Time-optimal steering of a point mass to a specified position with the required velocity

The planar problem of the time-optimal steering of a point mass to the origin of coordinates with a required velocity by means of a force of limited modulus is considered. The optimal control and the optimal time are investigated for different characteristic values of the positional radius vector and arbitrary values of the magnitude and direction of the initial velocity. The results are extended to the case of boundary conditions which are imposed on the magnitude and direction of the initial velocity in the form of inequalities and are transferred to a problem with control for a fixed time.