AN OSCILLATION THEOREM FOR SELF-ADJOINT DIFFERENTIAL-SYSTEMS AND THE RAYLEIGH PRINCIPLE FOR QUADRATIC FUNCTIONALS

In this note an oscillation theorem on self-adjoint differential syste ms of the form x = Ax + Bu, u = (C-lambda C-0) x - A(T)u is obtained, complementing, in particular, results of M. Morse. The application of this oscillation result yields the Rayleigh principle for quadratic fu nctionals, respectively, the existence theorem for corresponding self- adjoint eigenvalue problems, under the central assumptions that the pa ir (A,B) is controllable (or identically normal) and the triple (A, B, C-0) is strongly observable.