TO A TENSOR ANALYSIS OF THE SOLVABILITY OF THE PROBLEM OF REALIZATION OF A SECOND-ORDER BILINEAR SYSTEM WITH DELAY

Abstract

The analytical conditions (necessary and sufficient) are defined for the solvability of the problem of differential realization of a continuous beam of controlled trajectory curves in the class of bilinear nonautonomous ordinary differential equations (with delay and without it) of the second order in a real separable Hilbert space. The problem under consideration belongs to the type of inverse problems for an additive combination of nonstationary linear and bilinear operators of evolution equations in an infinite-dimensional Hilbert space. The meta-language of this theory is the constructions of tensor products of Hilbert spaces, the structures of lattices with ortho-complementation, and the functional apparatus of the nonlinear Rayleigh-Ritz operator. It is shown that in the case of a finite bundle of trajectories, the presence of a sublinearity-type property of this operator allows us to obtain sufficient conditions for the existence of such realizations. Along the way, the topological-metric conditions for the continuity of the projectivization of the nonlinear Rayleigh-Ritz functional operator with the calculation of the fundamental group of its image are justified. The results obtained provide the motivation for the development of a qualitative theory of nonlinear structural identification of higher-order multi-linear differential models (e.g. for processes, induced by the «brain–machine» interface-platform of the type of Neuralink).