Advances in Difference Equations
Volume 2006 (2006), Article ID 80757, 13 pages

On stability zones for discrete-time periodic linear Hamiltonian systems

Vladimir Răsvan

Department of Automatic Control, University of Craiova, Street A. I. Cuza no. 13, Craiova RO-200585, Romania

Received 18 June 2004; Revised 8 September 2004; Accepted 13 September 2004

Copyright © 2006 Vladimir Răsvan. 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.


The main purpose of the paper is to give discrete-time counterpart for some strong (robust) stability results concerning periodic linear Hamiltonian systems. In the continuous-time version, these results go back to Liapunov and Žukovskii; their deep generalizations are due to Kreĭn, Gel'fand, and Jakubovič and obtaining the discrete version is not an easy task since not all results migrate mutatis-mutandis from continuous time to discrete time, that is, from ordinary differential to difference equations. Throughout the paper, the theory of the stability zones is performed for scalar (2nd-order) canonical systems. Using the characteristic function, the study of the stability zones is made in connection with the characteristic numbers of the periodic and skew-periodic boundary value problems for the canonical system. The multiplier motion (“traffic”) on the unit circle of the complex plane is analyzed and, in the same context, the Liapunov estimate for the central zone is given in the discrete-time case.