# 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) z Δ = \cal S t z on 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 and Optimization Springer Journals

# Reid Roundabout Theorem for Symplectic Dynamic Systems on Time Scales

, Volume 43 (2) – Jan 1, 2001
18 pages

Publisher
Springer-Verlag
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
D.O.I.
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) z Δ = \cal S t z on 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 and OptimizationSpringer Journals

Published: Jan 1, 2001

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