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Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems

Lang, Jens ; Schmitt, Bernhard A. (2022):
Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems. (Publisher's Version)
In: Algorithms, 15 (9), MDPI, e-ISSN 1999-4893,
DOI: 10.26083/tuprints-00022452,
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Item Type: Article
Origin: Secondary publication DeepGreen
Status: Publisher's Version
Title: Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
Language: English
Abstract:

This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasize that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. In addition, as a benchmark method, the well-known backward differentiation formula BDF4, which is only A(73.3°)-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit A(84°)-stable method with nodes symmetric in [0, 1] to a common center that performs equally well. Numerical tests with four well established optimal control problems confirm the theoretical findings also concerning A-stability.

Journal or Publication Title: Algorithms
Volume of the journal: 15
Issue Number: 9
Place of Publication: Darmstadt
Publisher: MDPI
Collation: 30 Seiten
Uncontrolled Keywords: implicit Peer two-step methods, BDF-methods, nonlinear optimal control, first-discretize-then-optimize, discrete adjoints
Classification DDC: 500 Naturwissenschaften und Mathematik > 510 Mathematik
Divisions: 04 Department of Mathematics > Numerical Analysis and Scientific Computing
Date Deposited: 10 Oct 2022 12:47
Last Modified: 12 Oct 2022 05:39
DOI: 10.26083/tuprints-00022452
Corresponding Links:
URN: urn:nbn:de:tuda-tuprints-224527
Additional Information:

This article belongs to the Section Analysis of Algorithms and Complexity Theory

SWORD Depositor: Deep Green
URI: https://tuprints.ulb.tu-darmstadt.de/id/eprint/22452
PPN: 500268002
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