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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.
@article{COCV_2012__18_2_501_0,
author = {Kratz, Werner and Hilscher, Roman \v{S}imon},
title = {Rayleigh principle for linear hamiltonian systems without controllability},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {501--519},
publisher = {EDP-Sciences},
volume = {18},
number = {2},
year = {2012},
doi = {10.1051/cocv/2011104},
mrnumber = {2954636},
zbl = {1254.34120},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2011104/}
}
TY - JOUR AU - Kratz, Werner AU - Hilscher, Roman Šimon TI - Rayleigh principle for linear hamiltonian systems without controllability JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 501 EP - 519 VL - 18 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2011104/ DO - 10.1051/cocv/2011104 LA - en ID - COCV_2012__18_2_501_0 ER -
%0 Journal Article %A Kratz, Werner %A Hilscher, Roman Šimon %T Rayleigh principle for linear hamiltonian systems without controllability %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 501-519 %V 18 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2011104/ %R 10.1051/cocv/2011104 %G en %F COCV_2012__18_2_501_0
Kratz, Werner; Hilscher, Roman Šimon. Rayleigh principle for linear hamiltonian systems without controllability. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 501-519. doi: 10.1051/cocv/2011104
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