In this paper, the fixed-time pinning synchronization problem of an intermittently coupled complex network is investigated. An intermittently coupled complex network with delay is presented for the first time. A new fixed-time stability lemma is developed, which is less conservative than the existing results. A more economical controller is designed under intermittent pinning control strategy. Sufficient conditions are developed to realize fixed-time synchronization. Numerical simulations are conducted to verify the effectiveness and feasibility of the obtained results.
Hinweise
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
1 Introduction
During the last decades, complex networks have been employed to model multitudinous real natural and artificial systems, such as biological network [1], neural network [2], scientific network [3], communication network [4], etc. The dynamical behaviors of complex network mainly contain structural identification [5], diffusion [6], consensus [7], and synchronization [8]. Among them, the network synchronization has been massively researched by numerous scholars because of its precise interpretation of diverse natural phenomena and numerous potential applications in practical systems.
In fact, it is difficult for complex networks to realize self-synchronization through the coupling among network nodes. Fortunately, scholars proposed many productive control methods to reach synchronization, such as impulse control [9, 10], adaptive control [11, 12], feedback control [8, 13] and so on [14‐19]. Since real-world complex systems typically involve massive nodes and edges, it is a natural idea to decrease the number of controlled nodes to avoid controlling excessive nodes. Pinning control can apply some local feedback input to a fraction of nodes, and the state of other nodes is changed through the interaction among nodes. There are plenty of research achievements [20‐23] in this field, and it has received widespread application.
Anzeige
In the above control strategies, control actions are continuous, which may be infeasible in some practical systems with finite transmission bandwidth. In order to overcome this defectiveness, discontinuous control methods are produced. Particularly, intermittent control is classified as a discontinuous control strategy. It was first proposed by Zochowski in 2000 [24] and has already got close concern due to its low control cost and extensive application in engineering. A control period of intermittent control strategy is decomposed into rest time and work time. That is to say, it is deactivated during the rest time and activated during work time. However, it is conservative and unrealistic for periodicity intermittent control. Aperiodically intermittent strategy has broader applications due to its handy implementation. For instance, the intermittent control of wind power generation is obviously aperiodic [25]. Utilizing aperiodically intermittent control strategies, numerous achievements are obtained about synchronization in complex networks [26‐28].
It is noted that the above-proposed results are concerning on the asymptotic or exponential synchronization, in which the complex networks achieve synchronization in infinite horizon. In fact, the dynamical networks of practical engineering need to synchronize in a finite time [29]. Together with the faster convergence speed, interference suppression, and strong robustness, finite-time synchronization became popular. Nevertheless, the initial conditions of the considered systems are always not finite or known in advance, so the expected setting time could not be estimated. Under this circumstance, fixed-time synchronization [23, 30, 31] have been introduced to address the above problem, which can promise that settling time is not relevant to any incipient value. By employing different controllers [32], Xu Yuhua studied the fixed-time synchronization of complex systems. In [33], the mixed stochastic complex network with delay realized fixed-time synchronization by the proposed intermittent controller.
However, most intermittent controllers can essentially be regarded as semi-intermittent control strategies in the proposes of achieving fixed-time synchronization, in which a part of the controller must be activated constantly, such as the linear control term. The scheme of semi-intermittent control is inappropriate in the system of signal interruption. In [34‐37], the authors put forward a complete intermittent controller and achieved finite-time synchronization. Recently, complete intermittent controllers were used in quaternion-valued neural networks [36], complex-valued networks [37], and other kinds of systems to realize finite-time synchronization. Nevertheless, it will be difficult to solve the issues of fixed-time synchronization by the complete intermittent control because fixed-time control would involve sophisticated nonlinear analysis. Fortunately, a complete feasible intermittent controller was devised to reach fixed-time synchronization in [38].
