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@article{1705958, author = {Šepitka, Peter and Šimon Hilscher, Roman}, article_number = {February 2021}, doi = {http://dx.doi.org/10.1016/j.laa.2020.11.018}, keywords = {Linear Hamiltonian system; Left focal point; Right focal point; Comparative index; Principal solution; Sturmian theory}, language = {eng}, issn = {0024-3795}, journal = {Linear Algebra and Its Applications}, title = {Distribution and number of focal points for linear Hamiltonian systems}, url = {https://doi.org/10.1016/j.laa.2020.11.018}, volume = {611}, year = {2021} }
TY - JOUR ID - 1705958 AU - Šepitka, Peter - Šimon Hilscher, Roman PY - 2021 TI - Distribution and number of focal points for linear Hamiltonian systems JF - Linear Algebra and Its Applications VL - 611 IS - February 2021 SP - 26-45 EP - 26-45 PB - Elsevier SN - 00243795 KW - Linear Hamiltonian system KW - Left focal point KW - Right focal point KW - Comparative index KW - Principal solution KW - Sturmian theory UR - https://doi.org/10.1016/j.laa.2020.11.018 L2 - https://doi.org/10.1016/j.laa.2020.11.018 N2 - In this paper we consider the question of distribution and number of left and right focal points for conjoined bases of linear Hamiltonian differential systems. We do not assume any complete controllability (identical normality) condition. Recently we obtained the Sturmian separation theorem for this case which provides optimal lower and upper bounds for the numbers of left and right focal points of every conjoined basis in terms of the principal solutions at the endpoints of the interval. In this paper we show that for any two given integers within these bounds there exists a conjoined basis with these prescribed numbers of left and right focal points. We determine such conjoined bases by their initial conditions. Our approach is to transfer the problem through the comparative index into matrix analysis. The main results are new even for completely controllable linear Hamiltonian systems. As an application we extend a classical result for controllable systems by Reid (1971) about the existence of conjoined bases with an invertible first component. ER -
ŠEPITKA, Peter a Roman ŠIMON HILSCHER. Distribution and number of focal points for linear Hamiltonian systems. \textit{Linear Algebra and Its Applications}. Elsevier, 2021, roč.~611, February 2021, s.~26-45. ISSN~0024-3795. Dostupné z: https://dx.doi.org/10.1016/j.laa.2020.11.018.
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