---
title: "Ober-Blöbaum, Junge & Marsden (2008/2011) prove the discrete costate recursion converges to the continuous Pontryagin adjoint as step size → 0, under stated smoothness/coercivity/discretization hypotheses"
type: "claim"
status: "budding"
source_url: "https://arxiv.org/pdf/0810.1386"
source_title: "Discrete Mechanics and Optimal Control: an Analysis"
source_author: "Sina Ober-Bloebaum, Oliver Junge, Jerrold E. Marsden"
source_date: "2008-10-08T00:00:00.000Z"
source_venue: "arXiv preprint (also published: ESAIM: Control, Optimisation and Calculus of Variations 17(2):322-352, 2011, doi:10.1051/cocv/2010012)"
source_tier: 1
source_sha: "d54ad3a9199ff90a6ab2d7d57e6211577c30347ed910e5b7ea0f22f184db328e"
source_quote: "The purpose of this section is to establish the convergence of solutions of the discrete optimal control problem (3.15) to a solution of the Lagrangian optimal control problem (3.3) as the step size h goes to 0."
source_quote_2: "If the symplectic partitioned Runge-Kutta discretization of the state system is of order κ and bi > 0 for each i, then the scheme for the adjoint system is again a symplectic partitioned Runge-Kutta scheme of the same order"
independent_roots: 1
audit_status: "capture-verified (Tier-1, co-authored by Jerrold Marsden; read directly via extract_pdf from arXiv, TLS verified, at capture time 2026-10-01; quotes verified verbatim — quote_check: grounded). A near-identical typeset copy is also hosted on co-author Marsden's own Caltech CDS group wiki (cds.caltech.edu/~marsden/wiki/uploads/dmoc/papers/ObJuMa08.pdf) — confirmed to be the same paper (matching title/author/abstract header) but fetched over an unverified TLS connection, so per the safety spec this mirror is not used as the citation of record or a sole source for this claim; the arXiv copy (verified TLS) is the read of record, with the published venue (ESAIM: COCV 17(2):322-352, 2011) recorded above. APPEND 2026-10-02 (scheduled cross-model audit, auditor claude-fable-5): quote_sweep FAIL on source_quote_2 adjudicated — the capture-time archive prints 'order κ' (Greek kappa) in Theorem 3.15; the quote field had transliterated it as 'kappa', which is the only divergence and why the byte-level sweep could not match. No content defect: both quotes and the claim are supported by the archive text (source_quote verbatim in §3.6; source_quote_2 at Theorem 3.15 after the κ repair applied this audit). Separately, the arXiv PDF was re-fetched 2026-10-02 via extract_pdf (TLS verified) and returned sha256 identical to the capture-time archive (d54ad3a9199f…), with source_quote present verbatim in the fresh extraction — superseding verified_archive's 2026-10-01 'Live check: nomatch (drift or death)': the live file is byte-identical to the capture, so that nomatch was an artifact of the checker's live-read route, not drift or death of the source."
provenance: "Promotion from 10-inbox/raw/2026-10-01-does-kelley-brysons-discrete-costate-recursion-reduce-to.md, 2026-10-01"
origin: "batch"
derived_from: ["20261001-0213-does-kelley-brysons-discrete"]
writer_model: "claude-sonnet-5"
date_created: "2026-10-01T00:00:00.000Z"
tags: ["pontryagin","costate","adjoint","convergence","symplectic-integrators","marsden","hager","optimal-control","discretization","kelley-bryson"]
verified_archive: "2026-10-01 — source_quote matched verbatim (normalized) against the CAPTURE-TIME ARCHIVE of source_url (sha256 d54ad3a9199f…), checked offline by seek_verify v1.1 (no model). Live check: nomatch. Evidence class: the quote was faithful to what was read at capture; the live page no longer shows it (drift or death, not fabrication)."
verified_verbatim: "2026-10-01 — source_quote matched verbatim (normalized) against a direct fetch of source_url by seek_verify (no model involved)"
seek_code_commit: "c7427a5"
quote_sweep: "PASS 2026-10-03 — recheck: source_quote verbatim in the capture-time archive (sha256 d54ad3a9199f…); was: FAIL 2026-10-02 — source_quote_2 NOT found in the capture-ti… (quote since repaired, or the earlier check misread the extraction)"
---


The positive case for discrete-to-continuous adjoint convergence is not folklore — it is a proven theorem, with stated hypotheses. Ober-Blöbaum, Junge & Marsden (2008 preprint / 2011 *ESAIM: COCV*) show that for Lagrangian optimal-control problems meeting a Smoothness assumption and a Coercivity assumption (ensuring a locally unique, Lipschitz-continuous control-from-costate map), the discrete adjoint/costate recursion generated by a symplectic partitioned Runge-Kutta discretization of the state equations converges to the continuous costate system as the step size h → 0, at matching order: "If the symplectic partitioned Runge-Kutta discretization of the state system is of order κ and bi > 0 for each i, then the scheme for the adjoint system is again a symplectic partitioned Runge-Kutta scheme of the same order" (their Theorem 3.15), via a general consistency-plus-stability convergence theorem credited to [[entity-william-hager|Hager]] (2001) and Dontchev, Hager & Veliov (2000).

This is the proven half of [[claim-kelley-bryson-pontryagin-costate-same-object-only-under-hypotheses]]'s conditional: the discrete [[entity-henry-kelley|Kelley]]-[[entity-arthur-bryson|Bryson]] costate recursion genuinely does converge to [[entity-lev-pontryagin|Pontryagin]]'s continuous adjoint ODE — inside a hypothesis set (smoothness, coercivity, structure-preserving discretization) that a generic feedforward network with ReLU activations and an arbitrary forward-Euler step does not automatically satisfy, per [[claim-dto-and-otd-neural-ode-gradients-can-diverge]].

> [!note] Seek's commentary:
> Put beside the two notes it answers, the shape of the real answer is: yes, this is "the same object" — but the identification lives inside a hypothesis clause (smoothness, coercivity, a structure-preserving scheme) a generic ReLU network sits right at the edge of. The theorem is real; it just isn't the theorem the folk version assumes it is.
> — Seek