PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Existence and uniqueness of transport along a piecewise-smooth path

Statement

Let $n \geq 1$ and let $0 = t_{0} < t_{1} < \dots < t_{m} = 1$ be a finite partition. For each $j$ , let $A_{j} : [t_{j - 1}, t_{j}] \to M_{n} (R)$ be continuous. There is a unique continuous function $U : [0, 1] \to M_{n} (R)$ that is $C^{1}$ on each partition interval, satisfies $U^{'} (t) = - A_{j} (t) U (t)$ in its interior, and has $U (0) = I_{n}$ . Derivatives at interval endpoints are the corresponding one-sided derivatives.

Statement references

Linear matrix ODEs have unique global solutions on a fixed interval

Facts & Assumptions

Given: A positive integer $n$ , a finite partition of $[0, 1]$ , one continuous matrix coefficient on each closed subinterval, and the initial matrix $I_{n}$ .

[F1]

Linear matrix ODEs have unique global solutions on a fixed interval: a continuous matrix coefficient on a compact interval gives a unique global solution to the linear matrix initial-value problem.

Proof

technique · direct

1.1givenF1construct

Apply [F1] on $[t_{0}, t_{1}]$ with initial matrix $I_{n}$ to obtain the unique $C^{1}$ solution on the first interval.

2.1givenF1step 1.1construct

For each $j = 2, \dots, m$ , apply [F1] on $[t_{j - 1}, t_{j}]$ with initial value equal to the terminal value already constructed at $t_{j - 1}$ . Adjacent solutions agree at their shared endpoint by construction, so the finite family pastes to a continuous piecewise- $C^{1}$ solution on $[0, 1]$ .

3.1givenF1step 1.1step 2.1induction∎

Any other such function has initial value $I_{n}$ on the first interval, so it agrees there by [F1]. If it agrees through $t_{j - 1}$ , then it has the same initial value as the constructed solution on the $j$ th interval, so [F1] makes them agree there. Induction through the finite partition proves uniqueness on all of $[0, 1]$ .

Depends on

Linear matrix ODEs have unique global solutions on a fixed interval