TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The continuous ODE adjoint gives the parameter gradient with endpoint terms

Statement

Assume $T > 0$ . Let $u_{θ}$ solve the parameter-dependent ODE of Parameter-dependent ODE flows have the variational tangent on a common tube on $[0, T]$ . Let $J (θ) = ϕ (u_{θ} (T), θ) + \int_{0}^{T} ℓ (t, u_{θ} (t), θ) d t,$ where $ϕ$ and $ℓ$ are $C^{1}$ on a neighborhood of the trajectory tube and their first derivatives are bounded and uniformly continuous there. Let $S = D_{θ} u_{θ}$ be the tangent from that lemma. Let $p$ solve the terminal adjoint problem $- p^{'} (t) = D_{u} f (t, u_{θ} (t), θ)^{T} p (t) + D_{u} ℓ (t, u_{θ} (t), θ)^{T}, p (T) = D_{u} ϕ (u_{θ} (T), θ)^{T} .$ Then $\nabla_{θ} J = D_{θ} ϕ (u_{θ} (T), θ)^{T} + D_{θ} u_{0} (θ)^{T} p (0) + \int_{0}^{T} (D_{θ} ℓ (t, u_{θ}, θ)^{T} + D_{θ} f (t, u_{θ}, θ)^{T} p (t)) d t .$ This formula separates initial, terminal, running-cost and vector-field parameter terms. It applies only while the common-tube and differentiability hypotheses hold.

Facts & Assumptions

Given: $T > 0$ , the common-tube ODE, $C^{1}$ initial map, objective, and bounded uniformly continuous first derivatives in the Statement.

[F1]

Parameter-dependent ODE flows have the variational tangent on a common tube: the parameter flow is $C^{1}$ and $S^{'} = D_{u} f S + D_{θ} f$ , $S (0) = D_{θ} u_{0}$ .

[F2]

A tangent and adjoint pairing gives the ODE gradient boundary identity: if $S^{'} = A S + B$ and $p^{'} = - A^{T} p - g$ , then $p (T)^{T} S (T) - p (0)^{T} S (0) = \int_{0}^{T} (- g^{T} S + p^{T} B) d t$ .

[F3]

Linear matrix ODEs have unique global solutions on a fixed interval: a continuous coefficient linear matrix IVP has a unique solution on the full compact interval.

[F4]

The total-derivative chain rule differentiates a composition of $C^{1}$ Euclidean maps (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[A1]

All objective derivatives and ODE derivative coefficients are continuous and bounded on the common compact tube, and their first derivatives have the uniform continuity needed to differentiate the integral objective.

Proof

1.1F3givenconstruct

Fix $θ$ and write $A (t) = D_{u} f (t, u_{θ} (t), θ)$ and $g (t) = D_{u} ℓ (t, u_{θ} (t), θ)^{T}$ . The coefficient matrix $C (t) = (\begin{array}{cc} - A (t)^{T} & - g (t) \\ 0 & 0 \end{array})$ is continuous. Apply [F3] with initial time $T$ to the homogeneous matrix equation $Z^{'} = C (t) Z$ , choosing an initial matrix whose first column is $(D_{u} ϕ (u_{θ} (T), θ)^{T}, 1)^{T}$ and whose other columns are zero. Its first column is $(p (t), 1)^{T}$ and solves the terminal adjoint equation. Any other solution embeds as a column of such a matrix solution, so uniqueness in [F3] gives uniqueness of $p$ .

1.2F1F4A1given

By [F1] and the chain rule [F4], the terminal derivative is $D_{θ} ϕ^{T} + S (T)^{T} D_{u} ϕ^{T}$ . For the running term, a parameter increment $h$ gives $Δ u (t) = S (t) h + o (∥ h ∥)$ uniformly in $t$ by [F1]. Uniform continuity of the objective derivatives on the common compact tube gives a uniform first-order remainder $o (∥ h ∥)$ along the segment from $(u_{θ} (t), θ)$ to $(u_{θ + h} (t), θ + h)$ ; the common integrable majorant permits integration of this remainder. Applying [F4] pointwise and integrating therefore gives the running derivative $\int_{0}^{T} (D_{θ} ℓ^{T} + S^{T} D_{u} ℓ^{T}) d t$ . Consequently $\nabla_{θ} J = D_{θ} ϕ^{T} + S (T)^{T} D_{u} ϕ^{T} + \int_{0}^{T} (D_{θ} ℓ^{T} + S^{T} D_{u} ℓ^{T}) d t$ .

1.3F1F2given

In [F2], set $A = D_{u} f$ , $B = D_{θ} f$ , and $g = D_{u} ℓ^{T}$ . The adjoint equation in the Statement and tangent equation in [F1] satisfy its hypotheses. Transposing that row pairing identity and using $p (T) = D_{u} ϕ^{T}$ gives $S (T)^{T} D_{u} ϕ^{T} - D_{θ} u_{0}^{T} p (0) = \int_{0}^{T} (- S^{T} D_{u} ℓ^{T} + D_{θ} f^{T} p) d t .$

2.1step 1.2step 1.3algebra∎

Substitute step 1.3 into step 1.2; the terms involving $S^{T} D_{u} ℓ^{T}$ cancel, leaving the asserted continuous adjoint gradient formula.

Depends on

Used by

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

William W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, §3 (standard reference, not scraped)
Hannes Meinlschmidt, Optimization with Partial Differential Equations, complete lecture notes, Chapter 3 (standard reference, not scraped)
Michael B. Giles and Niles A. Pierce, An Introduction to the Adjoint Approach to Design, §2.4 (standard reference, not scraped)