Abstract
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 (Šepitka 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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Acknowledgements
The author is grateful to Professor Roman Šimon Hilscher for consultations regarding the subject of this paper. The author wish to thank an anonymous referee for detailed reading of the paper and valuable suggestions which improved the presentation of the results.
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This research was supported by the Czech Science Foundation under Grant GA16-00611S. The revised version of this paper was prepared under the support of the Grant GA19-01246S.
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Šepitka, P. Genera of Conjoined Bases for (Non)oscillatory Linear Hamiltonian Systems: Extended Theory. J Dyn Diff Equat 32, 1139–1155 (2020). https://doi.org/10.1007/s10884-019-09810-w
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DOI: https://doi.org/10.1007/s10884-019-09810-w
Keywords
- Linear Hamiltonian system
- Genus of conjoined bases
- Riccati differential equation
- Controllability
- Orthogonal projector