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This paper introduces a new technique for the analysis of time-optimal systems having multidimensional inputs. The method provides a cellular decomposition of the controllable set for small response times for a new generic class of systems called "minimally controllable systems." The decomposition permits a complete study of the time-optimal flow near the origin and forms the basis for the synthesis of closed-loop optimal control. The method, while generally applicable, is restricted here to the simplest, nontrivial case of third-order systems with two-dimensional controls.