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On stability zones for discrete-time periodic linear Hamiltonian systems. (English) Zbl 1133.39016
Summary: 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 Lyapunov 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 Lyapunov estimate for the central zone is given in the discrete-time case.

##### MSC:
 39A12 Discrete version of topics in analysis 37C20 Generic properties, structural stability of dynamical systems 37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
##### Keywords:
boundary value problems; canonical system
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##### References:
 [1] Ahlbrandt CD, Peterson AC: Discrete Hamiltonian Systems. Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Texts in the Mathematical Sciences. Volume 16. Kluwer Academic Publishers, Dordrecht; 1996:xiv+374. · Zbl 0860.39001 [2] Bohner, M, Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions, Journal of Mathematical Analysis and Applications, 199, 804-826, (1996) · Zbl 0855.39018 [3] Bohner, M; Došlý, O, Disconjugacy and transformations for symplectic systems, The Rocky Mountain Journal of Mathematics, 27, 707-743, (1997) · Zbl 0894.39005 [4] Bohner, M; Došlý, O; Kratz, W, An oscillation theorem for discrete eigenvalue problems, The Rocky Mountain Journal of Mathematics, 33, 1233-1260, (2003) · Zbl 1060.39003 [5] Eastham MSP: The Spectral Theory of Periodic Differential Equations, Texts in Mathematics. Scottish Academic Press, Edinburgh; 1973:X,130. [6] Ekeland I: Convexity Methods in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Volume 19. Springer, Berlin; 1990:x+247. · Zbl 0707.70003 [7] Erbe, LH; Yan, PX, Disconjugacy for linear Hamiltonian difference systems, Journal of Mathematical Analysis and Applications, 167, 355-367, (1992) · Zbl 0762.39003 [8] Erbe, LH; Yan, PX, Qualitative properties of Hamiltonian difference systems, Journal of Mathematical Analysis and Applications, 171, 334-345, (1992) · Zbl 0768.39001 [9] Erbe, LH; Yan, PX, Oscillation criteria for Hamiltonian matrix difference systems, Proceedings of the American Mathematical Society, 119, 525-533, (1993) · Zbl 0794.39006 [10] Erbe, LH; Yan, PX, On the discrete Riccati equation and its applications to discrete Hamiltonian systems, The Rocky Mountain Journal of Mathematics, 25, 167-178, (1995) · Zbl 0836.39003 [11] Gel’fand, IM; Lidskiĭ, VB, On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients, Uspekhi Matematicheskikh Nauk, 10, 3-40, (1955) [12] Halanay, A, An optimization problem for discrete-time systems, Probleme de Automatizare, V, 103-109, (1963) [13] Halanay A, Ionescu V: Time-Varying Discrete Linear Systems, Operator Theory: Advances and Applications. Volume 68. Birkhäuser Verlag, Basel; 1994:viii+228. · Zbl 0799.93035 [14] Halanay, A; Răsvan, Vl, Stability and boundary value problems for discrete-time linear Hamiltonian systems, Dynamic Systems and Applications, 8, 439-459, (1999) · Zbl 0943.39009 [15] Halanay, A; Răsvan, Vl, Oscillations in systems with periodic coefficients and sector-restricted nonlinearities, No. 117, 141-154, (2000), Basel · Zbl 0965.34021 [16] Hartman, P, Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity, Transactions of the American Mathematical Society, 246, 1-30, (1978) · Zbl 0409.39001 [17] Kratz W: Quadratic Functionals in Variational Analysis and Control Theory, Mathematical Topics. Volume 6. Akademie Verlag, Berlin; 1995:293. · Zbl 0842.49001 [18] Kreĭn, MG, On criteria of stable boundedness of solutions of periodic canonical systems, Prikladnaja Matematika i Mehanika, 19, 641-680, (1955) [19] Kreĭn, MG, The basic propositions of the theory of $$λ$$-zones of stability of a canonical system of linear differential equations with periodic coefficients, 413-498, (1955), Moscow [20] Kreĭn, MG; Jakubovič, VA, Hamiltonian systems of linear differential equations with periodic coefficents, 277-305, (1963), Kiev [21] Liapunov, AM, Sur une équation différentielle linéaire du second ordre, Comptes Rendus Mathématique Académie des Sciences paris, 128, 910-913, (1899) · JFM 30.0302.02 [22] Liapunov, AM, Sur une équation transcendente et LES équations différentielles linéaires du second ordre à coefficients périodiques, Comptes Rendus Mathématique Académie des Sciences paris, 128, 1085-1088, (1899) · JFM 30.0303.01 [23] Răsvan, Vl, Stability zones for discrete time Hamiltonian systems, Archivum Mathematicum (Brno), 36, 563-573, (2000) · Zbl 1090.39503 [24] Răsvan, Vl, Krein-type results for $$λ$$-zones of stability in the discrete-time case for 2nd order Hamiltonian systems, No. 13, 223-234, (2003), Brno · Zbl 1106.39306 [25] Tou JT: Optiumum Design of Digital Control Systems. Academic Press, New York; 1963:xi+186. [26] Yakubovich VA, Staržinskii VM: Linear Differential Equations with Periodic Coefficients. Nauka Publ. House, Moscow; 1972. (English version by J. Wiley, 1975) [27] Žukovskii, NE, Conditions for the finiteness of integrals of the equation $$d$$\^{}{2}$$y$$/dx\^{}{2} + py = 0, Matematicheskiĭ Sbornik, 16, 582-591, (1891/1893)
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