  We describe quantum controllability under the influences of the quantum
decoherence induced by the quantum control itself. It is shown that, when the
controller is considered as a quantum system, it will entangle with its
controlled system and then cause quantum decoherence in the controlled system.
In competition with this induced decoherence, the controllability will be
limited by some uncertainty relation in a well-armed quantum control process.
In association with the phase uncertainty and the standard quantum limit, a
general model is studied to demonstrate the possibility of realizing a
decoherence-free quantum control with a finite energy within a finite time. It
is also shown that if the operations of quantum control are to be determined by
the initial state of the controller, then due to the decoherence which results
from the quantum control itself, there exists a low bound for quantum
controllability.
