We consider the problem of controlling spatially structured states in reacting systems. The control objectives range from stabilization of linearly unstable solutions to prescribing the dynamics by direct manipulation of coherent structures through feedback.
An important first step in controller synthesis is model reduction of the original reaction-diffusion equations to a (small but accurate) system of ODEs. For this purpose, starting from traditional discretization methods (finite difference, pseudospectral) we proceed to recently introduced POD (Proper Orthogonal Decomposition)-Galerkin and nonlinear Galerkin methods. Linear state-space and modern geometric nonlinear control approaches are then applied to the resulting vectorfields.
POD-Galerkin as well as eigenvector-Galerkin and standard pseudospectral models are further reduced to nonlinear Galerkin models through an Approximate Inertial Manifold methodology; this methodology allows further reduction exploiting the separation of time scales between "higher" and "lower" modes in the hierarchy. These reduction techniques are compared both in terms of accuracy and computational efficiency in capturing the open- and closed-loop dynamics.
This is joint work with Stanislav Shvartsman and Edriss Titi.