The use of first integrals in the problem of optimal control of a linear system with a quadratic functional

A method for constructing an optimal control of a linear system in a variational problem with fixed time for the control process, fixed endpoints of the phase trajectory, and a quadratic functional is proposed. The method is based on the use of first integrals of the equations of unperturbed motion. Sufficient conditions for complete controllability of the linear nonstationary system have been obtained