A method of a control prolongation by annihilating polynomials is discussed which is based on use of linear systems algebraic properties and essentially bases on the Cayley-Hamilton theorem. The basic idea of the approach (with reference to discrete-time systems) is, that if exists an admissible finite control sequence for the given boundary conditions for system without restrictions on controls, there is an opportunity to put to it in conformity not a unique control sequence, which is the decision of the same problem, but with other time of the termination. One of the specified mapping (continuations) of control sequences quite is described by the linear operator generated by the annihilating polynomials of a matrix of system.

The opportunities of application of the control prolongation in various tasks with a zero terminal condition are analyzed.