Abstract: Optimal control problems,constrained by a state equation in the form of a partial differential equation (PDE),arise in many important applications, where one wants to steer the modelled process in order to have a solution close to some given target function.The discretized problems lead to a particular two-by-two block matrix form for which a very efficient preconditioner,leading to very tight eigenvalue bounds, will be presented. Various applications, such as in time-harmonic parabolic and Stokes equations and eddy current electromagnetic problems, will also be discussed.