A local feedback synthesis of time-optimal stabilizing controls in dimension three (Q1176544)

This paper is concerned with the problem of determining controls which are time-optimally stabilising. The equilibrium point in dimension three is considered. It is shown that there are two surfaces which are glued together along a singular arc. On this arc the optimal control is singular. Away from these two surfaces the optimal controls are piecewise constant with at most two switchings. Bang-bang trajectories with two switchings, but different switching orders, intersect in a non-trivial cut-locus. Optimality of trajectories ceases at this cut-locus. These facts are used to constructively determine the time-optimal control. The construction is based on an earlier result by the authors which characterises the small-time reachable set for an associated system to which time has been added as an extra coordinate.