Remark 4: Actually, there exist some results about impulsive delays. In ^[30,31], the time delays had to have strict upper and lower bounds, or to be smaller than the length of impulse interval ^[16,20]. Even if ^[19,21,32] relaxed impulsive delays, impulsive interval should meet that $\inf_{ u\in \mathcal{Z} _+}\left\lbrace t_ u-t_{ u-1} \right\rbrace\ge\varrho$ or $\sup_{ u\in \mathcal{Z} _+}\left\lbrace t_ u-t_{ u-1} \right\rbrace\leq\varrho$ for some $\varrho\ge0$. The length of impulsive interval is flexible in Theorem 1.

When the delays of the continuous dynamics are not to be considered, systems (2.1) and (2.2) can be represented as

and the resulting error system for systems (1) and (2) follows as

Corollary 1: Suppose that Assumption 1 holds. If there exists a function $V\in \mathcal{V} ^*$, scalars $p > 0$, $c$ with $p > c > 0$, $\sigma _ u = \ln\left({\sum_{i = 1}^{n}\max_j\left | d_{ij}^{(u)} \right | +\sum_{i = 1}^{n}\max_j\left | e_{ij}^{(u)} \right | }\right)$, $\sigma_ u$ with $p\rho_ u+\sigma_ u > 0$, $\sigma^* > 0$ satisfying condition Eq (3.10), $\Gamma _ u = \frac{\sum_{i = 1}^{n}\max_j\left | d_{ij}^{(u)} \right | }{\exp(\sigma_ u) }$, $\hat\Gamma _ u = \frac{\sum_{i = 1}^{n}\max_j\left | e_{ij}^{(u)} \right | }{\exp(\sigma_ u) }$. In that case the exponential synchronization of systems (3.11) and (3.12) can be achieved while Eq (3.1) holds with $\epsilon = q_1 = q_2 = 0$ and Eq (3.2) holds, and

Remark 5: Corollary 1 offers some criteria for exponential synchronization between system (2.1) and system (2.2) from the point of view of impulsive perturbation, which lowers the limitation on $\sigma_ u > 0$. Most of previous works ^[17,20,25] need $\sigma_ u < 0$ in the impulsive control case and $\sigma_ u > 0$ in the impulsive perturbation case. Corollary 1 presents condition $p\rho_ u+\sigma_ u > 0$ which makes $\sigma_ u$ is flexible. If $\sigma_ u > 0$, $p\rho_ u+\sigma_ u > 0$ always holds. If $\sigma_ u < 0$, we just need $\rho_ u > -\frac{\sigma_ u}{p }$ which ensures above condition to hold. It's worth noting that, in the impulsive perturbation problem, the smaller $\sigma_ u$ is, $\rho_ u$ must be larger to compensate.

3.2.   The case of impulsive control

In this subsection, from the perspective of impulsive control, we establish a number of criteria of exponential synchronization based on the concepts of AID and AIE. Moreover, we assume that $t_ u-t_{ u-1}\ge \rho > \rho_ u$ and $t_ u-t_{ u-1}\ge \eta > \rho_ u$, $u\in \mathcal{Z} _+$.

Theorem 2: Considering system (2.3) under Assumption 1. Suppose that there exists a function $V\in \mathcal{V} ^*$, scalars $b_1 = \max_{i} c_{ii}+\sum_{ i = 1}^{n}\max_{j, j\ne i}\left | c_{ij} \right |+ \sum_{ i = 1}^{n}\max_{j}\left | a_{ij} \right |$, $b_2 = \sum_{i = 1}^{n} r_i\max_{j} \left | g_{ij} \right |$ and $b_3 = \sum_{i = 1}^{n} \bar{r}_i\max_{j} \left | \hat g_{ij} \right |$, $\sigma _ u = -\ln\left({\sum_{i = 1}^{n}\max_j\left | d_{ij}^{(u)} \right | +\sum_{i = 1}^{n}\max_j\left | e_{ij}^{(u)} \right | }\right)$ with $\bar\sigma = \sup_{ u\in \mathcal{Z} _+}\sigma _ u > 0$, $\gamma > 0$ such that $\gamma > b_1+b_2\exp\left\lbrace \bar\sigma\right\rbrace +b_3\exp\left\lbrace \bar\sigma\right\rbrace$, $\Gamma _ u = \frac{\sum_{i = 1}^{n}\max_j\left | d_{ij}^{(u)} \right | }{\exp\left\lbrace \sigma_ u\right\rbrace }$, $\hat\Gamma _ u = \frac{\sum_{i = 1}^{n}\max_j\left | e_{ij}^{(u)} \right | }{\exp\left\lbrace \sigma_ u\right\rbrace }$, the impulsive estimation sequence $\left\lbrace \sigma_ u\right\rbrace$ satisfies the condition Eq (3.10), for every $t > 0$, the following inequalities hold:

