Joint work with Max D. Gunzburger, School of Computational Science, Florida State University. Optimization problems constrained by partial differential equations (PDEs) are particularly challenging from a computational point of view: the first order necessary conditions for optimality lead to a coupled system of PDEs. Specifically, for the solution of control problems constrained by a parabolic PDE, one needs to solve a system of PDEs coupled globally in time and space. For these, conventional time-stepping methods quickly reach their limitations due to the enormous demand for storage. For such a coupled PDE system, adaptive methods which aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities in the data or domain, with respect to both space and time, appear to be most promising. Here I propose an adaptive method based on wavelets. It builds on a recent paper by Schwab and Stevenson where a single linear parabolic evolution problem is formulated in a weak space-time form and where an adaptive wavelet method is designed for which convergence and optimal convergence rates (when compared to wavelet-best N term approximation) can be shown. Our approach extends this paradigm to control problems constrained by evolutionary PDEs for which we can prove convergence and optimal rates for each of the involved unknowns (state, costate, and control).