Linear Hamiltonian systems on time scales: Positivity of quadratic functionals

In this work, we consider a linear Hamiltonian system x(Delta) = A(t)x(sigma) + B(t)u, u(Delta) = -C(t)x(sigma) - A(t)(T)u (H) on an arbitrary time scale T, which allows one (among others) to treat both continuous and discrete linear Hamiltonian systems (as the sp ecial cases for T = R and T = Z) within one theory; to explain the discrepancies between these two theories while studying syst ems 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 T. We derive also th e corresponding Wronskian identity, Riccati equation in this general settin g on time scales. (C) 2000 Elsevier Science Ltd. All rights reserved.