In a communication problem the receiver generally tries to reconstruct something emitted elsewhere. In a control problem the controller chooses an input whose physical consequence becomes part of the next observation. This difference can disappear in simple linear calculations: a closed loop may admit an equivalent open-loop representation for a selected input–output relation. Feldbaum argues that it returns forcefully with nonlinear objects, constraints and statistically optimal control. The structural scheme must show who acts on what, what is observed and what remains uncertain.

His controlled object B receives a control input and an uncontrolled disturbance, then produces an output. Each quantity may be a vector. If the object has memory, its condition is represented by a point in an n-dimensional phase space. Complete information would include the object’s operator, its current state and the disturbance trajectory. The controller knows the command it issues, but usually lacks some or all of the remaining ingredients.

An open-loop controller asked to reproduce an exact desired trajectory would need extraordinary foresight: a complete object description, the initial state and future disturbance. Feedback relaxes that demand. By comparing observed output with the desired regime, a controller can compensate for disturbances it never measures directly, provided the object belongs to a suitable class. More subtly, the stream of commanded inputs and measured outputs supplies indirect evidence about the object itself.

This is not free knowledge. Noise corrupts the desired-signal, observation and actuation channels in Feldbaum’s general diagram. A finite experiment cannot normally identify one exact operator or disturbance history. The rational description is therefore a probability distribution over possibilities. Closed-loop experience can concentrate that distribution, but the rate and direction of concentration depend on which controls were applied.

This move matters because “unknown” can conceal several different situations. A fixed but unknown plant parameter differs from a fresh random disturbance at every step. A sensor’s noisy reading differs from uncertainty about the desired signal. Feldbaum draws separate channels so these sources do not collapse into one undifferentiated error term. In his later formal restrictions, independent noise sequences pass through known, memoryless combination rules, yielding conditional probabilities for observation given output and applied input given commanded input.

Suppose there are several hypotheses about the object operator. Each predicts a distribution of outputs after a proposed command. Bayes’s rule turns an observed output into revised probabilities. Repetition tends to concentrate weight around hypotheses closer to the actual object, but not all commands produce equally distinctive predictions. If two models respond almost identically near the current operating point, staying there can be excellent short-run regulation and terrible identification.

Feldbaum explicitly rejects a tempting shortcut: choose the most probable hypothesis, assign it probability one, assign every rival probability zero, and optimise as though the selection were certainly true. Such a controller may be easy to calculate, but in general it is not optimal in the broader problem. The probability distribution obtained from experience is itself information. Collapsing it erases both residual uncertainty and the prospective value of observations that would discriminate among surviving models.

The posterior is not merely a report for an engineer. In the optimal-control formulation it is part of the decision state. Today’s action changes both physical output and the likelihood of tomorrow’s observations. Tomorrow’s action should depend on those observations and their implied posterior. The controller is therefore selecting a tree of contingent moves, even when only one branch will be realised.

Open-loop study can be passive: watch an uncontrolled process and refine estimates as data arrive. A closed loop makes active study possible. The controller can vary its input specifically so the object’s response reveals its characteristics or a better way to control it. Feldbaum’s Russian imagery is tactile: the object is “felt out” or probed by cognitive actions. This is experimentation inside operation.

It would be easy to picture a neat superposition: a small, high-frequency test signal riding on a slow production command, or alternating periods labelled experiment and work. Feldbaum notes that automatic-search systems often permit such separation by frequency band or time. He then makes the more general claim. Separation is not necessary. One and the same action can be partly cognitive and partly directing, with consequences that cannot be assigned to two physically distinct inputs.

Consider a pharmacist adjusting a dose under uncertain sensitivity. A larger dose may move the patient towards the clinical target and reveal sensitivity more clearly. The informative consequence is not a decorative perturbation appended to the therapeutic act; it arises through the same intervention. This example also makes the ethical and safety boundary plain. Information has no absolute value. An experiment that reveals much but exposes the patient to unacceptable risk is not an admissible control.

The word “dual” here does not mean a mathematical dual problem, two separate controllers, or the duality between controllability and observability. It names two causal roles of action under uncertainty. Nor is every feedback response dual in the strong sense. If the plant is completely known, or if future policy cannot use the information, there is no reason to sacrifice present performance for learning. The dual effect becomes decision-relevant when action affects observations, observations revise uncertainty and revised uncertainty changes later action.

Feldbaum describes the extremes in practical language. An excessively hasty system directs the object without enough evidential support from study. An excessively cautious system waits too long, processing information while failing to move the object towards the required regime. Either can make the process poor or entirely unsuccessful. The task is to choose the most advantageous combination according to a performance criterion.

