Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations

In this paper, we present a discussion of the proper orthogonal decomposition (POD) as applied to simulation and feedback control of the one-dimensional heat equation. We provide two examples of input collections to which the POD process is applied. First, we apply POD directly to the finite element basis of linear B-splines. Next, we additionally include time snapshots. We show that although the second case provides better simulations, this POD basis is ill-suited for control problems. We provide a discussion of both the linear quadratic regulator (LQR) problem and the linear quadratic Gaussian (LQG) problem.