On Stability Zones for Discrete-time Periodic Linear Hamiltonian Systems

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 continuoustime version, these results go back to Liapunov and Žukovskii; their deep generalizations are due to Kreı̆n, Gel’fand, and Jakubovič and obtaining the discrete version is not an easy task since not all results migratemutatis-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.