Peak Estimation and Recovery with Occupation Measures

The paper’s Theorem 3.1 claims equality of the peak value and the measure program and “mutually recoverable solutions,” proving p*(9) ≥ p*(1) by constructing atomic measures from an attaining trajectory, and p*(9) ≤ p*(1) by noting μp is a probability measure and invoking Eν[p] ≤ sup p over the support (see Theorem 3.1 and the subsequent argument). However, the upper-bound step, as written, does not justify that μp’s support is precisely the set of points reached by admissible trajectories; without a superposition/characteristics argument, the inequality only yields ≤ supX p, not ≤ p*(1) (cf. the proof text around eqs. (10)–(11) in 3.1) . The candidate solution fills this gap cleanly by invoking the superposition principle for the Liouville/continuity equation, which certifies that feasible measures decompose into trajectory segments with a stopping time, making ⟨p, μp⟩ an expectation of p evaluated at reached points, hence ≤ p*. It also clarifies “mutual recoverability” via extremal/atomic selections. The paper’s appendix notes that attainment holds under compactness, but that assumption is not stated with Theorem 3.1; the model explicitly records the needed hypotheses and gives a complete proof path. Overall, the result is correct, but the paper’s proof sketch omits a crucial representability step and some assumptions, while the model provides them.