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PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)
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Zeros of the exact shooting residual are exactly BVP solutions

Statement

On an open set U of initial states, suppose each IVP has a unique solution ϕ(⋅;s) on [a,b] that is a C1 curve, and (s,ϕ(b;s))∈DB for every s∈U. Define the exact shooting residual by R(s)=B(s,ϕ(b;s)). Then R(s)=0 if and only if ϕ(⋅;s) is a BVP solution. Any BVP solution with initial state s equals this shot.

Facts & Assumptions

Given: The BVP data and an initial state s whose unique IVP solution exists on [a,b].

[F1]

Square nonlinear two-point boundary-value problems and endpoint residuals: a BVP solution is a continuously differentiable IVP curve satisfying B(y(a),y(b))=0, and R(s)=B(s,ϕ(b;s)).

Proof

1.1givenF1

If R(s)=0, then ϕ(⋅;s) already solves the IVP with initial state s and satisfies B(s,ϕ(b;s))=0 by the definition of R. It is continuously differentiable and lies in the BVP domain, so [F1] makes it a BVP solution.

1.2givenF1

Conversely, let y be any BVP solution with y(a)=s. It solves the same IVP as ϕ(⋅;s); the assumed uniqueness implies y(t)=ϕ(t;s) throughout [a,b]. The boundary equation then gives R(s)=B(s,ϕ(b;s))=B(y(a),y(b))=0.

2.1step 1.1step 1.2∎

The same uniqueness argument shows that any BVP solution with initial state s is exactly this trajectory, proving both the equivalence and the asserted trajectory identity.

Depends on

Used by

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Sources