  We extend the Runge-Gross theorem for a very general class of Markovian and
non-Markovian open quantum systems under weak assumptions about the nature of
the bath and its coupling to the system. We show that for Kohn-Sham (KS)
Time-Dependent Density Functional Theory, it is possible to rigorously include
the effects of the environment within a bath functional in the KS potential,
thus placing the interactions between the particles of the system and the
coupling to the environment on the same footing. A Markovian bath functional
inspired by the theory of nonlinear Schrodinger equations is suggested, which
can be readily implemented in currently existing real-time codes. Finally,
calculations on a helium model system are presented.
