  PDE-constrained optimization problems arise in a broad number of applications
such as hyperthermia cancer treatment or blood flow simulation. Discretization
of the optimization problem and using a Lagrangian approach result in a
large-scale saddle-point system, which is challenging to solve, and acquiring a
full space-time solution is often infeasible. We present a new framework to
efficiently compute a low-rank approximation to the solution by reformulating
the KKT system into a Sylvester-like matrix equation. This matrix equation is
subsequently projected onto a small subspace via an iterative rational Krylov
method and we obtain a reduced problem by imposing a Galerkin condition on its
residual. In our work we discuss implementation details and dependence on the
various problem parameters. Numerical experiments illustrate the performance of
the new strategy also when compared to other low-rank approaches.
