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Discrete Adjoint Implicit Peer Methods in Optimal Control

  • It is well known that in the first-discretize-then-optimize approach in the control of ordinary differential equations the adjoint method may converge under additional order conditions only. For Peer two-step methods we derive such adjoint order conditions and pay special attention to the boundary steps. For $s$-stage methods, we prove convergence of order s for the state variables if the adjoint method satisfies the conditions for order s-1, at least. We remove some bottlenecks at the boundaries encountered in an earlier paper of the first author et al. [J. Comput. Appl. Math., 262:73--86, 2014] and discuss the construction of 3-stage methods for the order pair (3,2) in detail including some matrix background for the combined forward and adjoint order conditions. The impact of nodes having equal differences is highlighted. It turns out that the most attractive methods are related to BDF. Three 3-stage methods are constructed which show the expected orders in numerical tests.

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Metadaten
Author:Jens Lang, Bernhard A. Schmitt
DOI:https://doi.org/https://doi.org/10.1016/j.cam.2022.114596
Document Type:Article
Language:English
Date of Publication (online):2021/02/27
Release Date:2021/02/27
Volume:Journal of Computational and Applied Mathematics
Issue:416:114596
Subprojects:B01
Licence (German):License LogoCreative Commons - CC0 1.0 - Universell - Public Domain Dedication