Lyapunov Stability of Compact Sets in Locally Compact Metric Spaces

The paper proves the equivalence (Theorem 21) between asymptotic stability of a compact set M and the existence of a strict Lyapunov function on a neighborhood of M, giving detailed proofs in both directions: Lyapunov ⇒ asymptotically stable via compact sublevel trapping and ω-limit arguments, and asymptotically stable ⇒ existence of a continuous strict Lyapunov function by first defining ℓ(x)=sup_{t≥0} d(φ(x,t),M) on A(M) and then regularizing with an integral to ensure strict decrease . The candidate solution (Phase 2) establishes the same equivalence; its forward direction mirrors the paper’s logic, while the converse uses an alternative construction L(x)=sup_{t≥0} e^{-at} d(φ(x,t),M), which is well known and valid, with continuity proved via tail truncation and uniform continuity on compact time windows, and strict decrease via the exponential discount. Minor presentational differences aside, both are correct; the proofs for the converse direction differ in the specific Lyapunov construction (integral vs discounted sup), but both satisfy the theorem’s conditions.