Riccati equations for linear Hamiltonian systems without controllability condition

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Authors

ŠEPITKA Peter

Year of publication 2019
Type Article in Periodical
Magazine / Source Discrete & Continuous Dynamical Systems - A
MU Faculty or unit

Faculty of Science

Citation
Web Full Text
Doi http://dx.doi.org/10.3934/dcds.2019074
Keywords Linear Hamiltonian system; Riccati differential equation; genus of conjoined bases; distinguished solution at infinity; principal solution at infinity; controllability
Description In this paper we develop new theory of Riccati matrix differential equations for linear Hamiltonian systems, which do not require any controllability assumption. When the system is nonoscillatory, it is known from our previous work that conjoined bases of the system with eventually the same image form a special structure called a genus. We show that for every such a genus there is an associated Riccati equation. We study the properties of symmetric solutions of these Riccati equations and their connection with conjoined bases of the system. For a given genus, we pay a special attention to distinguished solutions at infinity of the associated Riccati equation and their relationship with the principal solutions at infinity of the system in the considered genus. We show the uniqueness of the distinguished solution at infinity of the Riccati equation corresponding to the minimal genus. This study essentially extends and completes the work of W. T. Reid (1964, 1972), W. A. Coppel (1971), P. Hartman (1964), W. Kratz (1995), and other authors who considered the Riccati equation and its distinguished solution at infinity for invertible conjoined bases, i.e., for the maximal genus in our setting.
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