  Semilinear parabolic systems with bi-linear nonlinearities cover a lot of
applications and their optimal control leads to relatively simple optimality
conditions. An example is the incompressible Navier-Stokes system for
homogeneous fluids, which is however here modified towards a physically
reasonable model of slightly (so-called ``semi'') compressible liquids rather
than fully compressible gases. An optimal control problem optimizing also
pressure on the boundary is considered and, in the simple variant, analysed as
far as uniqueness of the control-to-state mapping and 1st-order optimality
conditions in the 2-dimensional case and outlined in a nonsimple variant for
the 3-dimensional case. Some other bi-linear parabolic systems as Cahn-Hilliard
diffusion or magneto-hydrodynamics can be treated analogously.
