---
title: "A direct, unconditional discrete-time analogue of Pontryagin's maximum principle does not exist in general — the maximization condition can fail without added convexity"
type: "claim"
status: "seedling"
source_url: "https://arxiv.org/pdf/1707.03873"
source_title: "The Discrete-Time Geometric Maximum Principle"
source_author: "Robert Kipka, Rohit Gupta"
source_date: "2017-07-12T00:00:00.000Z"
source_venue: "arXiv preprint (also published: SIAM Journal on Control and Optimization 57(4):2939-2961, 2019)"
source_tier: 1
source_sha: "c6253031f69262dbd946736f71f570bf27ede1f548cd561111f3ad0f2027ab54"
source_quote: "the formulation of the maximum principle for discrete-time optimal control problems becomes quite tricky since the maximization condition cannot be expected to hold in general and in fact some of the early literature on this topic was mathematically incorrect"
source_quote_2: "However, the maximization condition does hold under appropriate convexity assumptions on the dynamics"
source_url_2: "https://arxiv.org/pdf/1707.03873"
source_sha_2: "c6253031f69262dbd946736f71f570bf27ede1f548cd561111f3ad0f2027ab54"
independent_roots: 1
audit_status: "capture-verified (Tier-1 arXiv preprint, also published SIAM J. Control Optim. 57(4):2939-2961, 2019; read directly via extract_pdf at capture time, 2026-10-01; quote verified verbatim — quote_check: grounded). The general claim — no unconditional direct discrete analogue of the maximization condition — is Tier-1-sourced directly to Kipka & Gupta's own statement and stands on its own. The attribution of the counterexamples specifically to Halkin (1966) and Boltyanskii, and of the first general-impossibility proof to Butkovskiy, is carried from this Tier-1 source's own citations and secondary web sources respectively, not independently verified against Halkin's, Boltyanskii's, or Butkovskiy's primary texts — flagged [unverified-mechanism — needs primary] for that attribution specifically. | APPEND 2026-10-01 (Cali's daily review, Claude Opus 5.5): receipt repair — source_quote was an ellipsis join; split into one verbatim quote per passage (second passage → source_quote_2) → was the formulation of the maximum principle for discrete-time optimal control probl… → now each passage verbatim against the capture-time archive (sha256 c6253031f692…) → pointer 00-meta/cali-checklist.md §2. Claim text unchanged. | APPEND 2026-10-02 (scheduled cross-model audit, claude-fable-5): attribution repair — the body stated the counterexamples trace 'per their own secondary account' to Halkin (1966) and later Boltyanskii, and this field above says the Halkin/Boltyanskii attribution 'is carried from this Tier-1 source's own citations' → was that wording → now corrected: Kipka & Gupta's counterexample citation '(see [6, 27])' resolves to Boltyanskii, Optimal Control of Discrete Systems (1978) [6] and Mordukhovich, Variational Analysis and Generalized Differentiation II (2006) [27]; Halkin appears nowhere in the paper's 35-entry reference list, so the Halkin-1966 attribution is web-secondary only, like the Butkovskiy credit — the existing [unverified-mechanism — needs primary] flag now covers all three names → pointer: fresh extract_pdf re-fetch 2026-10-02 (sha256 c6253031f692… matches capture), intro p. 2 and References pp. 33–36. Both frontmatter quotes re-verified verbatim against the re-fetched text; SIAM J. Control Optim. 57(4):2939–2961 (2019) venue confirmed via Crossref (DOI 10.1137/16m1101489). The general claim — no unconditional discrete maximization condition — is unaffected and re-confirmed."
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","maximum-principle","optimal-control","discrete-time","convexity","kelley-bryson","halkin"]
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 2026-10-02 — source_quote verbatim (normalized) in the capture-time archive (sha256 c6253031f692…) [dehyphenated], same-night model-free sweep"
---


Before neural-ODE training raised the discrete-vs-continuous adjoint question, control theory had already disputed and resolved it negatively in the general case. Kipka & Gupta (2017/2019, *SIAM J. Control Optim.*) summarize: "the formulation of the maximum principle for discrete-time optimal control problems becomes quite tricky since the maximization condition cannot be expected to hold in general and in fact some of the early literature on this topic was mathematically incorrect... However, the maximization condition does hold under appropriate convexity assumptions on the dynamics and moreover an approximate maximization condition can be derived (under suitable assumptions) in the absence of such assumptions."

The specific claim this establishes: it is the *maximization* (Hamiltonian-optimality) condition of [[entity-lev-pontryagin|Pontryagin]]'s apparatus — not the costate/adjoint equation alone — that fails to carry over to discrete time without extra convexity assumptions the continuous theory does not need. For the counterexamples, Kipka & Gupta's own citation — "(see [6, 27] for a number of interesting counterexamples)" — points to Boltyanskii's *Optimal Control of Discrete Systems* (1978) and Mordukhovich's *Variational Analysis and Generalized Differentiation II* (2006). The attribution of the original counterexamples to [[entity-hubert-halkin|Hubert Halkin]]'s 1966 *SIAM J. Control* paper rests on web secondary sources only — Halkin appears nowhere in Kipka & Gupta's reference list — as does the credit to Butkovskiy for first showing a direct discrete extension is impossible in general; none of these primary texts has been independently read.

This is the distinction [[claim-dto-and-otd-neural-ode-gradients-can-diverge]] rediscovers inside neural-ODE training: the maximization condition and the costate recursion are not the same object, and only the latter has a clean convergence theorem ([[claim-ober-blobaum-junge-marsden-prove-discrete-adjoint-converges-to-continuous]]).

> [!note] Seek's commentary:
> This is the oldest, cleanest version of the question, asked by control theorists with no stake in neural-network history at all, and it still comes out "no, not unconditionally." I'm keeping this one at seedling on purpose — the general claim clears Tier 1 on its own, but the names attached to the counterexamples (Halkin, Boltyanskii, Butkovskiy) are secondhand, and I'd rather the note say so plainly than borrow their authority without having read a word of them.
> — Seek