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Suboptimality Estimates for the Semi‐Discretized LQR Problem for Parabolic PDEs
Author(s) -
Benner Peter,
Saak Jens
Publication year - 2010
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201010288
Subject(s) - linear quadratic regulator , mathematics , hilbert space , partial differential equation , discretization , galerkin method , parabolic partial differential equation , operator (biology) , optimal control , matrix (chemical analysis) , nonlinear system , quadratic equation , mathematical analysis , mathematical optimization , biochemistry , chemistry , physics , materials science , geometry , repressor , quantum mechanics , transcription factor , composite material , gene
The linear quadratic regulator problem (LQR) for parabolic partial differential equations (PDEs) has been understood to be an infinite‐dimensional Hilbert space equivalent of the finite‐dimensional LQR problem known from mathematical systems theory. The matrix equations from the finite‐dimensional case become operator equations in the infinite‐dimensional Hilbert space setting. A rigorous convergence theory for the approximation of the infinite‐dimensional problem by Galerkin schemes in the space variable has been developed over the past decades. Numerical methods based on this approximation have been proven capable of solving the case of linear parabolic PDEs. Embedding these solvers in a model predictive control (MPC) scheme, also nonlinear systems can be handled. Convergence rates for the approximation in the linear case are well understood in terms of the PDE's solution trajectories, as well as the solution operators of the underlying matrix/operator equations. However, in practice engineers are often interested in suboptimality results in terms of the optimal cost, i.e., evaluation of the quadratic cost functional. In this contribution, we are closing this gap in the theory. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)