Nevertheless, it is noted that the above-mentioned results are about continuously coupled networks. There are interferences such as delay, noise, and so on in the application of the network. Different components in the system may not always be connected, resulting in intermittent coupling between network nodes, that is, the topology of the network is dynamically changing. As discontinuous coupling networks, pulse-coupled networks [39] were introduced into the study of complex networks in 2008, in which coupling exists only in isolated moments and at other times, nodes are considered not interconnected. In addition to transient information communication, there is also intermittent information communication between different components. For example, certain multi-agent systems are required to exchange information within a predetermined time frame, such as memory resistor-based circuits. And in some multi-agent systems, each agent only shares the information with other agents on some disconnected time spans due to the limitation of sensing ranges, communication obstacles and equipment failures. Figure 1 depicts the intermittent coupling mechanism in the presented system: for any time span \((t_{2k} ,t_{2k + 2} ]\), \((t_{2k} ,t_{2k + 1} ]\) is the coupling time; and \((t_{2k + 1} ,t_{2k + 2} ]\) is the decoupling time. There is interaction among neighbors in communication or coupling time; in decoupling time, the communication among nodes is disappearing. In contrast with continuous coupling networks, the decoupling time of intermittent coupling is beneficial for the cooperation and communication of different nodes. Different from pulse coupled networks, information sharing among nodes occurs throughout all the coupling time interval, instead of at certain discrete time points. Nevertheless, there are few achievements about intermittently coupled networks. In intermittently coupled complex networks, Hu considered synchronization by using an adaptive controller [40, 41]. It is certainly worth exploring the fixed-time synchronization problems in an intermittently coupled network.
Fig. 1
The sketch of simplified intermittent coupling
×
Anzeige
Based on the objectivity of intermittent coupling and the differences with continuous or impulsive coupling, there are the following interesting and meaningful challenges: How to model intermittently coupled dynamical networks? How to design a complete intermittent and economical controller to achieve fixed-time synchronization for intermittently coupled networks? Unfortunately, there seem to be very few reported results at present to explore these challenging problems of intermittently coupled dynamical networks.
Inspired by the aforementioned analysis, we study the fixed-time pinning synchronization problem of an intermittently coupled complex network. The highlights are primarily captured in the following three folds:
(1)
Different from traditional continuously coupled complex network [30, 31, 36], an intermittently coupled dynamical network with time-varying delay is constructed in this paper. An index function \(\delta_{k} (t)\) is introduced to describe the discontinuities of coupling between nodes.
(2)
A new fixed-time stability lemma with complete aperiodically intermittent characteristic is proposed. The settling time is irrelevant to the initial values of our network systems and has superiority in increasing convergence speed and reducing convergence time than existing results.
(3)
An economical intermittent controller is proposed to reach fixed-time synchronization. A function \(sig( \cdot )\) is introduced to simplify the controller and the pinning method is considered to avoid imposing controllers to all node, which can reduce more control costs.
Notations: The set of nonnegative integers are represented by \(N_{ + }\), \(R^{n}\) and \(R^{n \times m}\) stand for \({\text{n}}\)-dimensional real column vector space and \(n \times m\) real matrices, individually. In describes unit matrix with n dimensions. \(\lambda_{\min } (A)\) is appointed to be the minimum eigenvalue of \(A \in R^{n \times m}\). \(\left\| x \right\|\) denotes the 2-norm of \(x \in R^{n}\). \({\text{diag}}\{ c_{1} ,c_{2} ,...,c_{n} \}\) describes the diagonal matrix, where ci denotes the \(i\) th diagonal element. Symbol \(\otimes\) denotes Kronecker product. \(C([ - \tau ,0],R^{n} )\) means continuous function space.
2 Model Description and Preliminaries
Considering a kind of intermittently coupled complex network, it can be depicted by
where \(x_{i} = (x_{i1} (t),x_{i2} (t),x_{i3} (t)\;...,x_{in} (t))^{T} \in R^{n}\) is the state vectors, nonlinear vector function \(f:R^{ + } \times R^{n} \times R^{n} \to R^{n}\) represents the node dynamics, time-varying function \(d(t)\) is time-varying delay appearing inside the dynamical nodes and meets \(0 \le d(t) \le d\), \(c > 0\) represents the coupling weight. \(B = [b_{ij} ] \in R^{N \times N}\) denotes weighted adjacency matrix. It satisfies: for \(i \ne j\), assume the \({\text{j}}\) th and \({\text{i}}\) th node are connect by an edge, it establishes \(b_{ij} = b_{ji} > 0\), or else, \(b_{ij} = b_{ji} = 0\); for \(i = j\), it establishes \(b_{ii} = 0\). \(u_{i} (t)\) is the controller for the \({\text{i}}\) th node. \(L = [l_{ij} ] \in R^{N \times N}\) is the corresponding Laplacian matrix, which is established by \(l_{ii} = \sum\nolimits_{j = 1,j \ne i}^{N} {b_{ij} }\), and \(l_{ij} = - b_{ij} (i \ne j)\).
