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.

Strong convexity converts transcription objective error into a control bound

Statement

Let $u^{*}$ minimize the LQR objective $J$ on a convex feasible set $F$ , with control weight $R (t) ⪰ r I$ for $r > 0$ . Let $u_{N} \in F$ and let $v_{N} \in F$ be a recovered comparison control. Suppose the computed objective $J_{N}$ satisfies $∣ J_{N} (w) - J (w) ∣ \leq δ_{N}$ for $w = u_{N}, v_{N}$ , and $J_{N} (u_{N}) \leq J_{N} (v_{N}) + η_{N}$ . If $J (v_{N}) \leq J (u^{*}) + ρ_{N}$ , then $J (v) - J (u^{*}) \geq \frac{r}{2} ∥ v - u^{*} ∥_{2}^{2}$ for every $v \in F$ , and $0 \leq J (u_{N}) - J (u^{*}) \leq 2 δ_{N} + η_{N} + ρ_{N}, ∥ u_{N} - u^{*} ∥_{L^{2}} \leq \sqrt{\frac{2 (2 δ_{N} + η_{N} + ρ_{N})}{r}} .$ The conclusion is conditional on the stated objective consistency and comparison bounds. A raw embedded RK error estimate is not a certified value of $δ_{N}$ .

Here are sufficient bounds that derive the two error terms from transcription errors and recovery. Set $C_{s} = e^{∥ A ∥_{\infty} T} ∥ B ∥_{\infty} \sqrt{T}$ , $M_{c} = \sup_{c \in U_{c}} ∥ c ∥_{2}$ , $M_{x} = e^{∥ A ∥_{\infty} T} (∥ x_{0} ∥_{2} + ∥ B ∥_{\infty} T M_{c})$ , $E = \sqrt{T} M_{x} + ∥ x_{d} ∥_{2}$ , and $M_{T} = M_{x} + ∥ x_{T} ∥_{2}$ . Suppose, for $w = u_{N}, v_{N}$ , a pointwise representative and sampling rule are fixed, $J_{N} (w)$ has the form $J_{N} (w) = \sum_{k} ω_{N k} {\tilde{ℓ}}_{N, w} (τ_{N k}) + \frac{1}{2} ({\tilde{x}}_{N} [w] (T) - x_{T})^{T} P ({\tilde{x}}_{N} [w] (T) - x_{T}),$ where ${\tilde{ℓ}}_{N, w} (t) = \frac{1}{2} (({\tilde{x}}_{N} [w] (t) - x_{d} (t))^{T} Q (t) ({\tilde{x}}_{N} [w] (t) - x_{d} (t)) + w (t)^{T} R (t) w (t))$ , $ω_{N k} \geq 0$ , $\sum_{k} ω_{N k} = T$ , and the chosen samples satisfy $∥ Q (τ_{N k}) ∥_{2} \leq ∥ Q ∥_{\infty}$ . Suppose $∥ {\tilde{x}}_{N} [w] - x_{w} ∥_{\infty} \leq e_{N}$ , $∥ x_{w} (τ_{N k}) - x_{d} (τ_{N k}) ∥_{2} \leq E_{N}$ , $∥ x_{w} (T) - x_{T} ∥_{2} \leq M_{T}$ , and $∣ \sum_{k} ω_{N k} ℓ_{w} (τ_{N k}) - \int_{0}^{T} ℓ_{w} (t) d t ∣ \leq q_{N},$ where $ℓ_{w} (t) = \frac{1}{2} ((x_{w} (t) - x_{d} (t))^{T} Q (t) (x_{w} (t) - x_{d} (t)) + w (t)^{T} R (t) w (t))$ . Then one may take $δ_{N} = q_{N} + \frac{T}{2} ∥ Q ∥_{\infty} (2 E_{N} + e_{N}) e_{N} + \frac{1}{2} ∥ P ∥ (2 M_{T} + e_{N}) e_{N} .$ If a feasible recovery sequence also satisfies $∥ v_{N} - u^{*} ∥_{2} \leq σ_{N}$ , then one may take $ρ_{N} = C_{J} σ_{N}$ , where $C_{J} = ∥ Q ∥_{\infty} E \sqrt{T} C_{s} + ∥ P ∥ M_{T} C_{s} + M \sqrt{T} M_{c},$ with $M$ any essential upper bound for $∥ R (t) ∥_{2}$ . In the control-only case, under Countable Choice the cell-average lemma supplies $v_{N} = P_{N} u^{*}$ and $σ_{N} \to 0$ . If $u^{*}$ is Lipschitz with constant $L_{u}$ , then $σ_{N} = \sqrt{T} L_{u} h_{N}$ is a valid rate for mesh size $h_{N}$ . For a general $L^{2}$ minimizer, no rate for $σ_{N}$ is asserted.

Facts & Assumptions

Given: The LQR model, $r > 0$ , a continuous minimizer, two feasible controls, and the finite objective inequalities in the Statement. For the explicit numerical-error estimate, also assume the stated pointwise sampling, state, and quadrature bounds.

[F1]

The state map is affine and bounded from $L^{2}$ to the supremum norm (The bounded measurable linear ODE has a unique affine state map).

