  Our aim is twofold: First, we rigorously analyse the generators of
quantum-dynamical semigroups of thermodynamic processes. We characterise a wide
class of GKSL-generators for quantum maps within thermal operations and argue
that every infinitesimal generator of (a one-parameter semigroup of) Markovian
thermal operations belongs to this class. We completely classify and visualise
them and their non-Markovian counterparts for the case of a single qubit.
  Second, we use this description in the framework of bilinear control systems
to characterise reachable sets of coherently controllable quantum systems with
switchable coupling to a thermal bath. The core problem reduces to studying a
hybrid control system ("toy model") on the standard simplex allowing for two
types of evolution: (i) instantaneous permutations and (ii) a one-parameter
semigroup of $d$-stochastic maps. We generalise upper bounds of the reachable
set of this toy model invoking new results on thermomajorisation. Using tools
of control theory we fully characterise these reachable sets as well as the set
of stabilisable states as exemplified by exact results in qutrit systems.
