A synthesis of time-optimal controls in the presence of saturated singular arcs

Standard necessary conditions for optimality in optimal control problems, like the maximum principle, are derived using approximations of the reachable set. In small time improved necessary conditions for optimality can be obtained by constructing the true reachable set. The construction relies heavily on explicit Lie-algebraic calculations to verify geometric properties. It is shown how these calculations, which were done earlier for piecewise constant controls, can be extended to piecewise smooth feedback controls. For a system we use this geometric approach to construct the small-time reachable set from a reference point p as a stratified CW-complex under codimension 1 assumptions which imply the existence of saturated singular arcs. As a Corollary, a local regular synthesis of time-optimal trajectories which stabilize an equilibrium of f in dimension 3 is given. 1991 Mathematics Subject Classification: 49K15, 49L05; 93C10.