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

Adiabatic theorem for bipartite quantum systems in weak coupling limit