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Approximations of linear control problems with bang-bang solutions

Title data

Alt, Walter ; Baier, Robert ; Lempio, Frank ; Gerdts, Matthias:
Approximations of linear control problems with bang-bang solutions.
In: Optimization. Vol. 62 (2013) Issue 1 . - pp. 9-32.
ISSN 0233-1934
DOI: https://doi.org/10.1080/02331934.2011.568619

Review:

Abstract in another language

We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the <i>L<sup>1</sup></i>-norm and with order 1/2 w.r.t. the <i>L<sup>2</sup></i>-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.

Further data

Item Type: Article in a journal
Refereed: Yes
Additional notes: CONTENTS:
1. Introduction
2. Euler Approximation
2.1 Discretization
2.2 Error estimates for the switching function
2.3 Error estimates for bang-bang controls
2.4 Numerical examples
3. Higher Order Approximations
Keywords: Linear optimal control; Bang-bang control; Discretization
Subject classification: Mathematics Subject Classification Code: 49J15 (49M25 49N05 49J30)
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Former Professors
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) > Chair Mathematics V (Applied Mathematics) - Univ.-Prof. Dr. Lars Grüne
Faculties
Result of work at the UBT: Yes
DDC Subjects: 500 Science
Date Deposited: 18 Feb 2021 11:23
Last Modified: 23 Mar 2021 13:02
URI: https://eref.uni-bayreuth.de/id/eprint/63114