# Rayleigh principle for linear Hamiltonian systems without controllability∗

Werner Kratz; Roman Šimon Hilscher

ESAIM: Control, Optimisation and Calculus of Variations (2012)

- Volume: 18, Issue: 2, page 501-519
- ISSN: 1292-8119

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topKratz, Werner, and Hilscher, Roman Šimon. "Rayleigh principle for linear Hamiltonian systems without controllability∗." ESAIM: Control, Optimisation and Calculus of Variations 18.2 (2012): 501-519. <http://eudml.org/doc/276370>.

@article{Kratz2012,

abstract = {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. },

author = {Kratz, Werner, Hilscher, Roman Šimon},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Linear Hamiltonian system; Rayleigh principle; self-adjoint eigenvalue problem; proper focal point; conjoined basis; finite eigenvalue; oscillation theorem; controllability; normality; quadratic functional; linear Hamiltonian system},

language = {eng},

month = {7},

number = {2},

pages = {501-519},

publisher = {EDP Sciences},

title = {Rayleigh principle for linear Hamiltonian systems without controllability∗},

url = {http://eudml.org/doc/276370},

volume = {18},

year = {2012},

}

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

DA - 2012/7//

PB - EDP Sciences

VL - 18

IS - 2

SP - 501

EP - 519

AB - 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.

LA - eng

KW - Linear Hamiltonian system; Rayleigh principle; self-adjoint eigenvalue problem; proper focal point; conjoined basis; finite eigenvalue; oscillation theorem; controllability; normality; quadratic functional; linear Hamiltonian system

UR - http://eudml.org/doc/276370

ER -

## References

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- M. Wahrheit, Eigenvalue problems and oscillation of linear Hamiltonian systems. Int. J. Difference Equ.2 (2007) 221–244.

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