Werbos's 1988 paper equates his 'dynamic feedback' backward recursion of ordered derivatives with control theory's time-dependent Lagrange multipliers, and calls it 'very similar to earlier methods from control theory'

werbosbackpropagationdynamic-feedbackordered-derivativesadjoint-methodoptimal-controlhistory-of-ml

Werbos defines first-order dynamic feedback as "the use of the chain rule for ordered derivatives, in order to calculate the derivatives of L with respect to the system inputs" (p. 342). It runs as a recursion from the last variable "backwards to x_{N-1}, x_{N-2}, on down to x_1" (p. 342). He then names the same object in other fields' vocabularies: "economists speak of impact multipliers, control theorists speak of variational derivatives, and many people speak of time-dependent Lagrange multipliers."

On control theory he is direct. Of the time-lag case: "This method of calculating derivatives is very similar to earlier methods from control theory" (p. 349). Of the recurrent generalization: it "could have been derived as a generalization of the 'adjoint' method used by Alsmiller et al. (1981)" (p. 348). On his own origin (p. 341): his 1974 work "also cited related work in control theory, which also used backwards flows of information to identify systems"; "The formulation to be given below could have been derived as an extension of control theory, but I found it easier simply to prove (9) directly"; and "a reviewer has suggested that Athans and Kalb (1966) came surprisingly close to the kind of approach presented here." He also says outright that "The J function comes from dynamic programming" (p. 345).

This is a different claim from claim-werbos-1988-rhw-special-case-of-dynamic-feedback. That note carries the p. 342 containment of RHW's networks inside his framework, a containment over network topology. This one carries his own identification of the operation with control theory's, an identification over mathematical object. The equivalence-not-transmission distinction of claim-recht-adjoints-equivalence-not-transmission applies: Werbos asserts the two are the same operation and does not say control theory caused his.

Together with claim-isaacs-p257-derives-optimal-paths-as-characteristics-of-value-pde-with-differentiated-half, this gives both ends of a shared mechanism (a scalar's gradient carried backward along a path) in each author's own words. Whether any line joins them is a separate question: claim-werbos-1988-reference-list-names-no-isaacs-bellman-pontryagin-kelley-or-bryson. See also claim-kelley-bryson-optimal-control-precursor and claim-backprop-special-case-kelley-bryson-formula.