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Mathematical Problems in Engineering
Volume 2011, Article ID 678575, 25 pages
Research Article

Commutativity of Linear Time-Varying Differential Systems with Nonzero Initial Conditions: A Review and Some New Extensions

1Computer Engineering Department, Halic University, Siracevizler Street, no. 29, 34363 Sisli, Istanbul, Turkey
2Department of Elementary Mathematics Education, Mevlana University, 42003 Konya, Turkey

Received 18 January 2011; Revised 21 June 2011; Accepted 12 July 2011

Academic Editor: P. Liatsis

Copyright © 2011 Muhammet Koksal and Mehmet Emir Koksal. 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.


Necessary and sufficient conditions for the commutativity of linear time-varying systems are derived in the case of nonzero initial conditions. It is shown that some commutative class of linear time-varying systems may not commute with arbitrary initial conditions. In this respect, commutativity of Euler differential systems is investigated. Explicit commutativity conditions for the fifth-order systems are solved. New results about the effects of commutativity on system sensitivity and disturbance properties are presented, which is very important for network design and industrial applications where many of the systems are composed of subsystems cooperating one after another in a chain. The results are supported by examples treated either analytically or numerically.