Then, drive system (2.1) can achieve exponential synchronization with response system (2.2) over the class $\mathscr{H}\left [\left \{ t_ u, \rho_ u \right \} \right]$.

Proof: Define $V(t) = V(t, z (t)) = \left \| z (t) \right \| = \sum_{i = 1}^{n}\left | z_i (t) \right |$, and $V_0 = \sup_{u\in[t_0-v, t_0]} {V(u)}$, such that

The proof is divided into three steps.

Step 1 : We will demonstrate that

where $\Omega _ u = \exp\left \{ -\sum_{j = 0}^{ u}\sigma _j \right \}$.

First, we need show that the following two situations for $u\in \mathcal{Z} ^0_+$.

(ⅰ). If $t^\bigtriangleup\in\left[t_0, t_1 \right)$, one has

(ⅱ). If $t^\bigtriangleup\in\left[t_ u, t_{ u-1} \right)$, $u\in Z_+$, one can obtain that

and

Then $D^+\Theta (t)|_{t = t^\bigtriangleup} < 0$, where

Construct an auxiliary function with $\iota > 0$

Without loss of generality, we assume $\rho\geq\eta$. First, provided that Eq (3.15) holds.

If $t^\bigtriangleup-\eta\geq t^\bigtriangleup-\rho\geq t_0$, based on Eq (3.16), one has

If $t^\bigtriangleup-\rho < t_0 < t^\bigtriangleup-\eta$ or $t^\bigtriangleup-\rho\leq t^\bigtriangleup-\eta < t_0$, we can derive similarly that

Next, suppose that Eqs (3.16) and (3.17) hold.

If $t^\bigtriangleup-\eta\ge t^\bigtriangleup-\rho\ge t_ u$, using Eq (3.16) we can found

If $t^\bigtriangleup-\rho < t_ u\leq t^\bigtriangleup-\eta$, by the fact $t_{ u-1}\leq t^\bigtriangleup-\rho < t_ u$, Eqs (3.16) and (3.17), it leads to

If $t^\bigtriangleup-\rho \leq t^\bigtriangleup-\eta < t_ u$, because of $t_{ u-1}\leq t^\bigtriangleup-\rho < t_ u$, Eqs (3.16) and (3.17), one has

According to the above situations and $\gamma > b_1 +b_2\exp(\bar\sigma)+b_3\exp(\bar\sigma)$, one can obtain that

It can be further deduced that

Then, we shall show that

We can easily get $\Theta(t)\le V_0$ when $t\in\left[t_0-v, t_0\right]$, so that $\Theta(t_0)\le V_0$. Suppose Eq (3.18) is false for $u$ = 0, then there exists $t_\bigtriangledown\in(t_0, t_1)$ makes $\Theta(t_\bigtriangledown) > V_0$, $\Theta(t)\leq V_0$ for $t\in (t_0-v, t_\bigtriangledown)$ and $D^+\Theta(t)|_{t = t_\bigtriangledown}\geq0$, which is contrary to $D^+\Theta(t)|_{t = t_\bigtriangledown} < 0$ in previous discussion. Provided that (3.18) holds for $u\leq U$, next we will prove that Eq (3.18) holds for $u = U+1$. Based on $\Theta(t)\leq\Omega_ U V_0exp\left\lbrace \gamma(t_ U-t_0)\right\rbrace$, $t\in\left[t_ U, t_{ U+1}\right)$, we have

