  We present a theoretical study of the optimal control of a qubit interacting
with a structured environment. We consider a model system in which the bath is
a bosonic reservoir at zero temperature and the qubit frequency is the only
control parameter. Using optimal control techniques, we show the extent to
which qubit population and relaxation effects can be manipulated. The reachable
qubit states by a shaped control with a fixed maximum intensity are found
numerically. We analyze the role of standard control mechanisms and the
structure of the set of reachable states with respect to the coupling strength
between the system and the environment. This investigation is used as a
starting point to explore the selectivity problem of two uncoupled qubits
interacting with their own baths and characterized by a specific coupling
strength. We numerically derive the optimal control solution for a wide range
of parameters and we show that the control law is close to a sinusoidal
function with a specific frequency in some peculiar cases.
