International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 508721, 28 pages
Research Article

Control Systems and Number Theory

Sanmenxia SuDa Communication Group, Sun Road, Sanmenxia Economical Development Zone, Sanmenxia, Henan, 472000, China

Received 3 October 2011; Accepted 16 November 2011

Academic Editor: Shigeru Kanemitsu

Copyright © 2012 Fuhuo Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We try to pave a smooth road to a proper understanding of control problems in terms of mathematical disciplines, and partially show how to number-theorize some practical problems. Our primary concern is linear systems from the point of view of our principle of visualization of the state, an interface between the past and the present. We view all the systems as embedded in the state equation, thus visualizing the state. Then we go on to treat the chain-scattering representation of the plant of Kimura 1997, which includes the feedback connection in a natural way, and we consider the 𝐻 -control problem in this framework. We may view in particular the unit feedback system as accommodated in the chain-scattering representation, giving a better insight into the structure of the system. Its homographic transformation works as the action of the symplectic group on the Siegel upper half-space in the case of constant matrices. Both of 𝐻 - and PID-controllers are applied successfully in the EV control by J.-Y. Cao and B.-G. Cao 2006 and Cao et al. 2007, which we may unify in our framework. Finally, we mention some similarities between control theory and zeta-functions.