This is richer than the slogan “explore versus exploit”. The working part of an action can itself yield information, and a nominal probe can also advance the plant. Present physical loss and future informational value share the same units only after they are carried through the objective. A bit of entropy reduction is not automatically worth a degree of temperature error, a metre of tracking deviation or an extra clinical hazard. The control problem supplies the exchange rate through expected future loss.

The conflict is also asymmetric in time. Today’s error cannot be undone merely because the controller learns later, while today’s information can improve several later commands. Longer horizons can therefore justify more probing, but only if uncertainty persists, observations are informative and subsequent decisions have room to adapt. Near the final step, the value of learning normally collapses because there are fewer opportunities to use it.

Constraints alter the bargain. A probe that would be optimal in an unconstrained quadratic model may be prohibited by actuator limits or safety envelopes. Conversely, a carefully chosen action inside the safe region may reveal enough to avoid repeated conservative operation. Dual control does not license reckless experiments; it formalises why information gained by admissible actions can have operational value.

Action risk measures loss because the current intervention makes the object depart, now or later, from the required or best attainable regime. It is the familiar physical cost of a command: deviation, effort, damage or delay. Learning risk measures loss because the present intervention is poor for obtaining information, causing subsequent interventions to be worse than they could have been. It is expressed not as abstract ignorance but as degraded later control.

The categories clarify two policies that can have the same immediate output. Imagine a robust conservative command that performs moderately under every hypothesis but reveals nothing. A more discriminating command may have the same expected first-stage tracking error, yet lead to different observable outputs under the competing models. The second can have lower learning risk because later decisions become tailored. Conversely, a violent probe may reduce uncertainty while creating so much immediate deviation that its action risk dominates.

The decomposition must not be treated as a universal algebraic identity lifted out of context. Part I states that equality for a defined class will be established later. The paper’s immediate contribution is to show why a current decision must account for both channels of consequence. A modern reconstruction should preserve this sequencing: concept first, formal conditions and solution later.

It also guards against a category error common in adaptive systems. Model-estimation error is not itself the final loss unless the application says so. Two parameter estimates can be statistically different yet recommend the same safe action; reducing their uncertainty has no control value at that moment. Another tiny distinction can flip a high-stakes command. Learning risk depends on how beliefs enter future decisions, not on a context-free accuracy score.

His first architecture contains a main controller A, an object B and an auxiliary controller A′. The main unit governs the object, in open or closed form. The auxiliary unit observes object input and output, estimates the object’s characteristics and uses a pre-specified algorithm to calculate better parameters for A. A further signal w installs those parameters. There are studying actions in the main loop, yet the parameter change is not found by automatic search; it is computed from the identified characteristics.

In the second architecture, automatic optimisation, the same input both studies B and directs its output towards the extremum of a quantity Q. Observed input–output responses are analysed to decide the directing component of the next intervention on the same object. Here the combination of studying and directing sits inside a genuine search loop.

The third architecture enlarges the object boundary. Treat controller A together with plant B as one complex object O. An auxiliary controller A′ studies the process and changes A’s control algorithm through w. Those algorithm changes have a dual character because they reveal how the closed system behaves and move its behaviour towards an admissible, favourable or optimal form. The lesson is architectural: “the object” is a modelling choice, and dual effects can arise at several layers.

This layered view anticipates modern adaptive control, experiment design and learning systems, but historical restraint matters. Feldbaum is not claiming that every parameter update solves the statistically optimal dual problem. A fixed identification-then-control routine may probe actively while remaining suboptimal because it schedules learning without pricing its effect on operational loss. The broader concept and the exact optimum are different claims.

Time is discrete but signal levels remain continuous. All variables are considered at instants s from zero to n, and n is fixed. Feldbaum notes that a passage towards continuous time may sometimes be made informally as the interval shrinks, while a rigorous limit is difficult. Infinite horizons and a variable, initially unknown stopping time are interesting but outside this series’ chosen formulation.

The problem is Bayesian: prior densities for random quantities are given. Feldbaum mentions minimax and “inductive probability” alternatives but does not solve them here. Desired-signal and disturbance functions depend on random parameter vectors with known priors. Noise sequences are independent under the stated assumptions. These choices define a probability model rather than asserting that all control problems naturally arrive with trustworthy priors.

