Computation of empirical eigenfunctions and order reduction for control of time-dependent parabolic PDEs

This article presents a methodology for the computation of empirical eigenfunctions and the construction of accurate low-dimensional approximations for control of nonlinear and time-dependent parabolic partial differential equations (PDE) systems. Initially, a representative (with respect to different initial conditions and input perturbations) ensemble of solutions of the time-varying PDE system is constructed by computing and solving a high-order discretization of the PDE. Then, the Karhunen-Loeve expansion is directly applied to the ensemble of solutions to derive a small set of empirical eigenfunctions (dominant spatial patterns) that capture almost all the energy of the ensemble of solutions. The empirical eigenfunctions are subsequently used as basis functions within a Galerkin's model reduction framework to derive low-order ordinary differential equation (ODE) systems that accurately describe the dominant dynamics of the PDE system. The method is applied to a diffusion-reaction process with nonlinearities, spatially-varying coefficients and time-dependent spatial domain, and is shown to lead to the construction of accurate low-order models and the synthesis of low-order controllers. The robustness of the predictions of the low-order models with respect to variations in the model parameters and different initial conditions, as well as the comparison of their performance with respect to low-order models which were constructed by using off-the-self basis function sets are successfully shown through computer simulations.