  Optimal control of two-qubit quantum systems attracts high interest due to
applications ranging from two-qubit gate generation to optimization of receiver
for transferring coherence matrices along spin chains. State preparation and
manipulation is among important tasks to study for such systems. Typically
coherent control, e.g. a shaped laser pulse, is used to manipulate two-qubit
systems. However, the environment can also be used $\unicode{x2013}$ as an
incoherent control resource. In this article, we consider optimal state
manipulation for a two-qubit system whose dynamics is governed by the
Gorini-Kossakowski-Sudarshan-Lindblad master equation, where coherent control
enters into the Hamiltonian and incoherent control into both the Hamiltonian
(via Lamb shift) and the superoperator of dissipation. We exploit two
physically different classes of interaction with coherent control and optimize
the Hilbert-Schmidt overlap between final and target density matrices,
including optimization of its steering to a given value. We find the conditions
when zero coherent and incoherent controls satisfy the Pontryagin maximum
principle, and in addition, when they form a stationary point of the objective
functional. Moreover, we find a case when this stationary point provides the
globally minimal value of the overlap. Using upper and lower bounds for the
overlap, we develop one- and two-step gradient projection methods operating
with functional controls.
