  We characterize and construct time-independent Markovian dynamics that drive
a finite-dimensional multipartite quantum system into a target (pure) entangled
steady state, subject to physical locality constraints. In situations where the
desired stabilization task can not be attained solely based on local
dissipative means, we allow for local Hamiltonian control or, if the latter is
not an option, we suitably restrict the set of admissible initial states. In
both cases, we provide algorithms for constructing a master equation that
achieves the intended objective and show how this can genuinely extend the
manifold of stabilizable states. In particular, we present quasi-local control
protocols for dissipatively engineering multipartite GHZ "cat" states and W
states on $n$ qubits. For GHZ states, we find that no scalable procedure exists
for achieving stabilization from arbitrary initial states, whereas this is
possible for a target W state by a suitable combination of a two-body
Hamiltonian and dissipators. Interestingly, for both entanglement classes, we
show that quasi-local stabilization may be scalably achieved conditional to
initialization of the system in a large, appropriately chosen subspace.
