  Efficiently controlling the trapping process, especially the trapping
efficiency, is central in the study of trap problem in complex systems, since
it is a fundamental mechanism for diverse other dynamic processes. Thus, it is
of theoretical and practical significance to study the control technique for
trapping problem. In this paper, we study the trapping problem in a family of
proposed directed fractals with a deep trap at a central node. The directed
fractals are a generalization of previous undirected fractals by introducing
the directed edge weights dominated by a parameter. We characterize all the
eigenvalues and their degeneracies for an associated matrix governing the
trapping process. The eigenvalues are provided through an exact recursive
relation deduced from the self-similar structure of the fractals. We also
obtain the expressions for the smallest eigenvalue and the mean first-passage
time (MFPT) as a measure of trapping efficiency, which is the expected time for
the walker to first visit the trap. The MFPT is evaluated according to the
proved fact that it is approximately equal to reciprocal of the smallest
eigenvalue. We show that the MFPT is controlled by the weight parameter, by
modifying which, the MFPT can scale superlinealy, linearly, or sublinearly with
the system size. Thus, this work paves a way to delicately controlling the
trapping process in the fractals.
