TOWARD A THEORY OF INVERSE PROBLEMS FOR BILINEAR DYNAMIC SYSTEMS DESCRIBED BY EIGHT-VARIANT SECOND-ORDER DIFFERENTIAL EQUATION: A GEOMETRIC APPROACH

In the development of the general theory of the realization of nonlinear dynamical systems, based on the geometric constructions of the tensor product of Hilbert spaces, system-theoretic foundations are constructed for the analytical study of the necessary and sufficient conditions for the existence of a differential realization of a continuous infinite-dimensional dynamical system (represented by a bundle of arbitrary power of controlled trajectories) in the class of bilinear nonstationary ordinary differential equations second order in a separable Hilbert space. The differential equation of state of the studied infinite-dimensional dynamic system allows modeling eight variants of the combined bilinear structure of nonlinearity, both from the trajectory itself and from the speed of movement on this trajectory. Along the way, for this dynamic realization, the topology-metric conditions for the continuity of the projectivization of the nonlinear functional Rayleigh-Ritz operator are analytically substantiated with the calculation of the fundamental group (the Poincaré group) of its image. The results obtained have the potential for the development of the mathematical theory of systems in substantiating the nonlinear theory of coefficient-operator inverse problems of nonautonomous differential models of polylinear controlled dynamic systems of higher orders.