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The study of algebraic extension of a countable family of controlled non-linear dynamic processes “inputoutput” having differential realization in the class of ordinary quasi-linear differential equations (with software-positional control and without) in a separable Hilbert space was conducted. This problem as a starting point for the development of the general theory of vector fields, simultaneously creating a reputation for it as a useful tool in precision mathematical modeling of complex dynamic systems. Key-Words:  non-linear dynamic processes, non-linear differential realization, non-stationary (A,B,B)2-model of vector field. Received: March 25, 2020. Revised: April 26, 2020. Accepted: May 15, 2020. Published: May 29, 2020

To Existence of a Nonstationary Quasi-Linear Vector Field Realizing the Expansion of a Control Trajectory Bundle in Hilbert Space