Wu Chai Wah
IBM T. J. Watson Research Center, P. O. Box 704, Yorktown Heights, New York 10598, USA.
Chaos. 2008 Sep;18(3):037103. doi: 10.1063/1.2944235.
This paper concerns pinning control in complex networks of dynamical systems, where an external forcing signal is applied to the network in order to align the state of all the systems to the forcing signal. By considering the control signal as the state of a virtual dynamical system, this problem can be studied as a synchronization problem. The main focus of this paper is to study how the effectiveness of pinning control depends on the underlying graph. In particular, we look at the relationship between pinning control effectiveness and the complex network asymptotically as the number of vertices in the network increases. We show that for vertex balanced graphs, if the number of systems receiving pinning control does not grow as fast as the total number of systems, then the strength of the control needed to effect pinning control will be unbounded as the number of vertices grows. Furthermore, in order to achieve pinning control in systems coupled via locally connected graphs, as the number of systems grows, both the pinning control and the coupling among all systems need to increase. Finally, we give evidence to show that applying pinning control to minimize the distances between all systems to the pinned systems can lead to a more effective pinning control.
本文关注动态系统复杂网络中的牵制控制，其中向网络施加外部强迫信号，以使所有系统的状态与强迫信号对齐。通过将控制信号视为虚拟动态系统的状态，该问题可作为同步问题进行研究。本文的主要重点是研究牵制控制的有效性如何取决于基础图。特别地，随着网络中顶点数量的增加，我们研究牵制控制有效性与复杂网络之间的渐近关系。我们表明，对于顶点平衡图，如果接受牵制控制的系统数量增长速度不如系统总数快，那么随着顶点数量的增加，实现牵制控制所需的控制强度将趋于无穷大。此外，为了在通过局部连接图耦合的系统中实现牵制控制，随着系统数量的增加，牵制控制以及所有系统之间的耦合都需要增加。最后，我们给出证据表明，应用牵制控制以使所有系统到被牵制系统的距离最小化，可以导致更有效的牵制控制。
