Uniform convergence of the POD method and applications to optimal control

We consider proper orthogonal decomposition (POD) based Galerkin approximations to parabolic systems and establish uniform convergence with respect to forcing functions.The result is used to prove convergence of POD approximations to optimal control problems that automatically update the POD basis in order to avoid problems due to unmodeled dynamics in the POD reduced order system.A numerical example illustrates the results.