The index function \(\delta_{k} (t)\) is defined as
The numerous excellent results of network synchronization are extensively concentrated on continuous coupling networks. This article investigates a new complex network with discontinuous coupling, namely intermittent coupling [40, 41]. The discontinuities of coupling between nodes are represented by the index function \(\delta_{k} (t)\). When \(\delta_{k} (t) = 1\), there are coupling among nodes, while \(\delta_{k} (t) = 0\), the coupling among nodes disappears. Moreover, \([t_{2k} ,t_{2k + 1} )\) and \([t_{2k + 1} ,t_{2k + 2} )\) are used to represent the time interval instead of \([t_{k} ,s_{k} )\) or \([s_{k} ,t_{k + 1} )\) ( in the time span \([t_{k} ,t_{k + 1} )\), \([t_{k} ,s_{k} )\) is the work time and \([s_{k} ,t_{k + 1} )\) is the rest time). The main purpose is that it will make the expression for setting time solutions more concise.
Accordingly, the isolated node performs the following dynamics.
$$ \dot{s}(t) = f(t,s(t),s(t - d(t))) $$
(3)
in which \(s(t) = (s_{1} (t)\;s_{2} (t)\;...\;s_{n} (t))^{T} \in R^{n}\).
The error system is defined to be \(e_{i} (t) = x_{i} (t) - s(t)\), then we have
in which \(\overline{f}(t,e_{i} (t),e_{i} (t - d(t))) = f(t,x_{i} (t),x_{i} (t - d(t))) - f(t,s(t),s(t - d(t)))\).
Definition 1
In our intermittently coupled complex network, if there can find a setting time Tf, which meets \(\mathop {\lim }\limits_{{t \to T_{f} }} \left\| {e_{i} (t)} \right\| = 0\) and \(e_{i} (t) = 0\), for \(\forall t \ge T_{f}\), where \(T_{f} \le T_{\max }\), and regardless of the initial conditions of (1) and (3), \(T_{\max }\) is not dependent on the incipient values in the system, then the fixed-time synchronization is achieved.
Assumption 1
For vector function \( y_{1} (t),y{\ominus }_{2} (t) \in R^{n} \), when \(t \in [0, + \infty )\), there exist.
in which both of \(\chi\), \(\gamma\) are known positive constant.
Assumption 2
The time-delay function \(d(t)\) conforms to \(\dot{d}(t) \le \tilde{d} < 1\).
Assumption 3
[38] In the pinning scheme, the isolated system can reach any nodes of network (1).
In order to determine how many nodes are pinned, a matrix \(\ell = diag\{ \ell_{1} ,\ell_{2} ,...,\ell_{n} \}\) is introduced, in which the element is defined as: if the ith node is pinned, \(\ell_{i} > 0\); otherwise \(\ell_{i} = 0\). If Assumption 3 holds, \(\Gamma_{\ell } = L + \ell\) is positive-definite matrix, and we have \(\lambda_{\min } (\Gamma_{\ell } ) > 0\).
