ON THE IMPLEMENTATION OF THE SUPERPOSITION PRINCIPLE FOR A FINITE BEAM OF INTEGRAL CURVE OF A SECOND-ORDER BILINEAR SYSTEM. I

Based on the projectivization of the non-linear Rayleigh-Ritz functional operator and the tensor product of real Hilbert spaces, for a finite bundle of integral curves of a controlled bilinear system of the second order, necessary and sufficient conditions for the existence of a differential realization of this bundle in the class of linear non-stationary ordinary differential equations of the second order are determined (including hyperbolic models) in a separable Hilbert space. In this case, the original bilinear structure models the nonlinearity of the system dynamics, both the trajectory itself and the speed of movement on this trajectory. The results obtained have applications in the theory of inverse problems of non-stationary controlled multilinear differential models of higher orders and the theory of optimal control using the technology of successive approximations in solving a two-point boundary value problem based on the quasi-linearization procedure using the Picard method.