  The control landscape of a quantum system $A$ interacting with another
quantum system $B$ is studied. Only system $A$ is accessible through time
dependent controls, while system B is not accessible. The objective is to find
controls that implement a desired unitary transformation on $A$, regardless of
the evolution on $B$, at a sufficiently large final time. The freedom in the
evolution on $B$ is used to define an \emph{extended control landscape} on
which the critical points are investigated in terms of kinematic and dynamic
gradients. A spectral decomposition of the corresponding extended unitary
system simplifies the landscape analysis which provides: (i) a sufficient
condition on the rank of the dynamic gradient of the extended landscape that
guarantees a trap free search for the final time unitary matrix of system $A$,
and (ii) a detailed decomposition of the components of the overall dynamic
gradient matrix. Consequently, if the rank condition is satisfied, a gradient
algorithm will find the controls that implements the target unitary on system
$A$. It is shown that even if the dynamic gradient with respect to the controls
alone is not full rank, the additional flexibility due to the parameters that
define the extended landscape still can allow for the rank condition of the
extended landscape to hold. Moreover, satisfaction of the latter rank condition
subsumes any assumptions about controllability, reachability and control
resources. Here satisfaction of the rank condition is taken as an assumption.
The conditions which ensure that it holds remain an open research question. We
lend some numerical support with two common examples for which the rank
condition holds.
