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Shooting error separates IVP integration and nonlinear residual error

Statement

Let the exact shooting residual $R (s) = B (s, ϕ (b; s))$ have a root $s^{*}$ whose trajectory is a BVP solution. For some $r > 0$ , suppose $R$ is $C^{2}$ on an open neighborhood of ${\overline{B}}_{2} (s^{*}, r)$ (twice continuously differentiable), $D R (s^{*})$ is invertible, and $D R$ is $L_{R}$ -Lipschitz on this ball, where $σ : = ∥ D R (s^{*})^{- 1} ∥_{o p}^{- 1} > 0$ and $(L_{R} / 2) r \leq σ / 2$ . Assume a certified evaluator from Certified differentiable IVP approximation interface for shooting succeeds on this ball with uniform trajectory error $δ_{I V P}$ , and that $B$ is $L_{B}$ -Lipschitz in its terminal-state argument between the exact and approximate endpoints there. For any candidate $\hat{s} \in {\overline{B}}_{2} (s^{*}, r)$ with computed residual $∥ B (\hat{s}, \tilde{Φ} (\hat{s})) ∥_{2} \leq τ_{r o o t}$ ,

$∥ \hat{s} - s^{*} ∥_{2} \leq \frac{2}{σ} (τ_{r o o t} + L_{B} δ_{I V P}) .$

If, in addition, $f$ is $C^{1}$ on an open field domain $Ω$ , and there are an open neighborhood $U \supseteq {\overline{B}}_{2} (s^{*}, r)$ and a compact convex $K \subseteq Ω$ such that for every $s \in U$ the IVP has a unique solution $ϕ (\cdot; s)$ on $[a, b]$ with graph in $K$ , and

$∥ f (t, y) - f (t, z) ∥_{2} \leq L ∥ y - z ∥_{2} whenever (t, y), (t, z) \in K,$

then the approximate trajectory from $\hat{s}$ differs uniformly from the exact BVP trajectory by at most

$δ_{I V P} + e^{L (b - a)} ∥ \hat{s} - s^{*} ∥_{2} .$

The integration and root-residual terms remain separate; the theorem makes no global solver-success claim.

Facts & Assumptions

Given: The BVP, exact shooting root $s^{*}$ , regularity and inverse bound, radius $r$ , certified evaluator with error $δ_{I V P}$ , terminal Lipschitz constant $L_{B}$ , candidate $\hat{s}$ and computed residual tolerance $τ_{r o o t}$ , and, when the trajectory estimate is used, the open set $U$ , common compact convex tube $K$ , $C^{1}$ field, and state-Lipschitz bound $L$ from the Statement.

[F1]

Square nonlinear two-point boundary-value problems and endpoint residuals: $R (s) = B (s, ϕ (b; s))$ is the exact endpoint residual.

[F2]

Certified differentiable IVP approximation interface for shooting: $\tilde{Φ} (s) = \tilde{ϕ} (b; s)$ and $\sup_{t} ∥ \tilde{ϕ} (t; s) - ϕ (t; s) ∥_{2} \leq δ_{I V P}$ .

[F3]

Finite-budget single shooting for a nonlinear two-point BVP: the computed root-tolerance status is only the residual bound for $B (s, \tilde{Φ} (s))$ .

[F4]

Vector Newton remainder under a Lipschitz Jacobian: apply its estimate on the convex ball with its evaluation point $x = s^{*}$ and its other point $x_{*} = s$ . Since $R (s^{*}) = 0$ , this gives $R (s) = D R (s^{*}) (s - s^{*}) + q$ with $∥ q ∥_{2} \leq (L_{R} / 2) ∥ s - s^{*} ∥_{2}^{2}$ .

[F5]

Gronwall's integral inequality with variable and constant coefficients: a difference bound $u (t) \leq u (a) + L \int_{a}^{t} u$ yields $u (t) \leq e^{L (t - a)} u (a)$ .

[F6]

Initial-data sensitivity of an ODE flow on a common compact tube: under the common-tube hypotheses in the Statement, exact solutions satisfy their Volterra integral equations.

Proof

1.1givenF1F4algebra

Put $e = \hat{s} - s^{*}$ . The vector remainder estimate [F4] and $R (s^{*}) = 0$ give $R (\hat{s}) = D R (s^{*}) e + q$ with $∥ q ∥_{2} \leq (L_{R} / 2) ∥ e ∥_{2}^{2}$ . Since $∥ e ∥_{2} \leq r$ and $∥ D R (s^{*}) e ∥_{2} \geq σ ∥ e ∥_{2}$ , the radius condition yields $∥ R (\hat{s}) ∥_{2} \geq σ ∥ e ∥_{2} - (σ / 2) ∥ e ∥_{2} = (σ / 2) ∥ e ∥_{2}$ .

2.1givenF2F3step 1.1algebra

The exact and computed residuals have the same first boundary argument. By the stated terminal-state Lipschitz bound and [F2], $∥ R (\hat{s}) - B (\hat{s}, \tilde{Φ} (\hat{s})) ∥_{2} \leq L_{B} δ_{I V P}$ . The computed residual assumption therefore gives $∥ R (\hat{s}) ∥_{2} \leq τ_{r o o t} + L_{B} δ_{I V P}$ . Combining with step 1.1 proves the initial-state estimate. If the candidate is the algorithm's root-tolerance output, [F3] supplies exactly the needed residual premise; other statuses supply no such premise.

3.1givenF2F5F6step 2.1algebra∎

Let $u (t) = ∥ ϕ (t; \hat{s}) - ϕ (t; s^{*}) ∥_{2}$ . The common-tube $C^{1}$ hypothesis makes the flows satisfy the Volterra equations by [F6], and the tube's Lipschitz bound gives $u (t) \leq ∥ e ∥_{2} + L \int_{a}^{t} u (r) d r$ . Applying [F5] yields $u (t) \leq e^{L (b - a)} ∥ e ∥_{2}$ for every $t \in [a, b]$ . The evaluator error [F2] and the triangle inequality add at most $δ_{I V P}$ , giving the displayed trajectory estimate. The given root trajectory is the exact BVP trajectory.

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Sources

R. Verfürth, Numerical Methods and Stochastics, Part I: Numerical Methods, Lecture Notes (standard reference, not scraped)
T. A. Driscoll and R. J. Braun, Fundamentals of Numerical Computation, Chapter 10 (standard reference, not scraped)