In this talk we discuss aspects of the physics and mathematics of a quantum control system interacting with its environment. In particular we discuss the control of an finite-dimensional Lindblad system by considering the geometry of its orbit and interorbit dynamics. This entails considering the geometry of flag manifolds, the structure of the Lindblad operator, and the convexity associated with the density equation. Applications are given to constructing pure states. This includes recent work with Rooney and Rangan.