  We investigate the possibility to control localization properties of the
asymptotic state of an open quantum system with a tunable synthetic
dissipation. The control mechanism relies on the matching between properties of
dissipative operators, acting on neighboring sites and specified by a single
control parameter, and the spatial phase structure of eigenstates of the system
Hamiltonian. As a result, the latter coincide (or near coincide) with the dark
states of the operators. In a disorder-free Hamiltonian with a flat band, one
can either obtain a localized asymptotic state or populate whole flat and/or
dispersive bands, depending on the value of the control parameter. In a
disordered Anderson system, the asymptotic state can be localized anywhere in
the spectrum of the Hamiltonian. The dissipative control is robust with respect
to an additional local dephasing.