To present the controller more concisely, a new function \(w_{i} (t)\) is introduced as:
There are four parts within working intervals: (1) the first part \(- c\ell_{i} e_{i} (t)\) stands for linear feedback term to satisfy Lyapunov stability; (2) the second part \(- c(\frac{\zeta }{{1 - \tilde{d}}}\int_{t - d(t)}^{t} {w_{i}^{T} (s)w_{i} (s)ds} )\frac{{w_{i} (t)}}{{\left\| {w_{i} (t)} \right\|^{2} }}\) is introduced to offset the impact from time delay; (3) the third part \(- \alpha_{1} sig^{{p_{1} }} (w_{i} (t))\)\(- \alpha_{1} (\frac{\zeta }{{1 - \tilde{d}}}\int_{t - d(t)}^{t} {w_{i}^{T} (s)w_{i} (s)ds} )^{{\frac{{p_{1} + 1}}{2}}} \frac{{w_{i} (t)}}{{\left\| {w_{i} (t)} \right\|^{2} }}\) is considered to drive synchronization error \(e_{i} (t)\) from 1 to 0; and (4) the last part \(- \alpha_{2} sig^{{p_{2} }} (w_{i} (t)) - \alpha_{2} (\frac{\zeta }{{1 - \tilde{d}}}\int_{t - d(t)}^{t} {w_{i}^{T} (s)w_{i} (s)ds} )^{{\frac{{p_{2} + 1}}{2}}} \frac{{w_{i} (t)}}{{\left\| {w_{i} (t)} \right\|^{2} }}\) can help ei(t) reach to 1, when \(e_{i} (t) > 1\). From (7), when \(t \in \left[ {t_{2k} ,t_{2k + 1} } \right)\), the controller is rewritten as: \(- c\ell_{i} e_{i} (t) - c(\frac{\zeta }{{1 - \tilde{d}}}\int_{t - d(t)}^{t} {w_{i}^{T} (s)w_{i} (s)ds} )\frac{{w_{i} (t)}}{{\left\| {w_{i} (t)} \right\|^{2} }} - \alpha sig^{\mu } (w_{i} (t)) - \alpha (\frac{\zeta }{{1 - \tilde{d}}}\int_{t - d(t)}^{t} {w_{i}^{T} (s)w_{i} (s)ds} )^{{\frac{\mu + 1}{2}}} \frac{{w_{i} (t)}}{{\left\| {w_{i} (t)} \right\|^{2} }}\). When \(e^{T} (t)(\Gamma_{\ell } \otimes I_{n} )e(t) + \frac{\zeta }{{1 - \tilde{d}}}\int_{t - d(t)}^{t} {e^{T} (s)} (\Gamma_{\ell } \otimes I_{n} )e(s)ds > 1\), it is known that \(sig( \cdot ) = 1\), and \(\mu = r + 1 > 1\), it can help \(e_{i} (t)\) reach to 1; when \(e^{T} (t)(\Gamma_{\ell } \otimes I_{n} )e(t) + \frac{\zeta }{{1 - \tilde{d}}}\int_{t - d(t)}^{t} {e^{T} (s)} (\Gamma_{\ell } \otimes I_{n} )e(s)ds\)\(< 1\), it implies \(sig( \cdot ) = - 1\), and \(0 < \mu = r - 1 < 1\), it can help ei(t) from 1 to 0. It can be intuitively known that the proposed controller has fewer terms and is simpler than (8). Compared with the economic controller proposed in [30, 31], on the one hand, the developed controller in this paper is designed to be intermittent one; on the other hand, the pinning control method is considered in the controller (7), which can avoid imposing controllers to all nodes. In other words, the controller (7) can be regarded as a more economic controller.
Remark 3
In this paper, the intermittent controller (7) is constructed in complete intermittent way. It is different from semi-intermittent controller [15, 33], in which the linear feedback term is added when \(t \in [t_{2k} ,\;t_{2k + 2} )\). This severely restricts their application in practical. However, the proposed controller (7) only appears in the work interval and dissipates during the rest interval. Therefore, this intermittent controller is not only more advantageous for practical application, but also more cost-effective for control implementation.
Based on the aforementioned substance, we rewrite the error system(4) by :
where \({\mathcal{P}} \in R^{n \times n}\) and \({\mathcal{Q}} \in R^{n \times n}\) should be symmetric matrix and positive definite matrix, respectively.
