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Journal of Applied Mathematics
Volume 2012, Article ID 182040, 24 pages
Research Article

Controllability and Observability Criteria for Linear Piecewise Constant Impulsive Systems

1Mathematics and Physics Department, Beijing Institute of Petrochemical Technology, Beijing 102617, China
2Center for Systems and Control, LTCS, and Department of Industrial Engineering and Management, Peking University, Beijing 100871, China
3School of Electrical and Electronic Engineering, East China Jiaotong University, Nanchang 330013, China

Received 14 May 2012; Accepted 22 July 2012

Academic Editor: Junjie Wei

Copyright © 2012 Hong Shi and Guangming Xie. 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.


Impulsive differential systems are an important class of mathematical models for many practical systems in physics, chemistry, biology, engineering, and information science that exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical processes. This paper studies the controllability and observability of linear piecewise constant impulsive systems. Necessary and sufficient criteria for reachability and controllability are established, respectively. It is proved that the reachability is equivalent to the controllability under some mild conditions. Then, necessary and sufficient criteria for observability and determinability of such systems are established, respectively. It is also proved that the observability is equivalent to the determinability under some mild conditions. Our criteria are of the geometric type, and they can be transformed into algebraic type conveniently. Finally, a numerical example is given to illustrate the utility of our criteria.