Publication details

Distribution and number of focal points for linear Hamiltonian systems

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Year of publication 2021
Type Article in Periodical
Magazine / Source Linear Algebra and Its Applications
MU Faculty or unit

Faculty of Science

Keywords Linear Hamiltonian system; Left focal point; Right focal point; Comparative index; Principal solution; Sturmian theory
Description In this paper we consider the question of distribution and number of left and right focal points for conjoined bases of linear Hamiltonian differential systems. We do not assume any complete controllability (identical normality) condition. Recently we obtained the Sturmian separation theorem for this case which provides optimal lower and upper bounds for the numbers of left and right focal points of every conjoined basis in terms of the principal solutions at the endpoints of the interval. In this paper we show that for any two given integers within these bounds there exists a conjoined basis with these prescribed numbers of left and right focal points. We determine such conjoined bases by their initial conditions. Our approach is to transfer the problem through the comparative index into matrix analysis. The main results are new even for completely controllable linear Hamiltonian systems. As an application we extend a classical result for controllable systems by Reid (1971) about the existence of conjoined bases with an invertible first component.
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