Complexity in linearly coupled dynamical networks: Some unusual phenomena in energy accumulation - Mathematics > Optimization and Control

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Abstract: This paper addresses the energy accumulation problem, in terms of the $H 2$norm, of linearly coupled dynamical networks. An interesting outer-couplingrelationship is constructed, under which the $H 2$ norm of the newlyconstructed network with column-input and row-output shaped matrices increasesexponentially fast with the node number $N$: it increases generally much fasterthan $2^N$ when $N$ is large while the $H 2$ norm of each node is 1. However,the $H 2$ norm of the network with a diffusive coupling is equal to $\gamma 2N$, i.e., increasing linearly, when the network is stable, where $\gamma 2$ isthe $H 2$ norm of a single node. And the $H 2$ norm of the network withantisymmetrical coupling also increases, but rather slowly, with the nodenumber $N$. Other networks with block-diagonal-input and block-diagonal-outputmatrices behave similarly. It demonstrates that the changes of $H 2$ norms indifferent networks are very complicated, despite the fact that the networks arelinear. Finally, the influence of the $H 2$ norm of the locally linearizednetwork on the output of a network with Lur-e nodes is discussed.

Autor: Zhisheng Duan, Jinzhi Wang, Guanrong Chen, Lin Huang

Fuente: https://arxiv.org/