Advances in Difference Equations
Volume 2006 (2006), Article ID 80757, 13 pages
doi:10.1155/ADE/2006/80757
On stability zones for discrete-time periodic linear
Hamiltonian systems
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.
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.