New existence results for conjoined bases of singular linear
Hamiltonian systems with given Sturmian properties

J 2025

New existence results for conjoined bases of singular linear Hamiltonian systems with given Sturmian properties

ŠEPITKA, Peter and Roman ŠIMON HILSCHER

Basic information

Original name

New existence results for conjoined bases of singular linear Hamiltonian systems with given Sturmian properties

Authors

ŠEPITKA, Peter (703 Slovakia, belonging to the institution) and Roman ŠIMON HILSCHER (203 Czech Republic, guarantor, belonging to the institution)

Edition

Linear Algebra and its Applications, Elsevier, 2025, 0024-3795

Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10101 Pure mathematics

Country of publisher

United States of America

Confidentiality degree

is not subject to a state or trade secret

References:

URL

Impact factor

Impact factor: 1.100 in 2024

Organization unit

Faculty of Science

DOI

https://doi.org/10.1016/j.laa.2024.11.017

UT WoS

001406712200001

EID Scopus

2-s2.0-85210300880

Keywords in English

Linear Hamiltonian system; Legendre condition; Sturmian separation theorem; Genus of conjoined bases; Comparative index; Dual comparative index; Riccati differential equation

Abstract

In the original language

In this paper we derive new existence results for conjoined bases of singular linear Hamiltonian differential systems with given qualitative (Sturmian) properties. In particular, we examine the existence of conjoined bases with invertible upper block and with prescribed number of focal points at the endpoints of the considered unbounded interval. Such results are vital for the theory of Riccati differential equations and its applications in optimal control problems. As the main tools we use a new general characterization of conjoined bases belonging to a given equivalence class (genus) and the theory of comparative index of two Lagrangian planes. We also utilize extensively the methods of matrix analysis. The results are new even for identically normal linear Hamiltonian systems. The results are also new for linear Hamiltonian systems on a compact interval, where they provide additional equivalent conditions to the classical Reid roundabout theorem about disconjugacy.

Links

GA23-05242S, research and development project

Name: Oscilační teorie na hybridních časových doménách s aplikacemi ve spektrální teorii a maticové analýze

Investor: Czech Science Foundation, Oscillation theory on hybrid time domains with applications in spectral and matrix analysis

Cite

ŠEPITKA, Peter and Roman ŠIMON HILSCHER. New existence results for conjoined bases of singular linear Hamiltonian systems with given Sturmian properties. Linear Algebra and its Applications. Elsevier, 2025, vol. 707, February, p. 187-224. ISSN 0024-3795. Available from: https://doi.org/10.1016/j.laa.2024.11.017.

@article{2456608,
   author = {Šepitka, Peter and Šimon Hilscher, Roman},
   article_number = {February},
   doi = {https://doi.org/10.1016/j.laa.2024.11.017},
   keywords = {Linear Hamiltonian system; Legendre condition; Sturmian separation theorem; Genus of conjoined bases; Comparative index; Dual comparative index; Riccati differential equation},
   language = {eng},
   issn = {0024-3795},
   journal = {Linear Algebra and its Applications},
   title = {New existence results for conjoined bases of singular linear Hamiltonian systems with given Sturmian properties},
   url = {https://www.sciencedirect.com/science/article/pii/S0024379524004373},
   volume = {707},
   year = {2025}
}

TY  - JOUR
ID  - 2456608
AU  - Šepitka, Peter - Šimon Hilscher, Roman
PY  - 2025
TI  - New existence results for conjoined bases of singular linear Hamiltonian systems with given Sturmian properties
JF  - Linear Algebra and its Applications
VL  - 707
IS  - February
SP  - 187-224
EP  - 187-224
PB  - Elsevier
SN  - 00243795
KW  - Linear Hamiltonian system
KW  - Legendre condition
KW  - Sturmian separation theorem
KW  - Genus of conjoined bases
KW  - Comparative index
KW  - Dual comparative index
KW  - Riccati differential equation
UR  - https://www.sciencedirect.com/science/article/pii/S0024379524004373
N2  - In this paper we derive new existence results for conjoined bases of singular linear Hamiltonian differential systems with given qualitative (Sturmian) properties. In particular, we examine the existence of conjoined bases with invertible upper block and with prescribed number of focal points at the endpoints of the considered unbounded interval. Such results are vital for the theory of Riccati differential equations and its applications in optimal control problems. As the main tools we use a new general characterization of conjoined bases belonging to a given equivalence class (genus) and the theory of comparative index of two Lagrangian planes. We also utilize extensively the methods of matrix analysis. The results are new even for identically normal linear Hamiltonian systems. The results are also new for linear Hamiltonian systems on a compact interval, where they provide additional equivalent conditions to the classical Reid roundabout theorem about disconjugacy.
ER  -

ŠEPITKA, Peter and Roman ŠIMON HILSCHER. New existence results for conjoined bases of singular linear Hamiltonian systems with given Sturmian properties. \textit{Linear Algebra and its Applications}. Elsevier, 2025, vol.~707, February, p.~187-224. ISSN~0024-3795. Available from: https://doi.org/10.1016/j.laa.2024.11.017.

Přehled o publikaci