[F2]

Expanding the LQR quadratic forms for $d = v - u$ gives $J (v) - J (u) = G_{u} [d] + \frac{1}{2} \int_{0}^{T} (δ x^{T} Q δ x + d^{T} R d) d t + \frac{1}{2} δ x (T)^{T} P δ x (T),$ where $δ x = x_{v} - x_{u}$ and $G_{u} [d] = \int_{0}^{T} ((x_{u} - x_{d})^{T} Q δ x + u^{T} R d) d t + (x_{u} (T) - x_{T})^{T} P δ x (T)$ ; this follows from [F1] and the definition of $J$ (Finite-horizon bounded-control linear-quadratic ODE problem, The bounded measurable linear ODE has a unique affine state map).

[F3]

For admissible controls, $∥ u ∥_{2} \leq \sqrt{T} M_{c}$ and $∥ x_{u} ∥_{\infty} \leq M_{x}$ by compactness of $U_{c}$ and the state-map bound (Finite-horizon bounded-control linear-quadratic ODE problem, The bounded measurable linear ODE has a unique affine state map).

[A1]

Countable Choice is assumed only for the control-only cell-average recovery specialization (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Cell averages give a strong recovery sequence for compact convex controls).

Proof

1.1F2givenalgebra

Fix $v \in F$ and put $d = v - u^{*}$ . Convexity gives $u^{*} + θ d \in F$ for $0 \leq θ \leq 1$ . By [F2], $J (u^{*} + θ d) - J (u^{*}) = θ G_{u^{*}} [d] + θ^{2} R_{d}$ , where $R_{d} = \frac{1}{2} \int_{0}^{T} (δ x_{d}^{T} Q δ x_{d} + d^{T} R d) d t + \frac{1}{2} δ x_{d} (T)^{T} P δ x_{d} (T)$ . Minimality makes this expression nonnegative for $0 < θ \leq 1$ ; divide by $θ$ and let $θ ↓ 0$ to get $G_{u^{*}} [d] \geq 0$ . Also $R_{d} \geq \frac{r}{2} ∥ d ∥_{2}^{2}$ since $Q, P ⪰ 0$ and $R ⪰ r I$ . Taking $θ = 1$ proves $J (v) - J (u^{*}) \geq \frac{r}{2} ∥ v - u^{*} ∥_{2}^{2}$ .

1.2givenF3algebra

For the stated sampled objective, subtract the quadrature of $ℓ_{w}$ evaluated at the exact state. At every quadrature node, the state-cost difference is bounded by $\frac{1}{2} ∥ Q ∥_{\infty} (2 E_{N} + e_{N}) e_{N}$ ; the control-cost summand is unchanged. Nonnegative weights with sum $T$ therefore bound the accumulated state perturbation by $\frac{T}{2} ∥ Q ∥_{\infty} (2 E_{N} + e_{N}) e_{N}$ . The exact-state quadrature discrepancy is at most $q_{N}$ . At $T$ , the terminal quadratic difference is at most $\frac{1}{2} ∥ P ∥ (2 M_{T} + e_{N}) e_{N}$ . Thus $∣ J_{N} (w) - J (w) ∣ \leq δ_{N}$ for $w = u_{N}, v_{N}$ with the displayed $δ_{N}$ .

2.1givenstep 1.1algebra

The assumed inequalities and consistency give $J (u_{N}) \leq J_{N} (u_{N}) + δ_{N} \leq J_{N} (v_{N}) + η_{N} + δ_{N} \leq J (v_{N}) + η_{N} + 2 δ_{N} \leq J (u^{*}) + ρ_{N} + η_{N} + 2 δ_{N}$ . Subtract $J (u^{*})$ and combine with step 1.1 to get the objective-gap bound.

3.1step 1.1step 2.1algebra

Apply the growth inequality of step 1.1 at $v = u_{N}$ . Substitute the upper objective-gap bound from step 2.1, divide by $r / 2 > 0$ , and take nonnegative square roots to obtain the control-error estimate.

4.1A1F1F2F3givenalgebra∎

For any feasible $v$ with $∥ v - u^{*} ∥_{2} \leq σ$ , [F2] and [F3] bound the absolute difference of the state running terms by $∥ Q ∥_{\infty} E \sqrt{T} C_{s} σ$ , the terminal terms by $∥ P ∥ M_{T} C_{s} σ$ , and the control terms by $M \sqrt{T} M_{c} σ$ . Hence $J (v) - J (u^{*}) \leq C_{J} σ$ . Under [A1] the cell-average lemma gives strong recovery for the control-only case. If $u^{*}$ is $L_{u}$ -Lipschitz, for each $t$ in a cell its distance from the cell average is at most $L_{u} h_{N}$ , so $∥ P_{N} u^{*} - u^{*} ∥_{2} \leq \sqrt{T} L_{u} h_{N}$ . Substitution gives the stated $ρ_{N}$ rate and completes the sufficient-estimate branch.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

38 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 §§13.1–13.4 (standard reference, not scraped)
J. Monnier, Optimal Control of ODEs: The Linear-Quadratic (LQ) Case, course notes, Chapter 2 §2.5 (standard reference, not scraped)