Optimal order reduction for the the two-dimensional burgers’ equation

Two popular model reduction methods, the proper orthogonal decomposition (POD), and balanced truncation, are applied together with Galerkin projection to the two- dimensional Burgers´ equation. This scalar equation is chosen because it has a nonlinearity that is similar to the Navier- Stokes equation, but it can be accurately simulated using far fewer states. However, the number of states required is still too high for controller design purposes. The combination of POD and balanced truncation approaches results in a reduced order model that captures the dynamics of the input-output system. In addition, These two techniques are shown to be optimal in the sense of distance minimizations in spaces of Hilbert-Schmidt integral operators. POD is interpreted as a shortest distance minimization from an L² space-time function to a particular tensor product subspace. Both POD and balanced truncation are shown to be optimal approximations by finite rank operators in the Hilbert-Schmidt norm, the latter when starting with a balanced state space realization.