The Galilei group in an optimal control problem

In the paper, results of studying an optimal control problem for the motion of a material point under control constraints are presented. The invariance of this problem with respect to the extended Galilei group is used. From the viewpoint of calculations, the symmetry allows us to construct a family of solutions through an extremal determined numerically. From the analytical viewpoint, the symmetry gives an opportunity to reduce system's dimension and to investigate properties of extremals.