  We study the adiabatic approximation of the dynamics of a bipartite quantum
system with respect to one of the components, when the coupling between its two
components is perturbative. We show that the density matrix of the considered
component is described by adiabatic transport formulae exhibiting
operator-valued geometric and dynamical phases. The present results can be used
to study the quantum control of the dynamics of qubits and of open quantum
systems where the two components are the system and its environment. We treat
two examples, the control of an atomic qubit interacting with another one and
the control of a spin in the middle of a Heisenberg spin chain.
