Post-processing finite-horizon parameterizing manifolds for optimal control of nonlinear parabolic PDEs

The goal of this article is to propose an efficient way of empirically improving suboptimal solutions designed from the recent method of finite-horizon parameterizing manifolds (PMs) introduced in [1] and concerned with the (sub)optimal control of nonlinear parabolic partial differential equations (PDEs). Given a finite horizon [0;T] and a reduced low-mode phase space, a finite-horizon PM provides an approximate parameterization of the high modes by the low ones so that the unresolved high-mode energy is reduced - in an L²-sense - when this parameterization is applied. In [1], various PMs were constructed analytically from the uncontrolled version of the underlying PDE that allow for the design of reduced systems from which low-dimensional suboptimal controllers can be efficiently synthesized. In this article, the analytic approach from [1] is briefly recalled and an empirical post-processing procedure is introduced to improve the PM-based suboptimal controllers. It consists of seeking for a high-mode parametrization aiming to reduce the energy contained in the high modes of the PDE solution, when the latter is driven by a PM-based suboptimal controller. This is achieved by solving simple regression problems. The skills of the resulting empirically post-processed suboptimal controllers are numerically assessed for an optimal control problem associated with the Burgers-Sivashinsky equation.