State and Control Spaces

To ensure the well-posedness of the problem, we must determine appropriate state and control spaces. First, we consider the weak formulation of the state equation (the Navier Stokes equations) which is given as follows:


This suggests the state space

To determine an appropriate control space, we observe that, by the trace theorem,

To avoid fractional Sobolev spaces in our numerical computation, we work with slightly more regular controls and select
as our control space .
The choice of the control space will show to be important for the numerical computation and the correct application of the optimization algorithm.