Low-dimensional approximation and control of periodic solutions in spatially extended systems

Abstract

Nonlinear model reduction is combined with numerical continuation and linear state-space control techniques to design regulators for periodic solutions in a spatially extended system. We address issues of construction and systematic evaluation of low-dimensional dynamic models using Galerkin projections on empirical orthogonal eigenfunctions (also known as proper orthogonal decomposition modes or Karhunen-Lo\`eve modes). The reduced order dynamical systems are used first to compute the open-loop bifurcation diagrams and then to design feedback controllers stabilizing unstable limit cycles. We outline the steps for discrete-time controller design and computational linear stability analysis of the resulting hybrid (continuous-discrete) closed-loop systems.