Existence of Complete Lyapunov Functions with Prescribed Orbital Derivative

The paper proves that for any compact K disjoint from the chain recurrent set RX and any negative C^l function g on a neighborhood of K, there exists a C^l complete Lyapunov function τK with τ̇K|K = g and τ̇K < 0 on U\RX, via a reduction to g ≡ −1 and a careful flowbox-based interpolation preserving completeness and (in their construction) the equality of the critical set with RX . The candidate solution gives a different, PDE-style construction: start from a smooth complete Lyapunov function, scale to achieve a strong descent margin near K, and then add a compactly supported correction h obtained by solving X·∇h to match g on K. This approach is sound in principle and aligns with the paper’s framework, though it needs a minor clarification when bounding τ̇K off K (one can bound |X·∇h| using a bounded-overlap cover or ensure X·∇h = ϕ on a neighborhood of K via a partition of unity constant along flow lines). With this small fix, the outline yields the stated theorem, so both are correct, with different proofs .