  In this work, we consider a model of two qubits driven by coherent and
incoherent time-dependent controls. The dynamics of the system is governed by a
Gorini-Kossakowski-Sudarshan-Lindblad master equation, where coherent control
enters into the Hamiltonian and incoherent control enters into both the
Hamiltonian (via Lamb shift) and the dissipative superoperator. We consider two
physically different classes of interaction with coherent control and study the
optimal control problem of state preparation formulated as minimization of the
Hilbert-Schmidt distance's square between the final density matrix and a given
target density matrix at some fixed target time. Taking into account that
incoherent control by its physical meaning is a non-negative function of time,
we derive an analytical expression for the gradient of the objective and
develop optimization approaches based on adaptation for this problem of
GRadient Ascent Pulse Engineering (GRAPE). We study evolution of the von
Neumann entropy, purity, and one-qubit reduced density matrices under optimized
controls and observe a significantly different behavior of GRAPE optimization
for the two classes of interaction with coherent control in the Hamiltonian.
