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Quadratic functionals: positivity, oscillation, Rayleigh's principle

Werner Kratz — 1998

Archivum Mathematicum

In this paper we give a survey on the theory of quadratic functionals. Particularly the relationships between positive definiteness and the asymptotic behaviour of Riccati matrix differential equations, and between the oscillation properties of linear Hamiltonian systems and Rayleigh’s principle are demonstrated. Moreover, the main tools form control theory (as e.g. characterization of strong observability), from the calculus of variations (as e.g. field theory and Picone’s identity), and from matrix...

Rayleigh principle for linear Hamiltonian systems without controllability

Werner KratzRoman Šimon Hilscher — 2012

ESAIM: Control, Optimisation and Calculus of Variations

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...

Rayleigh principle for linear Hamiltonian systems without controllability

Werner KratzRoman Šimon Hilscher — 2012

ESAIM: Control, Optimisation and Calculus of Variations

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...

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