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Rayleigh principle for linear Hamiltonian systems without controllability∗

Published online by Cambridge University Press:  22 July 2011

Werner Kratz  and
Roman Šimon Hilscher
Werner Kratz
Affiliation:
Department of Applied Analysis, Faculty of Mathematics and Economics, University of Ulm, 89069 Ulm, Germany. [email protected]
Roman Šimon Hilscher
Affiliation:
Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic; [email protected]
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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.

Keywords

Linear Hamiltonian systemRayleigh principleself-adjoint eigenvalue problemproper focal pointconjoined basisfinite eigenvalueoscillation theoremcontrollabilitynormalityquadratic functional
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 2 , April 2012 , pp. 501 - 519
Copyright
© EDP Sciences, SMAI, 2011

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