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

We propose 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. The method is based on the use of first integrals of the equations of unperturbed motion. We obtain sufficient conditions for complete controllability of the linear nonstationary system.