The Koopman Expectation: An Operator Theoretic Method for Efficient Analysis and Optimization of Uncertain Hybrid Dynamical Systems

The paper states the FP–Koopman adjointness 〈PS f, g〉 = 〈f, US g〉 and its expectation form E[g | PS f] = E[US g | f] as standard facts (their Eqs. (6)–(9)), and then rewrites the FP Expectation Form optimization (Eq. (27)) into the Koopman Expectation Form (Eq. (28)) by invoking this adjoint relationship, given S(I) ⊆ O and Qi(I) ⊆ Ci . The candidate solution provides a standard measure‑theoretic proof of adjointness (indicator→simple→L∞ limit) and then executes the same adjointness-based rewrite with explicit measurability/integrability assumptions and support arguments, matching the paper’s steps but with more detail. Thus both are correct and rely on the same adjointness principle; the model simply fills in technical details that the paper leaves implicit. Key definitions of PS and US used by both appear in Eqs. (3)–(5) , and the optimization spaces I, O, Ci are as defined in Eqs. (23)–(25) .