Exponential synchronization of complex dynamical network with mixed time-varying and hybrid coupling delays via intermittent control
© Botmart and Niamsup; licensee Springer. 2014
Received: 8 December 2013
Accepted: 22 April 2014
Published: 6 May 2014
In this paper, we shall investigate the problem of exponential synchronization for complex dynamical network with mixed time-varying and hybrid coupling delays, which is composed of state coupling, interval time-varying delay coupling and distributed time-varying delay coupling. The designed controller ensures that the synchronization of delayed complex dynamical network are proposed via either feedback control or intermittent feedback control. The constraint on the derivative of the time-varying delay is not required which allows the time-delay to be a fast time-varying function. We use common unitary matrices, and the problem of synchronization is transformed into the stability analysis of some linear time-varying delay systems. This is based on the construction of an improved Lyapunov-Krasovskii functional combined with the Leibniz-Newton formula and the technique of dealing with some integral terms. New synchronization criteria are derived in terms of LMIs which can be solved efficiently by standard convex optimization algorithms. Two numerical examples are included to show the effectiveness of the proposed feedback control and intermittent feedback control scheme.
Keywordsexponential synchronization complex dynamical network mixed time-varying delays hybrid coupling intermittent control
Complex dynamical network, as an interesting subject, has been thoroughly investigated for decades. These networks show very complicated behavior and can be used to model and explain many complex systems in nature such as computer networks , the world wide web , food webs , cellular and metabolic networks , social networks , electrical power grids etc. In general, a complex network is a large set of interconnected nodes, in which a node is a fundamental unit with specific contents. As an implicit assumption, these networks are described by the mathematical term graph. In such graphs, each vertex represents an individual element in the system, while edges represent the relations between them. Two nodes are joined by an edge if and only if they interact.
In the last decade, the synchronization of complex dynamic networks has attracted much attention of researchers in this field [7–18]. Because the synchronization of complex dynamical networks can well explain many natural phenomena observed and is one of the important dynamical mechanisms for creating order in complex dynamical networks, the synchronization of coupled dynamical networks has come be a focal point in the study of nonlinear science. Wang and Chen introduced a uniform dynamical network model and also investigated its synchronization [11–13]. They have shown that the synchronizability of a scale-free dynamical network is robust against random removal of nodes, and yet it is fragile to specific removal of the most highly connected nodes . The authors in [14, 15] investigated synchronization of general complex dynamical network models with coupling delays. Li and Chen  considered the synchronization stability of complex dynamical network models with coupling delays for both continuous- and discrete-time, and they derived some synchronization conditions for both delay-independent and delay-dependent asymptotical stabilities. By utilizing Lyapunov functional method. Wang et al.  introduced several synchronization criteria for both delay-independent and delay-dependent asymptotical stability. Li and Yi  investigated synchronization of complex networks with time-varying couplings, the stability criteria were obtained by using Lyapunov-Krasovskii function method and subspace projection method. Yue and Li  studied the synchronization stability of continuous and discrete complex dynamical networks with interval time-varying delays in the dynamical nodes and the coupling term simultaneously, delay-dependent synchronization stability are derived in the form of linear matrix inequalities.
It is well known that the existence of time-delay in a system may cause instability and an example of oscillations can be found in systems such as chemical engineering systems, biological modeling, electrical networks, physical networks, and many others [19–25]. The stability criteria for a system with time-delays can be classified into two categories: delay-independent and delay-dependent. Delay-independent criteria do not employ any information on the size of the delay; while delay-dependent criteria make use of such information at different levels. Delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the delay is small . Recently, the delay-dependent stability for interval time-varying delay was investigated in [6, 18, 20–22]. Interval time-varying delay is a time-delay that varies in an interval in which the lower bound is not restricted to be 0. Jiang and Han  considered the problem of robust control for uncertain linear systems with interval time-varying delay based on Lyapunov functional approach in which restriction on the differentiability of the interval time-varying delay was removed. Shao  presented a new delay-dependent stability criterion for linear systems with interval time-varying delay, and stability criteria are derived in terms of linear matrix inequalities without introducing any free-weighting matrices. In order to reduce further the conservatism introduced by the descriptor model transformation and bounding techniques, a free-weighting matrix method is proposed in [20, 26–29]. In , the synchronization problem has been investigated for continuous/discrete complex dynamical networks with interval time-varying delays. Based on a piecewise analysis method and the Lyapunov functional method, some new delay-dependent synchronization criteria are derived in the form of LMIs by introducing free-weighting matrices. It will be pointed out later that some existing results require more free-weighting matrix variables than our result.
