Rayleigh principle for linear Hamiltonian systems without controllability^∗

¹ Department of Applied Analysis, Faculty of Mathematics and Economics, University of Ulm, 89069 Ulm, Germany
werner.kratz@uni-ulm.de
² Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic
hilscher@math.muni.cz

Received: 25 October 2010

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.