LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

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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.

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A transverse hybrid event time is stable under dense-output and root error

Statement

Let $J = [a, b]$ with $a < b$ , let $y$ be an exact incoming mode trajectory on $J$ , and let $U$ be a dense reconstruction with $\sup_{t \in J} ∥ y (t) - U (t) ∥_{2} \leq η$ . Let $g (t, x)$ be $L_{g}$ -Lipschitz in $x$ on the certified tube, with declared crossing orientation $σ \in {- 1, + 1}$ . Suppose $h (t) = g (t, y (t))$ has a root $τ \in J$ and, throughout $J$ , $σ h^{'} (t) \geq ν$ for some $ν > 0$ . If a reported time $\hat{τ} \in J$ satisfies $∣ g (\hat{τ}, U (\hat{τ})) ∣ \leq ε$ , then $∣ \hat{τ} - τ ∣ \leq \frac{L_{g} η + ε}{ν} \leq \frac{C (η + ε)}{ν}, C : = \max {L_{g}, 1} .$ The same-sign derivative condition makes $τ$ the unique guard root in $J$ . The conclusion concerns a root matched to this transverse bracket; it gives no bound at grazing or across a change of event ordering.

Facts & Assumptions

Given: The incoming mode, a certified dense-output tube on $J$ , a guard $g$ , a true guard root $τ$ , uniform transversality on $J$ , and the residual tolerance at $\hat{τ}$ .

[F1]

Smooth hybrid ODE modes, guards, resets, and deterministic event priority: the incoming trajectory solves a smooth mode ODE, and an event guard has a declared orientation $σ$ .

[F2]

Certified dense-output and reference-guard enclosure data for hybrid events: an admissible provider supplies the valid uniform bound $∥ y (t) - U (t) ∥_{2} \leq η$ on the matched bracket and the guard state-Lipschitz constant $L_{g}$ on a tube containing both curves; these data need not come from a rational residual-tube checker.

[F3]

The derivative of a guard along an ODE mode is its Lie derivative: $h^{'} (t) = \partial_{t} g (t, y (t)) + D_{y} g (t, y (t)) f_{m} (t, y (t))$ .

[F4]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : a continuous scalar function on a closed interval and differentiable in its interior has an interior point whose derivative is its endpoint difference divided by interval length.

Proof

1.1givenF1F2algebra

By [F2] and the state-Lipschitz bound, $∣ h (\hat{τ}) - g (\hat{τ}, U (\hat{τ})) ∣ \leq L_{g} η$ . The residual hypothesis therefore gives $∣ h (\hat{τ}) ∣ \leq L_{g} η + ε$ , since $h (τ) = 0$ .

2.1givenF1F3F4step 1.1algebra

If $\hat{τ} = τ$ the desired bound is immediate. Otherwise apply [F4] to $h$ on the interval with endpoints $τ$ and $\hat{τ}$ . Since $σ h^{'} \geq ν$ throughout, the derivative has one fixed sign and magnitude at least $ν$ , so $∣ h (\hat{τ}) - h (τ) ∣ \geq ν ∣ \hat{τ} - τ ∣$ . Using $h (τ) = 0$ and step 1.1 proves $∣ \hat{τ} - τ ∣ \leq (L_{g} η + ε) / ν$ .

3.1givenF4step 2.1algebra∎

Since $L_{g} η + ε \leq \max {L_{g}, 1} (η + ε)$ , the second inequality follows with the stated $C$ . For any $s < t$ in $J$ , [F4] gives $h (t) - h (s) = h^{'} (c) (t - s)$ for some interior $c$ ; the fixed sign of $h^{'}$ makes this increment nonzero, so $h$ is strictly monotone and its root in $J$ is unique.

Depends on

Used by

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Sources

Ernst Hairer, course notes, Chapter III: Équations Différentielles Ordinaires (standard reference, not scraped)
Joel M. Esposito, Vijay Kumar, and George J. Pappas, Accurate Event Detection for Simulating Hybrid Systems, HSCC 2001, pp. 204–217 (standard reference, not scraped)