Locally optimal control in the problem of a time-optimal approach to a given target

The locally optimal control is constructed in the problem of a time-optimal approach to a target moving along the given trajectory by a controlled mass point, For the auxiliary problem of time-optimal attainment of the origin of coordinates by a mass point with a constraint on the absolute value of the control force, the locally optimal control is constructed and compared with the optimal. It is found that for some initial (start) conditions, optimal and locally optimal controls are close to each other. This closeness also lakes place for the construction of solar sail transfers to asteroids or a Mars orbit.