Therefore, Eq (3.18) holds for $t = t_{ U+1}$. Suppose that there are some $t\in \left(t_{ U+1}, t_{ U+2} \right)$ which leads to $\Theta(t)\geq\Omega_ U V_0\exp\left\lbrace \gamma(t_ u-t_0)\right\rbrace$, through the continuity of $V(t)$ in $\left(t_{ U+1}, t_{ U+2} \right)$, we can get $\hat t\in\left(t_{ U+1}, t_{ U+2} \right)$ such that $\Theta(\hat t) = \Omega_{ U+1}V_0\exp\left\lbrace \gamma(t_{ U+1}-t_0)\right\rbrace$, $\Theta(t) < \Omega_{ U+1}V_0\exp\left\lbrace \gamma(t_{ U+1}-t_0)\right\rbrace$, $t\in\left(t_{ U+1}, \hat t \ \right)$ and $D^+\Theta(t)|_{t = \hat t}\geq0$. When $s\in\left[t_{ U}, t_{ U+1} \right)$, it leads to

Thus, it follows from Eqs (3.16) and (3.17) that $D^+\Theta(t)|_{t = \hat t} < 0$, which is a contradiction with $D^+\Theta(t)|_{t = \hat t}\geq0$.

Then we can conclude that Eq (3.18) is true through using mathematical induction.

Step 2 : From Eqs (3.8) and (3.10), it can be derived that

Based on condition $\gamma\bar t-\sigma^* < 0$, inequality (3.19) and Assumption $l_1\left \| z \right \|^\alpha \le V(t, z)\le l_2\left \| z \right \|^\alpha$, one has

where $\chi = \left(\frac{l_2}{l_1}\exp\left\lbrace \sigma ^*N_0+\hat \sigma ^0\right\rbrace \right) ^{\frac{1}{\alpha } }$, $\lambda = \frac{\sigma^*-\gamma\bar t}{\alpha \bar t}$, which implies the system (2.2) can be exponentially synchronized with system (2.1) over the class $\mathscr{H}\left [\left \{ t_m, \rho_m \right \} \right]$.

Remark 6: Theorem 2 presents some criteria for exponential synchronization between systems (2.1) and (2.2) from the point of view of impulsive control. Compared to previous studies ^{[16,17,18,20]}, which have a common threshold for $\sigma > 0$ at each impulsive point, the results in this paper are less conservative and the rate coefficient $\sigma_ u$ is flexible here through the proposed concept of AIE.

4.   Numerical examples

In this section, we give illustrative examples to show the effectiveness of the obtained results.

Example 1: We consider the error system (2.3) with parameters as

$k(s) = \frac{1}{\mu }$, $0\leq\rho_ u\leq\bar\rho$, $v: = \max\left \{ \bar\rho, \rho, \mu \right \}$, $f(\cdot) = \bar f(\cdot) = 0.3tanh(\cdot)$, $\rho = 0.1$, $\mu = 0.4$, $t_ u = 0.7 u$ and

where $m\in\mathcal{Z} _+$, the matrices $D_ u$, $E_ u$ can be chosen by

$\ \ \ \ D_ u = \begin{bmatrix} 0.1 & 0 \\ 0 & 0.1 \end{bmatrix}$, $E_ u = \begin{bmatrix} 0.85 & 0 \\ 0 & 0.85 \end{bmatrix}$, when $\rho_ u = 0$;

$\ \ \ \ D_ u = \begin{bmatrix} 0.28 & 0 \\ 0 & 0.28 \end{bmatrix}$, $E_ u = \begin{bmatrix} 0.85 & 0 \\ 0 & 0.85 \end{bmatrix}$, when $\rho_ u = 0.1$;

$\ \ \ \ D_ u = \begin{bmatrix} 0.35 & 0 \\ 0 & 0.35 \end{bmatrix}$, $E_ u = \begin{bmatrix} 0.45 & 0 \\ 0 & 0.45 \end{bmatrix}$, when $\rho_ u = 1.1$.

Let $V(t) = \left \| z (t) \right \|$ in system (2.3), $c = 0.2$, $\epsilon = 0.7519$, it can be derived that $p = 0.2641$, and

Therefore, we can get $\bar t = 0.7$, $\rho^* = 0.4$, $\sigma_+^* = 0.0407$. Furthermore it can be obtained that $-c\bar t+p\rho^*+\sigma_+^* = -0.08166 < 0$. By Theorem 1, drive system (2.1) can achieve exponential synchronization with response system (2.2) over the class $\mathscr{H}\left [\left \{ t_ u, \rho_ u \right \} \right]$, see Figure 1.