Lemma 3
[31] Let \(\omega_{1} ,\omega_{2} ,...,\omega_{i} \ge 0\). Then, for \(0 < r_{1} < 1\) and \(r_{2} > 1\),
[38] In an undirected communication topology, if \({\rm M} \in R^{N \times N}\) is the Laplacian matrix, at the same time, if the pinning control matrix \(\ell = diag\{ \ell_{1} ,\ell_{2} ,...,\ell_{n} \}\) satisfies Assumption 3, then \(\lambda_{\min } (\Gamma_{\ell }^{ - 1} (\Gamma_{\ell } {\rm M} + {\rm M}\Gamma_{\ell } )) \le 0\) can be established.
Lemma 5
[38] Suppose that the time sequence \(t_{k}\) satisfies that \(\mathop {\lim }\limits_{k \to + \infty } t_{k} = + \infty\) and \(0 = t_{0} < t_{1} < ... < t_{k} < ...\), then the non-negative function \({\mathbb{V}}(t)\) fulfills.
in which \(0 < p_{1} < 1\), \(p_{2} > 1\), \({\upxi }\), \(\alpha_{1}\), \(\alpha_{2}\)\(\eta > 0\). Afterwards, when there is the condition that \(\frac{{{\upxi }(t_{2k + 1} - t_{2k} )}}{{\eta (t_{2k + 2} - t_{2k + 1} )}} = \lambda_{k} > 1\), \(\forall k \in N_{ + }\) holds, \({\mathbb{V}}(t) \equiv 0\), \(\forall t \ge T_{2}\) can be established, in which T2 denotes the setting time with the expression.
where \({\upxi }\), \(\alpha\), \(\eta\) are positive constants and \(1 < r < 2\), \(T_{con} (t_{{k_{1} }} ,t_{{k_{2} }} )\) and \(T_{res} (t_{{k_{1} }} ,t_{{k_{2} }} )\) are all the control and rest span length in \([t_{{k_{1} }} ,t_{{k_{2} }} )\), individually, and if \(\frac{{{\upxi }T_{con} (t_{2k} ,t_{2k + 2} )}}{{\eta T_{res} (t_{2k} ,t_{2k + 2} )}} = \lambda_{k} > 1,\forall k \in N_{ + }\) holds, there is \({\mathbb{V}}(t) \equiv 0\), \(\forall t \ge T_{2}\), in which \(T_{2}\) represents the setting time, and for any initial value, \(T_{2}\) is estimated by:
proving the existence of a settling-time T1, which makes inequality \({\mathbb{V}}(T_{1} ) = 1\) established, in addition, \({\mathbb{V}}(t) \le 1\) for all \(t \ge T_{1}\);
(2)
furthermore, proving the existence of a convergence time \(T_{2}\), which makes equality \({\mathbb{V}}(T_{2} ) = 0\) established, in addition, \({\mathbb{V}}(t) \equiv 1\) for all \(t \ge T_{2}\).
When \({\mathbb{V}}(t) \ge 1\) for all \(t > 0\), we rewrite the inequality (11) by
there is \(0 \le {\mathbb{V}}(t) \le \upsilon_{1} (t)\) when \(t \in [0, + \infty )\). Suppose there is a fixed time T1 that makes \(\upsilon_{1} (T_{1} ) \le 1\) tenable, \({\mathbb{V}}(T_{1} ) \le 1\) can be derived according to squeeze theorem. Simplify the expression of \(\upsilon_{1} (t)\) by introducing a new function \(U_{1} (t)\) to facilitate subsequent analysis and calculation, in which \(U_{1} (t) = \upsilon_{1}^{ - r} (t)\). The substituted inequality is represented as follows:
