Linear Hamiltonian systems on time scales: positivity of quadratic functionals

• Authors: HILSCHER Roman
• Year of publication: 2000
• Type: Article in Periodical
• Magazine / Source: Mathematical and Computer Modelling
• MU Faculty or unit: Faculty of Science
• Field: General mathematics
• Keywords: time scale; (continuous and discrete) linear Hamiltonian system; disconjugacy; principal solution; quadratic functional

Description

In this work we consider a linear Hamiltonian system (H)

x^\Delta = A_t x^\sigma + B_t u,
u^\Delta = -C_t x^\sigma - A_t^T u

on an arbitrary time scale T, which allows (among others)

• to treat both continuous and discrete linear Hamiltonian systems (as the special cases for T=R and T=Z) within one theory;
• to explain the discrepancies between these two theories while studying systems of the form (H).

As a main result we prove that disconjugacy of the system (H) is a sufficient condition for positive definiteness of the quadratic functional associated with (H). The principal tool is the Picone identity on T. We derive also the corresponding Wronskian identity, Riccati equation in this general setting on time scales.

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