For Part I the object is memoryless: its output at step s depends only on the disturbance and applied input at the same step. The relevant functions are assumed finite, single-valued, continuous and differentiable. Feldbaum stresses that memoryless objects include important model experiments: set initial conditions or parameters, run an experiment and register its result. Objects whose present output depends on earlier states, disturbances and controls are deferred to Part IV.

Signals are scalar, with multi-input and multi-output generalisation also deferred. The desired-signal, observation and actuation channels combine signal and noise through known fixed memoryless functions, so conditional densities can be obtained. The controller itself may have memory and may randomise. Its strategy density at step s can depend on step number, the preceding command and the histories of desired and observed signals. The synthesis task is to find the sequence of such strategy functions that minimises mean risk.

The chosen “simple” criterion adds stage losses across the fixed horizon. Its expected value is the mean risk, and the optimal system minimises that risk. A displayed example uses weighted squared deviation between actual and desired output. Other important criteria are non-additive and remain outside this initial formulation. The adjective “simple” describes the additive form, not the computational difficulty created by endogenous learning.

Decision loss gives information an operational price. An observation is valuable when it changes later action in a way that lowers expected loss. More mutual information can be worse control if it requires a damaging probe or resolves distinctions irrelevant to the permitted decisions. Less information can be sufficient when a robust action works across all plausible models. This does not diminish information theory; it identifies the criterion appropriate to the declared purpose.

The expectation also keeps rare possibilities visible. A low-probability model with catastrophic consequences can influence a risk-minimising policy if the loss is large enough. Conversely, a most-probable-model shortcut hides that tail by construction. The exact result depends on the loss function and priors, so declaring them is part of the engineering specification rather than clerical detail.

Expected loss is not a complete ethics or safety regime. If some outcomes are unacceptable regardless of probability, they should become constraints or be handled by an appropriate risk measure, not buried under a finite quadratic penalty. Feldbaum’s formulation supplies a disciplined baseline: state what is random, what is observed, what policies may depend on and what quantity is minimised.

For a two-step problem, work backwards. At the final step no later control can use new information, so choose the action with lowest posterior expected terminal loss. At the first step, enumerate a candidate action, its immediate expected loss and every possible observation. Update the posterior on each observation, insert the corresponding optimal terminal action, average those future losses and then compare first-step candidates. The selected first command may differ from the myopic optimum because it purchases a better-contingent second decision.

This is dynamic programming on an information state. In a finite hypothesis example the belief is a probability vector. With continuous uncertain parameters it is a density, usually an infinite-dimensional object. Feldbaum points to Bellman’s method for the general variational solution promised in the next part. The conceptual burden already appears in Part I: the strategy depends on history because history determines present knowledge.

A certainty-equivalent policy often removes the branching. It estimates one model or plugs in a mean parameter, then controls that surrogate. Such a method can work well when uncertainty is small, observations are nearly independent of action or robust commands vary little across hypotheses. It fails as a general optimum because it omits how present action changes future estimation and hence future control.

Randomised strategies are admitted for generality. In many ordinary finite expected-loss problems a deterministic minimiser exists at each belief, but ties, adversarial extensions or nonconvex restrictions can make randomisation relevant. The paper’s formulation does not prejudge that issue. It specifies a probability density over the current command conditional on the available information.

An ordinary thermostat reacts to error but normally does not select heat pulses for their value in identifying wall insulation. A dual controller can. It represents uncertainty, predicts how candidate actions alter observations, anticipates belief-dependent later decisions and trades present deviation against later improvement. This is a stronger functional signature than fixed feedback because behaviour depends on counterfactual informational consequences.

Yet the ends, hypotheses, channel laws, horizon and loss function can all be supplied from outside. Optimising within that frame does not show that the system formed its own goals, comprehends the experiment or experiences uncertainty. Even the paper’s “cognitive” or studying side is a control-theoretic role: an action improves statistical knowledge used by later decisions. Translating the term into phenomenal cognition would exceed the evidence.

For contemporary systems, the practical lesson is governance as much as optimisation. Declare which actions are allowed to probe, bound their physical and social cost, distinguish observations caused by policy from naturally arriving data, retain posterior uncertainty, and log the counterfactual reason a probing action was preferred. If learning crosses people, markets or production systems, experimentation may require consent and independent safeguards.

The intellectual achievement of Part I is the clean formulation of a problem that simpler separation principles evade. Identification and control cannot always be designed independently, because the controller helps generate its own future evidence. Feldbaum makes that coupling visible, gives it a name and places it inside a finite-horizon Bayesian decision problem. The later parts can solve and generalise what this first paper has carefully defined.