Abstract
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.