On bilinear differential realization of a continual beam of trajectory curves in the constructions of the Rayleigh-Ritz operator

We present functional and geometric conditions (necessary and suffrcient) for the existent of five non-stationary bilinear operators in a model of differential realization of a continuous bundle of controlled trajectory curves (dynamic processes of the power input / output type) in the dass of the second order bilinear non-autonomous ordinary differential equations (induding quasi-linear hyperbolic models) in a real parallel Hilbert space. The problem under consideration is a type of non-stationary nonlinear coefficient inverse problems for evolutionary equations in Hilbert space and is solved on the basis of a qualitative study of the continuity property of the Rayleigh-Ritz functional operator. It is shown that the strudure of the fundamental group of the image of this operator depends on the dimension of the projective space on which it acts. The results obtained are applied to the qualitative theory of non-linear structural identification of higher order multilinear non-stationary differential models.