  In the context of Galerkin discretizations of a partial differential equation
(PDE), the modes of the classical method of Proper Orthogonal Decomposition
(POD) can be interpreted as the ansatz and trial functions of a low-dimensional
Galerkin scheme. If one also considers a Galerkin method for the time
integration, one can similarly define a POD reduction of the temporal
component. This has been described earlier but not expanded upon -- probably
because the reduced time discretization globalizes time which is
computationally inefficient. However, in finite-time optimal control systems,
time \textit{is} a global variable and there is no disadvantage from using a
POD reduced Galerkin scheme in time. In this paper, we provide a newly
developed generalized theory for space-time Galerkin POD, prove its optimality
in the relevant function spaces, show its application for the optimal control
of nonlinear PDEs, and, by means of a numerical example with Burgers' equation,
discuss the competitiveness by comparing to standard approaches.
