Linear Hamiltonian systems on time scales: Positivity of quadratic functionals (Q1591021)

The subject of this paper is the linear Hamiltonian system in an arbitrary time scale. The author develops an approach which allows him to treat both the continuous form \[ \begin{cases} \chi'=A(t)\chi+B(t)u\\ u'=-C(t)\chi-A^T(t)u\end{cases} \] as well as the discrete form \[ \begin{cases} \Delta\chi_k=A_k\chi_{k+1}+B_ku_k\\ \Delta u_k=-C_k\chi_{k+1}-A^T_k u_k\end{cases} \] as particular cases of one general system in an arbitrary time scale. In either the continuous or discrete case, there is a version of the Reid Roundabout theorem that relates positivity of a certain quadratic functional (involving \(u\), \(\chi\), \(B\) and \(C\)) to oscillatory properties of the linear Hamiltonian system. The author develops a theory of time scales as well as definitions and properties of linear Hamiltonian systems on time scales. He introduces the concept of focal points for matrix solutions of linear Hamiltonian systems on time scales, and then proves a sufficient condition for positivity of a certain critical quadratic functional.