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Reid Roundabout Theorem for Symplectic Dynamic Systems on Time Scales

Reid Roundabout Theorem for Symplectic Dynamic Systems on Time Scales . The principal aim of this paper is to state and prove the so-called Reid roundabout theorem for the symplectic dynamic system (S) zΔ= \cal S tzon an arbitrary time scale \Bbb T , so that the well known case of differential linear Hamiltonian systems ( \Bbb T = \Bbb R ) and the recently developed case of discrete symplectic systems ( \Bbb T = \Bbb Z ) are unified. We list conditions which are equivalent to the positivity of the quadratic functional associated with (S), e.g. disconjugacy (in terms of no focal points of a conjoined basis) of (S), no generalized zeros for vector solutions of (S), and the existence of a solution to the corresponding Riccati matrix equation. A certain normality assumption is employed. The result requires treatment of the quadratic functionals both with general and separated boundary conditions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics & Optimization Springer Journals

Reid Roundabout Theorem for Symplectic Dynamic Systems on Time Scales

Applied Mathematics & Optimization , Volume 43 (2) – Jan 1, 2001

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References (35)

Publisher
Springer Journals
Copyright
Copyright © Springer-Verlag New York Inc. 2001
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s00245-001-0002-1
Publisher site
See Article on Publisher Site

Abstract

. The principal aim of this paper is to state and prove the so-called Reid roundabout theorem for the symplectic dynamic system (S) zΔ= \cal S tzon an arbitrary time scale \Bbb T , so that the well known case of differential linear Hamiltonian systems ( \Bbb T = \Bbb R ) and the recently developed case of discrete symplectic systems ( \Bbb T = \Bbb Z ) are unified. We list conditions which are equivalent to the positivity of the quadratic functional associated with (S), e.g. disconjugacy (in terms of no focal points of a conjoined basis) of (S), no generalized zeros for vector solutions of (S), and the existence of a solution to the corresponding Riccati matrix equation. A certain normality assumption is employed. The result requires treatment of the quadratic functionals both with general and separated boundary conditions.

Journal

Applied Mathematics & OptimizationSpringer Journals

Published: Jan 1, 2001

Keywords: Time scale; Symplectic system; Linear Hamiltonian system; Quadratic functional; Disconjugacy; Focal point; Principal solution; Riccati equation; Jacobi condition; Legendre condition; AMS Classification. 34C10, 39A10, 93C70

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