{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T04:08:43Z","timestamp":1771474123674,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,29]],"date-time":"2022-08-29T00:00:00Z","timestamp":1661731200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Mathematical modeling, simulation\n        and optimisation using the example of gas networks","award":["239904186"],"award-info":[{"award-number":["239904186"]}]},{"name":"Mathematical modeling, simulation\n        and optimisation using the example of gas networks","award":["TRR154\/2-2018"],"award-info":[{"award-number":["TRR154\/2-2018"]}]},{"name":"Mathematical modeling, simulation\n        and optimisation using the example of gas networks","award":["TP B01"],"award-info":[{"award-number":["TP B01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>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(73.35\u2218)-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\u2218)-stable method with nodes symmetric in [0,\u00a01] 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.<\/jats:p>","DOI":"10.3390\/a15090310","type":"journal-article","created":{"date-parts":[[2022,8,29]],"date-time":"2022-08-29T21:01:31Z","timestamp":1661806891000},"page":"310","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4603-6554","authenticated-orcid":false,"given":"Jens","family":"Lang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Technical University of Darmstadt, Dolivostra\u00dfe 15, 64293 Darmstadt, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5797-8024","authenticated-orcid":false,"given":"Bernhard A.","family":"Schmitt","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Philipps-Universit\u00e4t Marburg, Hans-Meerwein-Stra\u00dfe 6, 35043 Marburg, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"113133","DOI":"10.1016\/j.cam.2020.113133","article-title":"Symplectic Runge-Kutta discretization of a regularized forward-backward sweep iteration for optimal control problems","volume":"383","author":"Liu","year":"2021","journal-title":"J. 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