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


Chekroun, M. D., and H. Liu. 2016. “Post-processing finite-horizon parameterizing manifolds for optimal control of nonlinear parabolic PDEs.” 2016 IEEE 55th Conference on Decision and Control (CDC), 1411-1416. Las Vegas, USA: IEEE.


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 by Chekroun and Liu (Acta Appl. Math., 2015) 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 unexplained high-mode energy is reduced — in an L 2-sense — when this parameterization is applied. In Chekroun and Liu (Acta Appl. Math., 2015), 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 Chekroun and Liu (Acta Appl. Math., 2015) is recalled and a 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 a Burgers-type equation.

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Last updated on 12/23/2023