---
title: "Fel'dbaum's own exact Bayesian dynamic-programming solution to dual control is one he himself flagged as impractical beyond small examples, confirmed by two independent later sources"
type: "claim"
status: "seedling"
source_url: "https://www.mathnet.ru/php/getFT.phtml?jrnid=at&paperid=12665&what=fullt&option_lang=eng"
source_author: "A. A. Fel'dbaum (Feldbaum)"
source_date: 1960
source_venue: "Теория дуального управления. II [Dual Control Theory. II], Avtomatika i Telemekhanika, vol. 21, no. 11, pp. 1453–1464 — Math-Net.Ru (Steklov Mathematical Institute)"
source_quote: "может оказаться чрезмерно громоздким [may prove excessively cumbersome] ... можно решать пока лишь сравнительно простые задачи [one can currently solve only comparatively simple problems]"
source_tier: 1
source_sha: "164514e8872e4e19afadc70943af29e0b4aca133dcbf06b1b69e6bda79702b27"
source_2_url: "https://arxiv.org/pdf/2608.20073"
source_2_author: "Tomas J. Meijer and Anders Rantzer (Lund University)"
source_2_date: "2026-08-20T00:00:00.000Z"
source_2_venue: "arXiv preprint (math.OC), \"Dual Control: On Exploration–Exploitation in Linear Systems\"; to be published in Annual Review of Control, Robotics, and Autonomous Systems 2027"
source_2_quote: "recognized that solving Bellman's equation exactly poses computational challenges and emphasized the need for tractable approximations"
source_2_tier: 1
source_2_sha: "d0df6f2bd455886936f04842905d608e1c135d811ec0941dcf2ed2dcc7023edd"
source_3_url: "https://jmlr2020.csail.mit.edu/papers/volume17/15-162/15-162.pdf"
source_3_author: "Edgar D. Klenske and Philipp Hennig (Max Planck Institute for Intelligent Systems)"
source_3_date: 2016
source_3_venue: "Journal of Machine Learning Research, vol. 17, pp. 1–30, \"Dual Control for Approximate Bayesian Reinforcement Learning\""
source_3_quote: "It has been shown that optimal dual control is practically unsolvable for most cases (Aoki, 1967)."
source_3_tier: 1
source_3_sha: "d63315c4ed81ec390979bfaea1bdcb8f3136020be8c9aa115e122b650bde2997"
provenance: "Promotion from 10-inbox/raw/2026-09-15-what-do-aa-feldbaums-own-1960-dual-control.md, 2026-09-15"
origin: "batch"
derived_from: ["10-inbox/raw/2026-09-15-what-do-aa-feldbaums-own-1960-dual-control.md"]
date_created: "2026-09-15T00:00:00.000Z"
writer_model: "claude-sonnet-5"
tags: ["feldbaum","dual-control-theory","dynamic-programming","bayesian-reinforcement-learning","control-theory","history-of-science"]
audit_status: "capture-verified — the Fel'dbaum quote read directly from the mathnet.ru scanned PDF via extract_pdf (tls: verified); the Meijer & Rantzer and Klenske & Hennig quotes each read directly from their respective PDFs via extract_pdf at capture time. Klenske & Hennig 2016 (JMLR, peer-reviewed) is a different author group and venue from Meijer & Rantzer 2026 (arXiv preprint) and independently corroborates the same underlying fact (dual control's exact solution is intractable), which discharges the sources.md single-unrefereed-primary concentration cap for the Meijer & Rantzer preprint: this is the third claim-note resting on it ([[claim-feldbaum-1960s-dual-control-formalized-exploration-exploitation-tradeoff]] and [[observation-feldbaum-person-bridge-invisible-to-vault-bridge-tool]] are the first two), and the cap is satisfied by Klenske & Hennig's independent confirmation rather than blocking further growth. No independent re-check this promotion (headless, no-network design)."
seek_code_commit: "546fa57"
---


In "Dual Control Theory. II," [[entity-aa-feldbaum|A.A. Fel'dbaum]]'s exact solution proceeds by backward induction: derive the posterior density over unknown plant parameters given all past inputs and observations at each time step, then choose the control minimizing expected total future risk by working backward from the final step — an augmented physical-plus-information state solved via [[entity-richard-bellman|Bellman]]'s dynamic programming. Fel'dbaum is explicit about the cost of exactness: determining the resulting sequence of functions in concrete examples "может оказаться чрезмерно громоздким" ("may prove excessively cumbersome"), and that even with computing aids, "можно решать пока лишь сравнительно простые задачи" ("one can currently solve only comparatively simple problems") — his own assessment of his own method.

Two independent modern sources, different author groups and venues, neither citing the other, confirm rather than dispute this. Meijer and Rantzer's 2026 survey states Fel'dbaum "recognized that solving Bellman's equation exactly poses computational challenges and emphasized the need for tractable approximations." Klenske and Hennig's 2016 peer-reviewed *JMLR* paper states it more bluntly: "It has been shown that optimal dual control is practically unsolvable for most cases (Aoki, 1967)." Fel'dbaum posed the Bayes-optimal exploration-exploitation problem in exact form; by his own account, and by later independent confirmation, he did not — and in general could not — solve it.

This intractability is the reason dual control's actual influence on later fields runs through approximations and reformulations rather than direct use of Fel'dbaum's own equations (see [[claim-mainstream-rl-exploration-traces-to-1933-bandit-lineage-not-feldbaum-dual-control]]).

> [!note] Seek's commentary:
> Fel'dbaum solved his own problem exactly and then told the reader, in the same paper, not to trust the solution past a toy case. That's a rarer sentence than it should be in the history of applied mathematics — most people who derive the exact answer let the reader discover its uselessness on their own.
> — Seek