It can be seen that when \(\Lambda_{1} (T_{1} ) = 1\), one can acquire \(U_{1} (t) \ge 1\) based on \(U_{1} (t) = \upsilon^{ - r} (t)\), and \(\upsilon (T_{1} ) \le 1\) for \(U_{1} (t) = \upsilon^{ - r} (t)\), then, \({\mathbb{V}}(T_{1} ) \le 1\) can be acquired. Therefore, \({\mathbb{V}}(t)\) converges from \({\mathbb{V}}(t) > 1\) to \({\mathbb{V}}(t) \le 1\). Moreover, it can be verified that \(\Lambda_{1} (t_{{2k^{\prime}}} ) < 1\), \(\Lambda_{1} (t_{{2(k^{\prime} + 1)}} ) \ge 1\) and \(\mathop {\lim }\limits_{{t \to t_{{2(k^{\prime} + 1)}}^{ - } }} \Lambda_{1} (t) \ge \Lambda_{1} (t_{{2(k^{\prime} + 1)}} ) \ge 1\). when \(t \in [t_{{2k^{\prime} + 1}} ,t_{{2k^{\prime} + 2}} ]\), \(\Lambda_{1} (t)\) is a nonincreasing function, then, \(\Lambda_{1} (t_{{2k^{\prime} + 1}} ) \ge 1\) can be derived. Similarly, \(\mathop {\lim }\limits_{{t \to t_{{2k^{\prime} + 1}}^{ - } }} \Lambda_{1} (t) = \Lambda_{1} (t_{{2k^{\prime} + 1}} ) \ge 1\). Since \(\Lambda_{1} (t)\) is an increasing function within the interval \([t_{{2k^{\prime}}} ,t_{{2k^{\prime} + 1}} ]\), combined with the zero point theorem, we have \(T_{1} \in [t_{{2k^{\prime}}} ,t_{{2k^{\prime} + 1}} )\) such that \(\Lambda_{1} (T_{1} ) = 1\).
From the above analysis, the setting-time T1 can be estimated by
It is easy to see that \(0 \le {\mathbb{V}}(t) \le \upsilon_{2} (t)\) when \(t \ge T_{1}\). Assume that there is a time T2 that makes \(\upsilon_{2} (T_{2} ) = 0\) tenable, \({\mathbb{V}}(T_{2} ) = 0\) can be derived according to squeeze rule. When \(t \ge T_{2}\), \({\mathbb{V}}(T_{1} ) \equiv 0\). Simplify the expression of \(\upsilon_{2} (t)\) by introducing a new function \(U_{2} (t)\) to facilitate subsequent analysis and calculation, in which \(U_{2} (t) = \upsilon_{2}^{2 - r} (t)\). The substituted inequality is represented by:
The emphasis of the present work is achieving fixed-time synchronization by using aperiodic complete intermittent controller (7). Therefore, in rest interval \(t \in [t_{2k + 1} ,t_{2k + 2} )\), a relaxation condition \( {\dot{\mathbb{V}}}(t) \le \rho {\mathbb{V}}(t) \) is considered here. Lemma 6 provides a new fixed-time stability result to avoid the inconvenience by imposing controller (7).
Remark 5
According to the above meaning of \(\mu\) and \({\mathbb{V}}(t)\), \(\mu = r + sig({\mathbb{V}}(t) - 1)\) can be obtained. It is seen that \(\mu\) is related to \({\mathbb{V}}(t)\). Therefore, the part of μ that works can be selected by the value of \({\mathbb{V}}(t)\). If there is \({\mathbb{V}}(t) \ge 1\), \(r + sig({\mathbb{V}}(t) + 1) \ge 1\) is obtained. When \({\mathbb{V}}(t) < 1\), \(0 < r + sig({\mathbb{V}}(t) + 1) < 1\) can be obtained. Obviously, the value of μ varies within different ranges of \({\mathbb{V}}(t)\). In order to distinguish it from other cases, μ is represented by r.
Remark 6
In the Lemma 5 of [38], it was done by artificially ignoring the smaller term. For example, when \({\mathbb{V}}(t) > 1\), and \(0 < p_{1} < 1\), \(- \alpha {\mathbb{V}}^{{p_{1} }} (t)\) plays a smaller role than \(- \alpha_{2} {\mathbb{V}}^{{p_{2} }} (t)\), so it is obviously reasonable to ignore \(- \alpha_{1} {\mathbb{V}}^{{p_{1} }} (t)\). Fortunately, this method will shrink the differential inequality again. We introduce a sign function \(\mu = r + sig({\mathbb{V}}(t) - 1)\), where the value of \(\mu\) is determined by the relationship between \({\mathbb{V}}(t)\) and 1. That is to say, there is no effect of neglecting certain term, so a more accurate setting-time can be obtained. The setting-time in [38] is given by.
