Genera of Conjoined Bases for (Non)oscillatory Linear Hamiltonian Systems: Extended Theory
| Authors | |
|---|---|
| Year of publication | 2020 |
| Type | Article in Periodical |
| Magazine / Source | Journal of dynamics and differential equations. |
| MU Faculty or unit | |
| Citation | |
| Web | https://doi.org/10.1007/s10884-019-09810-w |
| Doi | http://dx.doi.org/10.1007/s10884-019-09810-w |
| Keywords | Linear Hamiltonian system; Genus of conjoined bases; Riccati differential equation; Controllability; Orthogonal projector |
| Description | In this paper we study the properties of conjoined bases of a general linear Hamiltonian system without any controllability condition. When the Legendre condition holds and the system is nonoscillatory, it is known from our previous work that conjoined bases with eventually the same image form a special structure called a genus. In this work we extend the theory of genera of conjoined bases to arbitrary systems, for which the Legendre condition is not assumed and/or the system may be oscillatory. We derive a classification of all genera of conjoined bases and show that they form a complete lattice. These results are based on the relationship between subspaces of solutions of a linear control system and orthogonal projectors satisfying a certain Riccati type differential equation. The presented theory is applied in our paper (Sepitka in Discrete Contin Dyn Syst 39(4):1685-1730,2019) to general Riccati matrix differential equations for possibly uncontrollable linear Hamiltonian systems. |
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