We investigate the time-optimal control of the purification of a qubit interacting with a 
structured environment, consisting of a strongly coupled two-level defect in interaction with 
a thermal bath. On the basis of a geometric analysis, we show for weak and strong 
interaction strengths that the optimal control strategy corresponds to a qubit in resonance 
with the reservoir mode. We investigate under which conditions qubit coherence and 
correlation between the qubit and the environment can speed up the control process.