Control of open quantum systems in a Bosonic bath

We consider the Lindblad-Kossakowski quantum master equation describing the dynamics of an open quantum systems in the form originally proposed by Davies and Spohn. This equation contains dissipative corrections, accounting for the interaction with the environment, whose expression strongly depends on the adopted Markov approximation. In the case where a control is present, the rigorous derivation of the Markov approximation in the standard case, the weak coupling limit, shows that the control appears not only in the coherent part of the equation but also in the dissipative correction. This complicates the analysis of the dynamics but also offers the opportunity of indirectly affecting the interaction with the environment through the control. In this paper we study this scenario for a finite dimensional quantum system interacting with a (Bosonic) bath of harmonic oscillators. We prove several control theoretic properties of this system and discuss how the control can be used to effectively shape the influence of the environment and obtain desired features of the dynamics.