  The work considers an open qutrit system whose density matrix $\rho(t)$
evolution is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master
equation with simultaneous coherent (in the Hamiltonian) and incoherent (in the
superoperator of dissipation) controls. Incoherent control makes the
decoherence rates depending on time in a specific controlled manner and within
clear physical mechanics. We consider the problem of maximizing the
Hilbert-Schmidt overlap between the system's final state $\rho(T)$ and a given
target state $\rho_{\rm target}$ and the problem of minimizing the squared
Hilbert-Schmidt distance between these states. For the both problems, we
perform their realifications, derive the corresponding Pontryagin function,
adjount system (with the two cases of transversality conditions in view of the
two terminal objectives), and gradients of the objectives, adapt the one-,
two-, three-step gradient projection methods. For the problem of maximizing the
overlap, we also adapt the regularized first-order Krotov method. In the
numerical experiments, we analyze, first, the methods' operation and, second,
the obtained control processes, in respect to considering the environment as a
resource via incoherent control.
