Converse Barrier Functions via Lyapunov Functions

Jun Liu

arxiv: 2007.11086 · v2 · submitted 2020-07-21 · 🧮 math.OC · cs.SY· eess.SY

Converse Barrier Functions via Lyapunov Functions

Jun Liu This is my paper

Pith reviewed 2026-05-24 14:28 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords barrier functionsLyapunov functionsrobust safetyconverse theoremsset stabilitydynamical systemscontinuous timediscrete time

0 comments

The pith

Robustly safe dynamical systems always admit a barrier function that doubles as a Lyapunov function for set stability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that robust safety of a dynamical system implies the existence of a barrier function constructed from a Lyapunov function for set stability. It reaches this conclusion by first showing that the closure of the forward reachable set from a robustly safe set is itself robustly asymptotically stable. Converse Lyapunov theorems then supply the required barrier function. The argument is given separately for continuous-time and discrete-time systems. A reader would care because the result supplies a systematic way to obtain safety certificates from existing stability theory.

Core claim

We prove a robust converse barrier function theorem via the converse Lyapunov theory. We show that the closure of the forward reachable set of a robustly safe set must be robustly asymptotically stable under mild technical assumptions. As a result, all robustly safe dynamical systems must admit a robust barrier function in the form of a Lyapunov function for set stability. We present the results in both continuous-time and discrete-time settings and remark on connections with various barrier function conditions.

What carries the argument

Robust asymptotic stability of the closure of the forward reachable set, which permits direct application of converse Lyapunov theorems to produce barrier functions.

If this is right

Every robustly safe system possesses a Lyapunov function that also serves as a robust barrier function.
The same link between safety and stability holds in both continuous-time and discrete-time dynamics.
Standard converse Lyapunov constructions can be reused to certify robust safety.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Control designs that already use Lyapunov functions for stability can be checked for safety with little extra work.
The result suggests that existing numerical methods for computing Lyapunov functions might be repurposed for barrier-function synthesis.
Similar unification arguments could be attempted for other certificate types such as contraction metrics or sum-of-squares relaxations.

Load-bearing premise

The closure of the forward reachable set of a robustly safe set must be robustly asymptotically stable under mild technical assumptions.

What would settle it

A concrete robustly safe system whose forward reachable set closure fails to be robustly asymptotically stable would disprove the theorem.

read the original abstract

We prove a robust converse barrier function theorem via the converse Lyapunov theory. While the use of a Lyapunov function as a barrier function is straightforward, the existence of a converse Lyapunov function as a barrier function for a given safety set is not. We establish this link by a robustness argument. We show that the closure of the forward reachable set of a robustly safe set must be robustly asymptotically stable under mild technical assumptions. As a result, all robustly safe dynamical systems must admit a robust barrier function in the form of a Lyapunov function for set stability. We present the results in both continuous-time and discrete-time settings and remark on connections with various barrier function conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The abstract sketches a converse result linking robust safety to Lyapunov-type barrier functions via reachable-set stability, but that central step can't be checked from what's here.

read the letter

The main point is that the paper claims a robust converse barrier theorem: if a set is robustly safe, then its forward reachable set closure is robustly asymptotically stable under mild assumptions, so standard converse Lyapunov theory gives a barrier that is also a Lyapunov function for set stability. It states this for both continuous and discrete time and notes links to other barrier conditions. That framing is the actual new piece on offer. If the argument works, it gives a clean theoretical bridge between safety and stability that control people might use. The paper does well to make the robustness step explicit rather than hand-waving it. The soft spot is exactly where the stress-test note says: the claim that the reachable-set closure is robustly asymptotically stable is load-bearing, and the abstract gives no proof details or precise statement of the mild assumptions. Without those, it is impossible to tell whether the step holds for natural classes of systems or whether the assumptions turn out to be restrictive. The soundness of the whole result therefore cannot be judged yet. Citation comparisons to prior converse theorems are also missing. This is a paper for control theorists already working on barrier functions and safety verification. A reader who knows the standard Lyapunov converse results might find the perspective useful if the details check out. It deserves a serious referee once the full text is available, because the topic is substantive and the idea is not obviously circular, but right now there is not enough to decide.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove a robust converse barrier function theorem by linking robust safety to the existence of a Lyapunov function serving as a robust barrier. The argument proceeds by establishing that, under mild technical assumptions, the closure of the forward reachable set of a robustly safe set is itself robustly asymptotically stable; standard converse Lyapunov theory for set stability then supplies the barrier function. The result is stated for both continuous-time and discrete-time dynamical systems, with additional remarks on connections to other barrier function conditions.

Significance. If the central robustness argument holds under natural assumptions, the result would provide a rigorous bridge between robust safety verification and converse Lyapunov theory for set stability. This could allow existing stability tools to be repurposed for barrier-function synthesis and would strengthen the theoretical foundations of safety-critical control. The approach via reachable-set closure is a potentially useful technical step if the mild assumptions prove to be broadly satisfied.

major comments (1)

[Abstract] Abstract: the load-bearing step—that the closure of the forward reachable set of a robustly safe set is robustly asymptotically stable under mild technical assumptions—cannot be verified from the provided abstract. Without the precise statement of those assumptions and the details of the robustness argument, it is impossible to assess whether the implication holds for the intended class of systems or whether counter-examples exist.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful summary and for highlighting the importance of clearly conveying the technical assumptions. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the load-bearing step—that the closure of the forward reachable set of a robustly safe set is robustly asymptotically stable under mild technical assumptions—cannot be verified from the provided abstract. Without the precise statement of those assumptions and the details of the robustness argument, it is impossible to assess whether the implication holds for the intended class of systems or whether counter-examples exist.

Authors: We agree that the abstract, as a concise summary, does not state the precise technical assumptions or sketch the robustness argument. The full manuscript develops these elements in detail (Sections 3 and 4). To make the central claim more readily assessable from the abstract itself, we will revise the abstract to include a brief enumeration of the mild technical assumptions under which the reachable-set closure is shown to be robustly asymptotically stable. revision: yes

Circularity Check

0 steps flagged

No circularity: standard converse Lyapunov application with external robustness step

full rationale

The abstract presents a derivation that first establishes (via a robustness argument under mild assumptions) that the closure of the forward reachable set of a robustly safe set is robustly asymptotically stable, then invokes standard converse Lyapunov theory to obtain a Lyapunov function that serves as the barrier. No equations, fitted parameters, self-definitions, or self-citations appear in the provided text that would reduce the central claim to its inputs by construction. The cited converse Lyapunov theory is described as standard (external to the paper), and the new robustness step is presented as an independent argument rather than a renaming or tautology. With only the abstract available and no load-bearing self-referential steps exhibited, the derivation does not meet any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard converse Lyapunov theory for set stability and a robustness argument applied to forward reachable sets of safe sets.

axioms (2)

domain assumption Converse Lyapunov theory applies to set stability for the reachable set
Invoked to conclude existence of Lyapunov function from robust asymptotic stability.
domain assumption Mild technical assumptions on the system and safety set hold
Required for the reachable set to be robustly asymptotically stable.

pith-pipeline@v0.9.0 · 5599 in / 1125 out tokens · 28685 ms · 2026-05-24T14:28:47.977868+00:00 · methodology

Converse Barrier Functions via Lyapunov Functions

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)