  This article is devoted to developing an approach for manipulating the von
Neumann entropy $S(\rho(t))$ of an open two-qubit system with coherent control
and incoherent control inducing time-dependent decoherence rates. The following
goals are considered: (a) minimizing or maximizing the final entropy
$S(\rho(T))$; (b) steering $S(\rho(T))$ to a given target value; (c) steering
$S(\rho(T))$ to a target value and satisfying the pointwise state constraint
$S(\rho(t)) \leq \overline{S}$ for a given $\overline{S}$; (d) keeping
$S(\rho(t))$ constant at a given time interval. Under the Markovian dynamics
determined by a Gorini--Kossakowski--Sudarshan--Lindblad type master equation,
which contains coherent and incoherent controls, one- and two-step gradient
projection methods and genetic algorithm have been adapted, taking into account
the specifics of the objective functionals. The corresponding numerical results
are provided and discussed.