In comparison with [38], Lemma 6 can pursue a smaller setting-time, which is consistent with previous speculations.
3 Main Results
In this section, we shall explore fixed-time synchronization issue in the intermittently coupled network (1) under controller (7). The following content expounds the main results.
Theorem 1.
Based on the premise that the Assumption 1–3 hold, if there exists constant \(\zeta \ge 2\gamma > 0\) such that.
in which \({\upxi } = - 2\chi + 2c\lambda_{\min } (\Gamma_{\ell } ) - \frac{\zeta }{{1 - \tilde{d}}} > 0\), \(\eta = 2\chi + \frac{\zeta }{{1 - \tilde{d}}}\), then the fixed-time synchronization of network (1) could be obtained under controller (7). Additionally, the setting-time T2 can be estimated as
where \( \zeta > 2\gamma \) and \(\eta = 2\chi + \frac{\zeta }{{1 - \tilde{d}}}\).
Finally, combining Lemma 6 with (32), (33), (45), (50) and (49), there are \({\mathbb{V}}(t) \equiv 1\) for all \(t \ge T_{2}\). From \(\left\| {e(t)} \right\| \le \sqrt {\frac{{{\mathbb{V}}(t)}}{{\lambda_{\min } (\Gamma_{\ell } )}}}\), for all \(t \ge T_{2}\), it can be concluded that \(\left\| {e_{i} (t)} \right\| \equiv 0\). From Definition 1, the fixed-time synchronization of intermittently coupled network (1) can be achieved under aperiodic intermittent pinning controller (7). Furthermore, the settling-time is apparently small and more accurate than the estimated value T2 defined in (35). The proof is completed.
Remark 7
During the partial analytical process of above certification, we discuss \( - 2\alpha e^{T} (t)(\Gamma _{\ell } \otimes I_{n} )sig^{{r + sig(V(t) + 1)}} (w(t)) - 2\alpha e^{T} (t)(\Gamma _{\ell } \otimes I_{n} ){\mho } \) in two cases: \({\mathbb{V}}(t) \ge 1\) and \({\mathbb{V}}(t) < 1\). On the basis of the definition \(\rho = r + sig({\mathbb{V}}(t) - 1)\), when \({\mathbb{V}}(t) \ge 1\), \(\rho \ge 1\); when \({\mathbb{V}}(t) < 1\), \(0 < \rho < 1\). Furthermore, for the two different cases of \(p_{1} > 1\), \(0 < p_{2} < 1\), the scaling of \(\sum\nolimits_{i = 1}^{n} {\omega_{i}^{p} }\) in Lemma 3 is not entirely the same. Therefore, it is necessary to discuss the \({\mathbb{V}}(t)\) partition case.
Remark 8
For controller (7), the pinned nodes are selected as follows: firstly, through Tarjan’s algorithm, decide components which are composed of all connecting nodes; then, choose one node from each component, randomly. The chosen nodes are pinned nodes, Specially, if a node is isolated, it should be pinned. The pinned nodes satisfy Assumption 3. Under the condition of ensuring that the network can achieve synchronization, there are minimal number of pinned nodes.
Consider the periodic intermittent coupling case, the index function defined by
in which, \(T > 0\) and \({\mathfrak{m}} > 0\) represent control period and represents control rate, respectively. Then, a periodic intermittent controller is devised as
To measure the extent of synchronization is achieved, we introduce synchronization square error \(E(t) = \sum\nolimits_{i = 1}^{5} {\left\| {e_{i} (t)} \right\|^{2} }\) as a quantity index. When the control actions are removed, the trajectory of E(t) is drawn by Fig. 3, which shows that synchronization cannot be reached.
