  A quantum control landscape is defined as the expectation value of a target
observable $\Theta$ as a function of the control variables. In this work
control landscapes for open quantum systems governed by Kraus map evolution are
analyzed. Kraus maps are used as the controls transforming an initial density
matrix $\rho_{\rm i}$ into a final density matrix to maximize the expectation
value of the observable $\Theta$. The absence of suboptimal local maxima for
the relevant control landscapes is numerically illustrated. The dependence of
the optimization search effort is analyzed in terms of the dimension of the
system $N$, the initial state $\rho_{\rm i}$, and the target observable
$\Theta$. It is found that if the number of nonzero eigenvalues in $\rho_{\rm
i}$ remains constant, the search effort does not exhibit any significant
dependence on $N$. If $\rho_{\rm i}$ has no zero eigenvalues, then the
computational complexity and the required search effort rise with $N$. The
dimension of the top manifold (i.e., the set of Kraus operators that maximizes
the objective) is found to positively correlate with the optimization search
efficiency. Under the assumption of full controllability, incoherent control
modelled by Kraus maps is found to be more efficient in reaching the same value
of the objective than coherent control modelled by unitary maps. Numerical
simulations are also performed for control landscapes with linear constraints
on the available Kraus maps, and suboptimal maxima are not revealed for these
landscapes.
