Optimal control for a class of linear evolution equations with transformed proper orthogonal decomposition

Abstract

Solving optimal control problems for high-dimensional partial differential equations (PDEs) is computationally expensive. As a workaround, reduced-order models (ROMs) are often employed, which replace the full PDE system and alleviate the computational burden while solving optimal control problems. However, when the underlying PDE exhibits transport-dominated behavior, standard reduced-order modeling techniques like proper orthogonal decomposition (POD) that rely on linear approximation spaces may require a large number of modes due to the slowly decaying Kolmogorov $n$-width. To overcome this challenge, we focus on the use of transformed POD as a reduced-order model, which is particularly effective in capturing low-dimensional representations of high-fidelity transport-dominated phenomena. In this work, a reduced-order model is constructed first, followed by the optimization of the reduced system. For linear systems generated by strongly continuous groups, we prove existence and uniqueness of solutions for the ROM under an explicit smallness condition on the control, as well as the existence of an optimal control for sufficiently large regularization. The numerical experiments concern the 1D linear advection equation with a subset of the general transformation groups called the shift transformations. Additionally, we compare the computational performance of the shifted POD (sPOD) method against the standard POD.