If the delays $\rho$, $\mu$ and the rate coefficient $\rho$ are chosen as $\rho = 0.8$, $\mu = 1.4$ and $\rho = 0.3122$. We can find that all conditions in Theorem 1 are satisfied so that drive system (2.1) can achieve exponential synchronization with response system (2.2), see Figure 2. According to Remark 3, synchronization rate drops as the delay $\rho$ or $\mu$ grows, which agrees well with the simulation result in Figure 2.

If $D_ u$, $E_ u$ are selected as

$\ \ \ \ \ D_ u = \begin{bmatrix} 0.36 & 0 \\ 0 & 0.35 \end{bmatrix}$, $E_ u = \begin{bmatrix} 1.31 & 0 \\ 0 & 1.46 \end{bmatrix}$, when $\rho_ u = 0$;

$\ \ \ \ D_ u = \begin{bmatrix} 0.51 & 0 \\ 0 & 0.51 \end{bmatrix}$, $E_ u = \begin{bmatrix} 0.86 & 0 \\ 0 & 0.86 \end{bmatrix}$, when $\rho_ u = 0.1$;

$\ \ \ \ D_ u = \begin{bmatrix} 0.31 & 0 \\ 0 & 0.31 \end{bmatrix}$, $E_ u = \begin{bmatrix} 0.82 & 0 \\ 0 & 0.82 \end{bmatrix}$, when $\rho_ u = 1.1$,

where $u\in Z_+$, then by calculation we have

and $\sigma_ u = 0.3452$. Under this circumstance, $-c\bar t+p\rho^*+\sigma_+^* = 0.2228 > 0$, this is contrary to condition Eq (3.4). Therefore, system (2.2) may not be able to achieve exponential synchronization with system (2.1), see Figure 3.

Example 2: We consider the following error system:

where $k(s) = \frac{1}{1-e^{-\mu }} e^{-s}$, $\rho = 0.8$, $\mu = 0.4$, $t_ u = u$ and $\tau_ u$, $c_ u$, $d_ u$ can be selected by

Let $V(t) = \left \| z (t) \right \|$ in system (4.1), there are

and $\sigma^* = 0.4326$, $\bar t = 1$. Then we can choose $\gamma = 0.42 > 0.3+0.03+0.08$ makes condition $\gamma\bar t-\sigma^* = -0.0126 < 0$ is established. From Theorem 1, drive system (2.1) can achieve exponential synchronization with response system (2.2) over the class $\mathscr{H}\left [\left \{ t_ u, \rho_ u \right \} \right]$, see Figure 4 (solid red line). It is clear that, owing to the idea of AIE, it is not necessary that $\sigma_ u$ is positive for every $u\in \mathcal{Z} _+$.

If $d_ u$, $e_ u$ are

then it can be figured out that

therefore, $\sigma^* = 0.1246$. Under this circumstance, $\gamma\bar t-\sigma^* = 0.2954 > 0$, which contradicts one of the conditions of Theorem 2, i.e., $\gamma\bar t-\sigma^* < 0$. Consequently, system (2.2) may not be able to achieve exponential synchronization with system (2.1), see Figure 4 (dotted blue line).

5.   Conclusions

In this paper, we have explored the issue of synchronization for a class of impulsive systems with discrete and distributed delay. Sufficient Lyapunov conditions for the synchronization of the considered system under impulse perturbation and impulse control are established, respectively. It is worth noting that the concepts of AIE and APIE, which are presented in this paper, make impulsive estimation more flexible and relax the constraint of a common threshold. Theoretical results show that time delay size of continuous dynamics is variable and does not have a strict magnitude relationship with impulsive delay. In addition, the obtained display inequalities indicate that time delay of a continuous system may have a potential effect on synchronization. However, from the perspective of impulsive control, there exists a limitation between system time delay size and impulsive interval length, which is an issue to be discussed in subsequent work.

Use of AI tools declaration

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

Conflict of interest

The authors declare that there are no conflicts of interest.