  In this paper we analyze an optimization problem with limited observation
governed by a convection--diffusion--reaction equation. Motivated by a Schur
complement approach, we arrive at continuous norms that enable analysis of
well-posedness and subsequent derivation of error analysis and a preconditioner
that is robust with respect to the parameters of the problem. We provide
conditions for inf-sup stable discretizations and present one such
discretization for box domains with constant convection. We also provide a
priori error estimates for this discretization. The preconditioner requires a
fourth order problem to be solved. For this reason, we use Isogeometric
Analysis as a method of discretization. To efficiently realize the
preconditioner, we consider geometric multigrid with a standard Gauss-Seidel
smoother as well as a new macro Gauss-Seidel smoother. The latter smoother
provides good results with respect to both the geometry mapping and the
polynomial degree.
