##
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