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

Under generic conditions a local feedback synthesis for the problem of time-optimally stabilizing an equilibrium point in dimension three is constructed. There exist two surfaces which are glued together along a singular are on which the optimal control is singular. Away from these 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 nontrivial cut-locus and optimality of trajectories ceases at this cut-locus. The construction is based on an earlier result by Krener and Schättler which gives the precise structure of the small-time reachable set for an associated system to which time has been added as an extra coordinate.