  We propose a general framework for investigating a large class of
stabilization problems in Markovian quantum systems. Building on the notions of
invariant and attractive quantum subsystem, we characterize attractive
subspaces by exploring the structure of the invariant sets for the dynamics.
Our general analysis results are exploited to assess the ability of open-loop
Hamiltonian and output-feedback control strategies to synthesize Markovian
generators which stabilize a target subsystem, subspace, or pure-state. In
particular, we provide an algebraic characterization of the manifold of
stabilizable pure states in arbitrary finite-dimensional Markovian systems,
that leads to a constructive strategy for designing the relevant controllers.
Implications for stabilization of entangled pure states are addressed by
example.
