tailieunhanh - ON STABILITY ZONES FOR DISCRETE-TIME PERIODIC LINEAR HAMILTONIAN SYSTEMS ˘ VLADIMIR RASVAN Received

ON STABILITY ZONES FOR DISCRETE-TIME PERIODIC LINEAR HAMILTONIAN SYSTEMS ˘ VLADIMIR RASVAN Received 18 June 2004; Revised 8 September 2004; Accepted 13 September 2004 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 continuousˇ time version, these results go back to Liapunov and Zukovskii; their deep generalizations are due to Kre˘n, Gel’fand, and Jakuboviˇ and obtaining the discrete version is not an ı c easy task since not all results migrate mutatis-mutandis from continuous time to discrete time, that is, from ordinary differential to difference equations. Throughout. | ON STABILITY ZONES FOR DISCRETE-TIME PERIODIC LINEAR HAMILTONIAN SYSTEMS VLADIMIR RASVAN Received 18 June 2004 Revised 8 September 2004 Accepted 13 September 2004 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 Zukovskii their deep generalizations are due to Krein Gel fand and Jakubovic 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. Copyright 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction motivation and problem statement A Stability analysis of linear Hamiltonian systems with periodic coefficients goes back to Liapunov 21 and Zukovskii 27 . If the simplest case of the second-order scalar equation is considered y A2 p t y 0 where p t is T-periodic then we call A0 a A-point of stability of if for A A0 all solutions of are bounded on R. If moreover all solutions of any equation of type but with p t replaced by p1 t sufficiently close to p t in some sense are also bounded for A A0 then A0 is called a A-point of strong robust stability. Remark that we might take p1 t Ap t with A A0. In this case it was established by Liapunov himself 21 that the set of the A-points of strong stability of is open and if it is .