| KRATZ, Werner a Roman ŠIMON HILSCHER. Rayleigh principle for linear Hamiltonian systems without controllability. ESAIM: Control, Optimisation and Calculus of Variations. Les Ulis: EDP Sciences, 2012, roč. 18, č. 2, s. 501-519. ISSN 1292-8119. doi:10.1051/cocv/2011104. |
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@article{924428,
author = {Kratz, Werner and Šimon Hilscher, Roman},
article_location = {Les Ulis},
article_number = {2},
doi = {http://dx.doi.org/10.1051/cocv/2011104},
keywords = {Linear Hamiltonian system; Rayleigh principle; Self-adjoint eigenvalue problem; Proper focal point; Conjoined basis; Finite eigenvalue; Oscillation theorem; Controllability; Normality; Quadratic functional},
language = {eng},
issn = {1292-8119},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
title = {Rayleigh principle for linear Hamiltonian systems without controllability},
volume = {18},
year = {2012}
}
TY - JOUR
ID - 924428
AU - Kratz, Werner - Šimon Hilscher, Roman
PY - 2012
TI - Rayleigh principle for linear Hamiltonian systems without controllability
JF - ESAIM: Control, Optimisation and Calculus of Variations
VL - 18
IS - 2
SP - 501-519
EP - 501-519
PB - EDP Sciences
SN - 12928119
KW - Linear Hamiltonian system
KW - Rayleigh principle
KW - Self-adjoint eigenvalue problem
KW - Proper focal point
KW - Conjoined basis
KW - Finite eigenvalue
KW - Oscillation theorem
KW - Controllability
KW - Normality
KW - Quadratic functional
N2 - In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools are the extended Picone formula, which is proven here for this general setting, results on piecewise constant kernels for conjoined bases of the Hamiltonian system, and the oscillation theorem relating the number of proper focal points of conjoined bases with the number of finite eigenvalues. As applications we obtain the expansion theorem in the space of admissible functions without controllability and a result on coercivity of the corresponding quadratic functional.
ER -
KRATZ, Werner a Roman ŠIMON HILSCHER. Rayleigh principle for linear Hamiltonian systems without controllability. \textit{ESAIM: Control, Optimisation and Calculus of Variations}. Les Ulis: EDP Sciences, 2012, roč.~18, č.~2, s.~501-519. ISSN~1292-8119. doi:10.1051/cocv/2011104.
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