Control synthesis in the problem of the time-optimal intersection of a sphere

The classical problem of the time-optimal control of the motion of a point mass is considered, with the control accomplished by applying a force of limited magnitude. Open- and closed-loop control laws are constructed that ensure intersection with a sphere (from outside or inside) in a coordinate space of arbitrary dimensions. The maximum principle is used to show that the control is of maximum magnitude and maintains a constant direction, while the optimal trajectories are parabolas; the general situation of a multi-dimensional space is equivalent to the two-dimensional (plane) case. The feedback controls and optimal time of motion are constructed as functions of the phase coordinates, the Bellman function of the problem is analysed in detail, qualitative properties of the control and the optimal time of motion in the neighbourhood of the critical phase states of the system are determined. Mathematical simulation is used to construct a section of the Bellman function over a large range of motion parameters; the results are compared with those of analytic and asymptotic studies. (C) 1997 Elsevier Science Ltd.