Converging approximations of attractors via almost Lyapunov functions and semidefinite programming

The paper’s Theorem 2 proves p*_5 = λ(A) for LP (11) and derives the quantitative bound λ(K \ A) ≤ ∫_X w + ε λ(X) − p*_5, using the key construction of a function v with βv − ∇v·f = 0, v = 0 on M+, and v < 0 on X \ M+ (citing [21]) to show p*_5 ≤ λ(A); this, together with p*_5 ≥ p*_4 = λ(A), yields equality and then the bound (13) follows from w ≥ 1 − ε on K (Theorem 2 and its proof) . The candidate solution correctly establishes v ≥ 0 on M+ and the positive invariance of J^{-1}([0, ε]), and gives the lower bound p*_5 ≥ λ(A), but its proof of the crucial upper bound p*_5 ≤ λ(A) is invalid: it attempts to deduce this from a near-optimal feasible sequence and the bound of item (4), which itself implicitly requires p*_5 = λ(A). Consequently, items (4)–(5) in the candidate solution are circular (they use the equality p*_5 = λ(A) before proving it), whereas the paper’s argument closes the loop correctly by constructing the needed v and then deriving (13) .