Linear Hamiltonian systems on time scales: Positivity of quadratic functionals

Abstract

In this work, we consider a linear Hamiltonian system x Δ = A t x σ + B t u, u Δ =− C t x σ − A t T u (H) on an arbitrary time scale ℸ, which allows one (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 form (H). As a main result, we prove that disconjugacy of system (H) is a sufficient condition for positive definiteness of the quadratic functional associated with (H), The principal tool is the Picone identity on ℸ. We derive also the corresponding Wronskian identity, Riccati equation in this general setting on time scales.