To a General Theory of Differential Realization of Bilinear Nonautonomous Evolutionary Equations of the Second Order with Delay

In this paper, a characteristic feature of solvability of the problem of differential realization of a continuous bundle of controlled trajectory curves in the class of bilinear non–autonomous ordinary differential equations (with and without delay) of the second order in a separable Hilbert space is established. The problem under consideration belongs to the type of inverse problems for an additive combination of non–stationary linear and bilinear operators of an evolution equation in a Hilbert space. The metalanguage of this theory is the construction of tensor products of Hilbert spaces, the structure of lattices with orthocomplement, the theory of extension of M2–operators and the functional apparatus of the nonlinear Rayleigh–Ritz operator. It is shown that in the case of a fi nite bundle of trajectory curves, the presence of the sublinearity property of this operator allows us to obtain suffi cient conditions for the existence of such realizations. Along the way, topological–metric conditions for the continuity of the projectivization of the nonlinear Rayleigh–Ritz operator are substantiated with the calculation of the fundamental group of its image.