This problem is motivated by a medical research project, Kent-Andre Mardal is involved in, where the waste clearance in the human brain is investigated. There is the hypothesis that the waste clearance is controlled by the circulation of cerebrospinal fluid (CSF) through diffusive processes, potentially also by advection or chemical reactions. In clinical studies, tracer liquid is injected in the cerebrospinal fluid of human patients and its concentration is then measured by magnetic resonance imaging (MRI). The behavior of the tracer can be modeled with a parabolic partial differential equation (PDE), depending on the mentioned processes (diffusion, advection, reaction). This gives rise to a PDE-constrained optimization problem with limited observation in time (the MRI images). We restrict ourselves to a model problem only involving diffusion. The solution to this problem is characterized by the first order optimality system (Karush-Kuhn-Tucker conditions), which is a linear system with saddle point structure. We solve this system with a Krylov space solver and a robust preconditioner. The preconditioner is based on a Schur complement formulation for the state variable, which leads to a formulation that requires H^2-regularity in space and H^1-regularity in time. A conforming discretization would be required to have C^1 smooth basis functions. While this would need quite some effort with standard finite element methods, in Isogeometric Analysis such basis functions can be set up with ease. Since the observation is done only for certain points in time, but always for the whole spatial domain, we can use fast diagonalization to efficiently realize the space-time preconditioner.