DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Time meshes and piecewise-constant control parameterization

Definition

For the horizon $[0, T]$ of Finite-horizon bounded-control linear-quadratic ODE problem, a time mesh is a finite partition $0 = t_{0} < t_{1} < \dots < t_{N} = T$ with $N \geq 1$ . Its cell widths are $h_{j} = t_{j + 1} - t_{j} > 0$ and its mesh size is $∣ Δ ∣ = \max_{0 \leq j < N} h_{j}$ . A piecewise-constant control vector is $(u_{0}, \dots, u_{N - 1})$ and represents the measurable control $u^{Δ} (t) = u_{j}$ for $t \in [t_{j}, t_{j + 1})$ ; its value at $T$ is immaterial in $L^{2}$ . If $u_{j} \in U_{c}$ for every $j$ , then $u^{Δ} \in U_{c}$ . A mesh family is refining when its mesh sizes tend to zero.

Depends on

Finite-horizon bounded-control linear-quadratic ODE problem

Used by

Dependency tree · two levels

4 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

S. Gros and M. Diehl, Numerical Optimal Control, draft course manuscript, Chapter 13 §§13.1–13.3 (standard reference, not scraped)