Summary: Several difficulties in controller design for infinite-dimensional systems arise from using an approximation for the state of the system. In this paper it is demonstrated that the graph topology is an appropriate framework in which to discuss convergence of approximations used for controller design. It is also shown that Galerkin type approximations to large class of problems possess the required convergence properties and can be used to design controllers which will perform as desired when implemented on the original infinite-dimensional system. An \({\mathcal H}_ \infty\)-controller design problem is used to illustrate this approach.