Journal of Applied Mathematics
Volume 2012 (2012), Article ID 876120, 12 pages
Research Article

Geometric Analysis of Reachability and Observability for Impulsive Systems on Complex Field

Shouwei Zhao,1,2,3 Jitao Sun,2 and Hai Lin3,4

1College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, China
2Department of Mathematics, Tongji University, Shanghai 200092, China
3Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576
4Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA

Received 21 October 2011; Revised 8 December 2011; Accepted 8 December 2011

Academic Editor: Nazim I. Mahmudov

Copyright © 2012 Shouwei Zhao et al. 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.


Nowadays, quantum systems have become one of the focuses of the ongoing research and they are typical complex systems, whose state variables are defined on the complex field. In this paper, the issue of reachability and observability is addressed for a class of linear impulsive systems on complex field, for simplicity, complex linear impulsive systems. This kind of time-driven impulsive systems allows free impulsive instants, which leads to the limitation of using traditional definitions of reachability and observability directly. New notations about the span reachable set and unobservable set are proposed. Sufficient and necessary conditions for span reachability and observability of such systems are established. Moreover, the explicit characterization of span reachable set and unobservable set is presented by geometric analysis. It is pointed out that the geometric conditions are equivalent to the algebraic ones in known results for special cases. Numerical examples are also presented to show the effectiveness of the proposed methods.