Comparison theorems for conjoined bases of linear Hamiltonian differential systems and the comparative index

Abstract

In this paper we present comparison results for focal points of conjoined bases Y(t),Yˆ(t) of two linear Hamiltonian differential systems under the majorant condition H(t)−Hˆ(t)≥0 for their Hamiltonians H(t),Hˆ(t). Both systems are considered without controllability (or normality) assumptions and under the Legendre condition for Hˆ(t). The main result of the paper connects the difference between the number of proper focal points of Y(t),Yˆ(t) with the number of proper focal points of the transformed conjoined basis Zˆ−1(t)Y(t), where Zˆ(t) is a symplectic fundamental solution matrix of the Hamiltonian system associated with Hˆ(t). Focal points of this transformed basis coincide with focal points of the matrix Wronskian of Y(t),Yˆ(t). The main tool of the paper is the comparative index theory for discrete symplectic systems generalized to the continuous case.