Publication details

Focal points and principal solutions of linear Hamiltonian systems revisited

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Year of publication 2018
Type Article in Periodical
Magazine / Source Journal of Differential Equations
MU Faculty or unit

Faculty of Science

Field General mathematics
Keywords Linear Hamiltonian system; Proper focal point; Principal solution; Antiprincipal solution; Controllability
Description In this paper we present a novel view on the principal (and antiprincipal) solutions of linear Hamiltonian systems, as well as on the focal points of their conjoined bases. We present a new and unified theory of principal (and antiprincipal) solutions at a finite point and at infinity, and apply it to obtain new representation of the multiplicities of right and left proper focal points of conjoined bases. We show that these multiplicities can be characterized by the abnormality of the system in a neighborhood of the given point and by the rank of the associated T-matrix from the theory of principal (and antiprincipal) solutions. We also derive some additional important results concerning the representation of T-matrices and associated normalized conjoined bases. The results in this paper are new even for completely controllable linear Hamiltonian systems. We also discuss other potential applications of our main results, in particular in the singular Sturmian theory.
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