Publication details

Principal and antiprincipal solutions at infinity of linear Hamiltonian systems



Year of publication 2015
Type Article in Periodical
Magazine / Source Journal of Differential Equations
MU Faculty or unit

Faculty of Science

Field General mathematics
Keywords Linear Hamiltonian system; Antiprincipal solution at infinity; Principal solution at infinity; Minimal principal solution; Controllability; Normality; Conjoined basis; Order of abnormality; Genus of conjoined bases; Moore-Penrose pseudoinverse
Description The concept of principal solutions at infinity for possibly abnormal linear Hamiltonian systems was recently introduced by the authors. In this paper we develop the theory of antiprincipal solutions at infinity and establish a limit characterization of the principal solutions. That is, we prove that the principal solutions are the smallest ones at infinity when they are compared with the antiprincipal solutions. This statement is a generalization of the classical result of W. T. Reid, P. Hartman, or W. A. Coppel for controllable linear Hamiltonian systems. We also derive a classification of antiprincipal solutions at infinity according to their rank and show that the antiprincipal solutions exist for any rank in the range between explicitly given minimal and maximal values. We illustrate our new theory by several examples.
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