On a geometric theory of the realization of nonlinear controlled dynamic processes in the class of second-order bilinear models

In the development of the general theory of inverse problems of implementation of nonlinear dynamical systems on the basis of geometric constructions of the tensor product of Hilbert spaces, system-theoretic foundations are built for the analytical study of necessary and sufficient conditions for the existence of a differential implementation of a continuous infinite-dimensional dynamical system (represented by a beam of any power of controlled trajectories) in the class of bilinear nonstationary ordinary second order differential equations in a separable Hilbert space. The bilinear eight-variant structure of the differential equations of state of the infinite-dimensional dynamic system under study models the combined nonlinearity of both the trajectory itself and the speed of movement on this trajectory. Along the way, for this dynamic implementation, the topological-metric conditions for the continuity of the projectivization of the nonlinear Rayleigh - Ritz functional operator are analytically substantiated with the calculation of the fundamental group of its image. The results obtained have the potential for the development of geometric systems theory in substantiating the nonlinear analysis of coefficient-operator inverse problems of non-autonomous differential models of multilinear controllable dynamic systems of higher orders.