In this paper, the problem of pinning control for synchronization of complex dynamical networks is discussed. A cost function of the controlled network is defined by the feedback gain and the coupling strength of the network. An interesting result is that lower cost is achieved by the control scheme of pinning nodes with smaller degrees. Some rigorous mathematical analysis is presented for achieving lower cost in the synchronization of different star-shaped networks. Numerical simulations on some non-regular complex networks generated by the Barabasi-Albert model and various star-shaped networks are shown for verification and illustration.
Complex networks are currently being studied across many fields of sciences, including physics, chemistry, biology, mathematics, sociology and engineering [1,2,3,5,9,15,17,19]. A complex network is a large set of interconnected nodes, in which a node is a fundamental unit with specific contents. Examples of complex networks include the Internet, food webs, cellular neural networks, biological neural networks, electrical power grids, telephone cell graphs, etc. Recently, synchronization of complex networks of dynamical systems has received a great deal of attention from the nonlinear dynamics community [10,12,16,18,20]. A special control strategy called pinning control is used to achieve synchronization of complex networks; that is, only a fraction of the nodes or even a single node is controlled over the whole network [4,6,11,21]. This control method has become a common technique for control, stabilization and synchronization of coupled dynamical systems. In general, different nodes have different degrees in a network, thus a natural question is how different the effect would be when nodes with different degrees are pinned.
Consider a dynamical network consisting of N identical and diffusively coupled nodes, with each node being an n-dimensional dynamical system. The state equations of the network are
where f (•) is the dynamical function of an isolated node, x i = (x i1 , x i2 , • • • , x in ) ∈ R n are the state variables of node i, constant c > 0 represents the coupling strength, and Γ ∈ R N ×N is the inner linking matrix. Moreover, the coupling matrix A = (a ij ) ∈ R N ×N represents the coupling configuration of the network: If there is a connection between node i and node j (i = j), then a ij = a ji = 1; otherwise, a ij = a ji = 0 (i = j); the diagonal entries of A are defined by
Suppose that the network is connected in the sense of having no isolated clusters. Then, the coupling matrix A is irreducible. From Lemma 2 of [20], it can be proved that zero is an eigenvalue of A with multiplicity one and all the other eigenvalues of A are strictly negative.
Network (1) is said to achieve (asymptotical) synchronization if
where, because of the diffusive coupling configuration, s(t) is a solution of an isolated node, which can be an equilibrium, a periodic or a chaotic orbit. As shown in [4,6,11,21], this can be achieved by controlling several nodes (or even only one node) of the network. Without loss of generality, suppose that the controllers are added on the last Nk nodes of the network, so that the equations of the controlled network can be written as
where the feedback gains ε i are positive constants. It can be seen that synchronizing all states x i (t) to s(t) is determined by the dynamics of an isolated node, the coupling strength c > 0, the inner linking matrix Γ, the feedback gains ε i ≥ 0, and the coupling matrix A.
As discussed in [6,11,21], to achieve synchronization of complex dynamical networks, the controllers are generally preferred to be added to the nodes with larger degrees. However, it is also known that, to achieve a certain synchronizability of the network, the feedback gains ε i usually have to be quite large. In [4], when a single controller is used, the coupling strength c has to be quite large in general. From the view point of realistic applications, these are not expected and sometimes cannot be realized. Practically, a designed control strategy is expected to be effective and also easily implementable. In this paper, for various star-shaped networks and non-regular complex networks, a new concept of cost function is introduced to evaluate the efficiency of the designed controllers. It is found that surprisingly the cost can be much lower by controlling nodes with smaller degrees than controlling nodes with larger degrees. As will be seen, moreover, both the feedback gains ε i and the coupling strength c can be much smaller than those used in [4,6,11,21].
The outline of this paper is as follows. In Section 2, a new definition of cost function and some mathematical preliminaries are given. Stability of different star-shaped networks controlled by pinning some nodes with small degrees are analyzed in Sections 3 and 4, respectively, where some simulated examples of dynamical networks are compared for illustration and verification.
In Section 5, pinning control of non-regular complex dynamical networks of chaotic oscillators is studied through numerical simulations. Finally, Section 6 concludes the paper.
Denote e i (t) = x i (t)s(t), where s(t) satisfies ṡ(t) = f (s(t)). Then, the error equations of network (1) can be written as
while the error equations of the controlled network (4) can be written as
where ãii = a ii -
and denote e(t) = (e 1 (t), e 2 (t),
By analyzing the matrix ÃT , it is easy to see that all the eigenvalues of ÃT are negative, which are denoted by
There exists an orthogonal matrix U such that ÃT = U JU -1 , where
Let ẽ(t) = e(t)U . Then, from ( 6),
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