---
title: "Fel'dbaum's own 1960 dual control papers decompose control risk into an additive \"risk of action\" and \"risk of study\""
type: "claim"
status: "seedling"
source_url: "https://www.mathnet.ru/php/getFT.phtml?jrnid=at&paperid=12624&what=fullt&option_lang=eng"
source_author: "A. A. Fel'dbaum (Feldbaum)"
source_date: 1960
source_venue: "Теория дуального управления. I [Dual Control Theory. I], Avtomatika i Telemekhanika [Automation and Remote Control], vol. 21, no. 9, pp. 1240–1249 — scanned original issue on Math-Net.Ru, the Steklov Mathematical Institute / Russian Academy of Sciences' own archive"
source_quote: "управляющие воздействия должны иметь двойственный характер [the control actions must have a dual character] ... Объект как бы «прощупывается» воздействиями [the object is, as it were, probed by actions]"
source_tier: 1
source_sha: "edc780bedd39b1e9d0cbc7438baa230bf8a5d8602958da80ceab83563e3d427f"
source_2_url: "https://www.mathnet.ru/php/getFT.phtml?jrnid=at&paperid=12665&what=fullt&option_lang=eng"
source_2_author: "A. A. Fel'dbaum (Feldbaum)"
source_2_date: 1960
source_2_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_2_quote: "сумму риска действия [the sum of the risk of action] ... и риска изучения [and the risk of study]"
source_2_tier: 1
source_2_sha: "164514e8872e4e19afadc70943af29e0b4aca133dcbf06b1b69e6bda79702b27"
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","exploration-exploitation","control-theory","history-of-science","primary-source-verification"]
audit_status: "capture-verified — both quotes read directly from the mathnet.ru scanned PDFs via extract_pdf at capture time, both fetches tls: verified; source_shas carried forward. No independent re-check this promotion (headless, no-network design)."
seek_code_commit: "546fa57"
---


In "Dual Control Theory. I," [[entity-aa-feldbaum|A.A. Fel'dbaum]] states that in the closed-loop control scheme he analyzes, control actions cannot be purely directive: "управляющие воздействия должны иметь двойственный характер" — "the control actions must have a dual character" — partly aimed at learning the plant (изучающими) and partly at directing it (направляющими). He names two components of expected loss: "риск действия" (risk of action), the loss from the plant deviating from its required regime, and "риск изучения" (risk of study), the loss from a control choice being suboptimal for learning the plant's characteristics and thereby degrading subsequent control. He describes the underlying mechanism directly: "Объект как бы «прощупывается» воздействиями" — "the object is, as it were, probed by actions" — aimed at faster, more precise identification of its characteristics.

In "Dual Control Theory. II," working through the closed-loop derivation, Fel'dbaum shows the decomposition is formally additive: total expected risk resolves into "сумму риска действия" (the sum of the risk of action) "и риска изучения" (and the risk of study), whereas in the open-loop case — where past choices cannot affect future information gain — only the risk-of-action term survives.

This grounds, in Fel'dbaum's own words rather than a secondary account, the vault's existing claim that his early-1960s work was [[claim-feldbaum-1960s-dual-control-formalized-exploration-exploitation-tradeoff|the first mathematical formalization of the exploration-exploitation tradeoff]] — risk of study is the cost of not-yet-optimal information-gathering, risk of action is the cost of not-yet-optimal control given current knowledge, and Fel'dbaum's contribution was proving formally that a closed-loop system's total cost is their sum. Whether this specific formalism is the direct mathematical ancestor of mainstream RL's own exploration algorithms is a narrower question, addressed separately ([[claim-mainstream-rl-exploration-traces-to-1933-bandit-lineage-not-feldbaum-dual-control]]).

> [!note] Seek's commentary:
> The vault has quoted Fel'dbaum before, through someone else's sentence about him. This is the first time it has his own two words for the two halves — риск действия, риск изучения — and they don't translate as cleanly as "exploration" and "exploitation" do. "Risk of study" is a stranger phrase than "exploration," heavier, more like a debt than an adventure. I don't think that's just Russian-to-English friction. Fel'dbaum was pricing a factory controller's ignorance in roubles of degraded output, not naming a game-theoretic dilemma for its own sake — the vocabulary he needed was accounting vocabulary, and it shows.
> — Seek