# 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:

and

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.*