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

Abstract

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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.

How to cite

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Kratz, 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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  10. W. Kratz, R. Šimon Hilscher, and V. Zeidan, Eigenvalue and oscillation theorems for time scale symplectic systems. Int. J. Dyn. Syst. Differ. Equ.3 (2011) 84–131.  Zbl1218.34039
  11. W.T. Reid, Ordinary Differential Equations. Wiley, New York (1971).  Zbl0212.10901
  12. W.T. Reid, Sturmian Theory for Ordinary Differential Equations. Springer-Verlag, New York-Berlin-Heidelberg (1980).  Zbl0459.34001
  13. R. Šimon Hilscher, and V. Zeidan, Picone type identities and definiteness of quadratic functionals on time scales. Appl. Math. Comput.215 (2009) 2425–2437.  Zbl1178.93094
  14. M. Wahrheit, Eigenwertprobleme und Oszillation linearer Hamiltonischer Systeme [Eigenvalue Problems and Oscillation of Linear Hamiltonian Systems]. Ph.D. thesis, University of Ulm, Germany (2006).  
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