Open Access

On stability zones for discrete-time periodic linear Hamiltonian systems

Advances in Difference Equations20062006:080757

DOI: 10.1155/ADE/2006/80757

Received: 18 June 2004

Accepted: 13 September 2004

Published: 16 January 2006

Abstract

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.

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Authors’ Affiliations

(1)
Department of Automatic Control, University of Craiova

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Copyright

© Vladimir Răsvan 2006

This article is published under license to BioMed Central Ltd. 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.