Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions
type: Item
instance of: scholarly article
title: Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions
zbMATH Open document ID: 1259.49003
author: Walter Alt
author: R. Baier
author: Matthias Gerdts
author: Frank Lempio
published in: Numerical Algebra, Control and Optimization
publication date: 2012-11-13T00:00:00Z
review text: In the paper, the authors investigate a class of linear-quadratic optimal control problems. The aim of the article is to derive error estimates for the Euler approximations of the considered problems. Such estimates were previously known only in the case of Lipschitz continuous controls. Since an optimal control for the given problems typically has a bang-bang structure, the assumption that the optimal control is Lipschitz continuous is not satisfied. After the Euler discretization for the linear-quadratic control problems, the authors show convergence of order \(h\) for the optimal values of the objective function, where \(h\) is the mesh size of the discretization. Assuming that the optimal control is of bang-bang type, it is shown then that the discrete and the continuous controls coincide except on a set of measure \(O(\sqrt{h})\). Here the authors use an approach based on a second-order optimality condition known from the stability analysis of bang-bang control. Finally, under a slightly stronger assumption on the smoothness of the coefficients of the system equation the error estimates for the discretized solutions are improved to order \(O(h)\). The authors also give a numerical example which confirms the error estimates.
Mathematics Subject Classification ID: 49J15
Mathematics Subject Classification ID: 49M25
Mathematics Subject Classification ID: 49N10
Mathematics Subject Classification ID: 49J30
zbMATH DE Number: 6104988
zbMATH Keywords: linear-quadratic optimal control problems
zbMATH Keywords: bang-bang control
zbMATH Keywords: discretization
MaRDI profile type: Publication
full work available at URL: naco.2012.2.547
OpenAlex ID: W2329217170
DOI: 10.3934/NACO.2012.2.547
Recommended article: Error bounds for euler approximation of a state and control constrained optimal control problem1
Recommended article: Error estimates for linear-quadratic control problems with control constraints
Recommended article: Error estimates for the approximation of multibang control problems
Recommended article: Q4427719
Recommended article: Error Analysis for the Implicit Euler Discretization of Linear-Quadratic Control Problems with Higher Index DAEs and Bang–Bang Solutions
Recommended article: Improved Error Estimate for an Implicit Discretization Scheme for Linear-Quadratic Control Problems with Bang-Bang Solutions
Recommended article: Q5323191
Recommended article: Approximations of linear control problems with bang-bang solutions
Recommended article: Large-Scale Scientific Computing
Recommended article: Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang-bang solutions
description: scientific article; zbMATH DE number 6104988