Locating Extremal Periodic Orbits for the Planar Circular Restricted Three Body Problem using Polynomial Sum-of-Squares Optimization

The paper correctly states the auxiliary-function/SOS framework and the generalized S-procedure with the key polynomial D(a)=F·∇V+(Φ−L)R+σH−∑s_jG_j, and it uses δ-suboptimal localization via sublevel sets S_ε, matching the model’s derivations. However, for PCR3BP the paper replaces the true dynamics with rational approximations of radical terms and then optimizes D as if it were polynomial without quantifying approximation error or certifying that R>0 and F=Rf hold exactly; thus the bound is not rigorously tied to the original ODE. The model’s step (1) and (2) are mathematically sound in the exact setting, but (3) asserts near-attainment by periodic orbits and asymptotic tightness without establishing the required assumptions or error control for the non-polynomial PCR3BP. Hence both are incomplete: the paper for unquantified approximation/rigor gaps, and the model for over-claiming existence/near-attainment. Key paper statements: the S-procedure and SOS relaxation (equations and discussion around D in 20–21), the generalized S-procedure (19) on S={H=0,G_j≥0}, the localization set S_ε (15–16) and the auxiliary-function equality (9–12), the PCR3BP energy constraint and semialgebraic region (26–27), and the empirical extremality claim at e=−1.589070 (Section 6).