Fel'dbaum's own 1960 dual control papers decompose control risk into an additive "risk of action" and "risk of study"

feldbaumdual-control-theoryexploration-exploitationcontrol-theoryhistory-of-scienceprimary-source-verification

In "Dual Control Theory. I," 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 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).