Quadratic Functionals: Positivity, Oscillation, Rayleigh's Principle

Werner Kratz

Address. Universitaet Ulm, Abteilung Mathematik V, D-89069 Ulm, Germany

E-mail: kratz@mathematik.uni-ulm.de

Abstract. 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 analysis (as e.g. l'Hospital's rule for matrices) are discussed.

AMS classification. 49N10, 34A30, 34C10, 93B07, 93C05

Key words. Quadratic functional, Hamiltonian system, Riccati equation, oscillation, observability, Rayleigh's principle, eigenvalue problem, linear control system