This work considers various families of quantum control landscapes (i.e.
objective functions for optimal control) for obtaining target unitary
transformations as the general solution of the controlled Schr\"odinger
equation. We examine the critical point structure of the kinematic landscapes
J_F (U) = ||(U-W)A||^2 and J_P (U) = ||A||^4 - |Tr(AA'W'U)|^2 defined on the
unitary group U(H) of a finite-dimensional Hilbert space H. The parameter
operator A in B(H) is allowed to be completely arbitrary, yielding an objective
function that measures the difference in the actions of U and the target W on a
subspace of state space, namely the column space of A. The analysis of this
function includes a description of the structure of the critical sets of these
kinematic landscapes and characterization of the critical points as maxima,
minima, and saddles. In addition, we consider the question of whether these
landscapes are Morse-Bott functions on U(H). Landscapes based on the intrinsic
(geodesic) distance on U(H) and the projective unitary group PU(H) are also
considered. These results are then used to deduce properties of the critical
set of the corresponding dynamical landscapes.Comment: 15 pages, 3 figure