Fig. 3
Trajectory of E(t) without controller
×
Next, the complete intermittent controller (7) is added to network (54) with the parameters \(\alpha = 1\), \(\zeta = 0.2\), \(r = 1.5\), \(\tilde{d} = 0\), \(c{ = 50}\) and \(\ell = {\text{diag}}\{ 1,0,0,0,1\}\) which is pinning control matrix. By some simple calculation, if \(\lambda_{\min } (\Gamma_{\ell } ) = 0.1981\) and \(\gamma = - 0.2971\), Assumption 3 will hold. And the estimated setting-time is \(T_{f} = 6.7558\). The trajectory of synchronization square error (denoted by E0(t)) with controller (7) is depicted in Fig. 4. It is obviously to see that the synchronization of intermittently coupled complex network can be realized via our proposed aperiodically intermittent pinning controller. Furthermore, according to Fig. 4, it is apparent that the synchronization can be achieved when \(t = 2.3{\text{s}}\). This is considerably smaller than \(T_{f} = 6.7558{\text{s}}\).
Fig. 4
Trajectory of E0(t) with controller (7)
×
In the following, it will be verified the estimated setting-time of fixed-time synchronization is not dependent of the incipient conditions. Therefore, we magnify the initial values by a factor of ten to observe the setting-time. The synchronization square error (denoted by E1(t)) is portrayed in Fig. 5. Moreover, based on comparing the synchronization time shown in Figs. 4 and 5, it can be concluded that the setting-time is similar for different initial conditions. Thus, it can be quite evident that the characteristics of fixed-time synchronization are verified.
Fig. 5
Trajectory of E1(t) with controller (7)
×
4.2 Comparisons of the Other Results
Example 2
In [38], the authors investigated fixed-time synchronization in complex networks with continuous coupling via controller (8).
We describe the network [38] with 5 nodes in the following way:
The same parameters and initial values as in SubSect. 4.1 and [38] are used in this example. For the purpose of comparison, the designed controller (7) and controller (8) proposed in [38] are used to complex network (54), respectively. According to [38], the parameter of controller (8) are given as: \(c = 50\), \(\alpha_{1} = 1\), \(\alpha_{2} = 1\), \(p_{1} = 0.5\), \(p_{2} = 1.5\), and \(\zeta = 0.2\).
Figure 6 plots the synchronization square error (expressed by E01(t)) with controller (7) and the synchronization square error (expressed by E2(t)) with controller (8). According to Fig. 6, It is not difficult to conclude that the synchronization square error E2(t) converges faster and has small error, which shows the effectiveness of our controller (7).
Fig. 6
Trajectories of E01(t) and E2(t) under controller (7) and controller in [38]
×
Example 3
In this example, the controller (8) proposed in [38] is applied to the intermittently coupled network (1) with \(c = 50\), \(\alpha_{1} = 1\), \(\alpha_{2} = 1\), \(p_{1} = 0.5\), \(p_{2} = 1.5\) and \(\zeta = 0.2\). The trajectories of synchronization square error under controller (7) (denoted by E0(t)) and controller (8) in [38] (denoted by E3(t)) are shown by Fig. 7. It could be found that the synchronization under controller in [38] is realized at about \(t = 5{\text{s}}\), while the synchronization time is \(t = 2.3{\text{s}}\) by using controller (7). Obviously, our proposed method can possess smaller convergence time. Moreover, it is noted that the synchronization square error E0(t) can be smaller than E3(t).
Fig. 7
Trajectories of E0 (t) and E3(t) under controller (7) and controller in [38]
×
5 Conclusion
This paper investigates the fixed-time synchronization of intermittently coupled network with time-varying delay under complete intermittent controller. A novel fixed-time stability lemma is proven. An aperiodically intermittent economical controller is designed. Several fixed-time synchronization conditions have been exported and the settling-time is independent from any initial values. The obtained results have been demonstrated by numerical simulations. It is noted that the index function is defined to be identical to all nodes in this work, the case of non-uniformity for the index function can be further considered. Moreover, the estimation of setting-time is related to controller parameters and model parameters, the preassigned-time synchronization problem can be further considered to address it. These challenging problems bring the future research direction.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (62003077), Major Science and Technology Project of Shanxi Province (202201090301013), the Key Research and Development Program of Shanxi Province (202202150401005, 202202100401002), Shanxi Provincial Technology Innovation Center (202104010911033) and the Fundamental Research Program of Shanxi Province (20210302123210, 202203021222186).
Declarations
Competing interests
The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