Intermittent control is one of discontinuous control and has a nonzero control width. It is an engineering approach that has been widely used in engineering fields, such as manufacturing, air-quality control, transportation, and communication in practice. However, results using intermittent control to study exponential synchronization are few. In recent years, several synchronization criteria for complex dynamical networks with or without time-delays via feedback control or intermittent control have been presented; see [30–41] and the references therein. Synchronization of a complex dynamical network with delayed nodes by pinning periodically intermittent control was also reported in . A periodically intermittent control was applied to the complex dynamical networks with both time-varying delays dynamical nodes and time-varying delays coupling in [32, 33]. In , the authors investigated exponential synchronization of a complex network with nonidentical time-delayed dynamical nodes by applying open-loop control to all nodes and adding some intermittent controllers to partial nodes. The authors in  investigated synchronization of a general model of complex delayed dynamical networks. The periodically intermittent control scheme is introduced to drive the network to achieve synchronization. Based on the Lyapunov stability theory and pinning control method, some novel synchronization criteria for such dynamical network are derived. To the best of the authors’ knowledge, the problem of exponential synchronization for a complex dynamical network with mixed time-varying delays in the network hybrid coupling and time-varying delays in the dynamical nodes has not been fully investigated yet and remains open.
In this paper, inspired by the above discussions, we shall investigate the problem of exponential synchronization for a complex dynamical network with mixed time-varying and hybrid coupling delays, which is composed of constant coupling, interval time-varying delay coupling, and distributed time-varying delay coupling. The designed controller ensures that the synchronization of a delayed complex dynamical network is proposed via either feedback control or intermittent feedback control. The constraint on the derivative of the time-varying delay is not required, which allows the time-delay to be a fast time-varying function. We use common unitary matrices, and the problem of synchronization is transformed into the stability analysis of some linear time-varying delay systems. Based on the construction of an improved Lyapunov-Krasovskii functional is combined with the Leibniz-Newton formula and the technique of dealing with some integral terms. New synchronization criteria are derived in terms of LMIs which can be solved efficiently by standard convex optimization algorithms. Two numerical examples are included to show the effectiveness of the proposed feedback control and intermittent feedback control scheme.
The organization of the remaining part is as follows. In Section 2, a class of general complex dynamical network model with mixed time-varying and hybrid coupling delays and some useful lemmas are given. In Section 3, synchronization stability in complex dynamical network with mixed time-varying and hybrid coupling delays via feedback control and intermittent feedback control are investigated. Numerical examples illustrated the obtained results are given in Section 4. The paper ends with conclusions in Section 5.
2 Network model and mathematic preliminaries
It is assumed that network (1) is connected in the sense that there are no isolated clusters, that is, A, B, C are irreducible matrices.
Definition 2.1 
The initial condition function denotes a continuous vector-valued initial function of .
In this paper, we assume that is an orbitally stable solution of the above system. Clearly, the stability of the synchronized states (3) of network (1) is determined by the dynamics of the isolate node, the coupling strength , , and , the inner-coupling matrices , , and , and the outer-coupling matrices A, B, and C.
The following lemmas are used in the proof of the main result.
Lemma 2.2 
Let A, B be a family of diagonalizable matrices. Then A, B is a commuting family (under multiplication) if and only if it is a simultaneously diagonalizable family.
Lemma 2.3 
Lemma 2.4 (Cauchy inequality )
3 Synchronization of delayed complex dynamical network via delayed feedback control and intermittent control
is the Jacobian of at with the derivative of respect to ,
is the Jacobian of at with the derivative of respect to ,
is the Jacobian of at with the derivative of respect to .
then the dynamical networks (5) is exponentially stable, and then exponential synchronization of the controlled dynamical networks (1) is achieved.
where , , . In addition, with (2) and the irreducible feature of A, B, and C we can select with such that , .
are exponentially stable, then will tend to the origin exponentially, which is equivalent to the synchronization of the dynamical networks (5) being exponentially stable. This completes the proof. □
then the dynamical networks (4) is exponentially stable, then exponential synchronization of the controlled dynamical networks (1) is achieved.