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Acta Applicandae Mathematicae

, Volume 135, Issue 1, pp 81–144 | Cite as

Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs

  • Mickaël D. Chekroun
  • Honghu Liu
Article

Abstract

This article proposes a new approach for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. The approach fits into the long tradition of seeking for slaving relationships between the small scales and the large ones (to be controlled) but differ by the introduction of a new type of manifolds to do so, namely the finite-horizon parameterizing manifolds (PMs). Given a finite horizon [0,T] and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the high modes by the controlled low ones so that the unexplained high-mode energy is reduced—in a mean-square sense over [0,T]—when this parameterization is applied.

Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes introduced in Chekroun et al. (2014). These formulas allow for an effective derivation of reduced systems of ordinary differential equations (ODEs), aimed to model the evolution of the low-mode truncation of the controlled state variable, where the high-mode part is approximated by the PM function applied to the low modes. The design of low-dimensional suboptimal controllers is then obtained by (indirect) techniques from finite-dimensional optimal control theory, applied to the PM-based reduced ODEs.

A priori error estimates between the resulting PM-based low-dimensional suboptimal controller \(u_{R}^{\ast}\) and the optimal controller u are derived under a second-order sufficient optimality condition. These estimates demonstrate that the closeness of \(u_{R}^{\ast}\) to u is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with the suboptimal controller \(u_{R}^{\ast}\) and the optimal controller u ; and (ii) the energy kept in the high modes of the PDE solution either driven by \(u_{R}^{\ast}\) or u itself.

The practical performances of such PM-based suboptimal controllers are numerically assessed for optimal control problems associated with a Burgers-type equation; the locally as well as globally distributed cases being both considered. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results.

Keywords

Parabolic optimal control problems Low-order models Error estimates Burgers-type equation Backward–forward systems 

Notes

Acknowledgements

We are grateful to Monique Chyba and to Bernard Bonnard for their interest in our works on parameterizing manifolds, which led the authors to propose this article. MDC is also grateful to Denis Rousseau and Michael Ghil for the unique environment they provided to complete this work, at the CERES-ERTI, École Normale Supérieure, Paris. This work has been partly supported by the National Science Foundation grant DMS-1049253 and Office of Naval Research grant N00014-12-1-0911.

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Hawai‘i at MānoaHonoluluUSA
  2. 2.Department of Atmospheric & Oceanic SciencesUniversity of CaliforniaLos AngelesUSA

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