A Converse Control Lyapunov Theorem for Joint Safety and Stability

Jun Liu; Maxwell Fitzsimmons; Thanin Quartz

REVIEW 3 major objections 4 minor 31 references

A Converse Control Lyapunov Theorem for Joint Safety and Stability

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that a strictly compatible control Lyapunov–barrier pair exists if and only if a single smooth Lyapunov function certifies both asymptotic stability and safety.

desk verdict A genuinely new converse CLBF construction that is likely right, but Proposition 1's proof as written flips the sign of the CBF condition, so the supporting controller does not actually guarantee safety. read the letter →

arxiv 2509.12182 v2 pith:DSMKJ7BD submitted 2025-09-15 math.OC eess.SY

Thanin Quartz , Maxwell Fitzsimmons , Jun Liu This is my paper · ORCID

classification math.OCeess.SY MSC 93D3093D1593C10

keywords controlLyapunovfunctionbarrierLyapunov-barrierconversetheoremsafetystabilityhittingtimeforwardinvariance

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes a converse Lyapunov theorem for joint safety and stability. It shows that, under mild assumptions, having a pair of certificates—one for stability (a control Lyapunov function) and one for safety (a control barrier function) that are strictly compatible at the safe set's boundary—is exactly equivalent to having one smooth Lyapunov function that does both jobs at once. The construction produces the combined function explicitly as a ratio of the original Lyapunov function evaluated along trajectories up to the time they hit the safe set's boundary. This matters because it turns a two-certificate design problem into a single-function certificate with a PDE characterization and prescribed boundary conditions, and it reveals when the two goals are fundamentally in conflict.

What carries the argument

The hitting time T(x) to the safe set boundary, whose continuous differentiability follows from the implicit function theorem because the closed-loop field points strictly inward at ∂C. Using T, the paper forms the ratio W(x) = V(x)/V(φ(T(x),x)); this ratio is constant along trajectories, splits the domain into W > 1 outside C, W < 1 inside, and W = 1 on the boundary, and solves the PDE ∇W·F = −ω₁. The regularity of W rests on the estimate |∇T(x)| = O(1/|x|) obtained under a stabilizable linearization.

What would settle it

Take any control-affine system satisfying Assumptions 1 and 2, explicitly compute T(x) by backward integration and W(x) = V(x)/V(φ(T(x),x)); the central theorem would be refuted if W is not continuously differentiable on the domain of attraction (excluding the origin as appropriate), or if {x: W(x) ≤ 1} differs from the safe set C by more than a set of measure zero, or if ∇W·F = −ω₁ fails off the origin.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: if a control system of the form ẋ = f(x) + g(x)u admits a strictly compatible pair—a control Lyapunov function for asymptotic stability and a control barrier function for forward invariance of a compact safe set C, with strict inequality at ∂C—then there is a single smooth function W whose sublevel set {W ≤ 1} is exactly C, whose boundary {W = 1} is ∂C, and which decreases along the closed-loop flow everywhere off the origin. The proof constructs W(x) = V(x)/V(φ(T(x),x)), where T(x) is the unique time at which the trajectory from x crosses ∂C; this ratio automatically satisfies a PDE ∇W·F = −ω₁ and the level-set conditions. Under a stabilizable linearization a

Load-bearing premise

The assumption that at every boundary point some control makes the barrier's Lie derivative strictly positive (the strict inward-pointing condition) carries the whole construction; if only the usual non-strict barrier inequality holds, the hitting time T(x) may not be differentiable and the ratio W may fail to be a valid CLBF.

Editorial extensions

If this is right

A strict CLF–CBF compatibility condition on the boundary of the safe set is both necessary and sufficient for the existence of a single smooth CLBF.
The combined certificate W satisfies a PDE with prescribed boundary conditions, making it a candidate for PDE-based learning or verification of safe-stabilizing functions.
If a specification admits no smooth single Lyapunov-barrier certificate, then every CLF–CBF pair is conflict-ridden: no pair can be simultaneously satisfied in a robust sense.
The constructed W, together with a universal feedback formula, yields a smooth (resp. continuous) safe-stabilizing controller, with exponential stability when the linearization is stabilizable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: The ratio construction suggests a time-reparametrization interpretation—W measures how much Lyapunov value is consumed before reaching the safe boundary—which may connect to minimum-time or reach-avoid cost interpretations.
Inference: Because strict inward pointing at the boundary is load-bearing, systems whose barrier condition is only non-strict may require a qualitatively different certificate; exploring boundary tangency cases could extend the theorem.
Inference: A computational recipe follows: integrate the closed-loop flow backward to find T(x), evaluate V at the boundary crossing, and form W; this can be tested numerically on low-dimensional polynomial systems as a fast falsifier for the theorem.
Inference: The equivalence implies that verifying joint safety and stability can be reduced to checking a single function's level sets and PDE, which may simplify controller synthesis for safety-critical robotics.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the construction W=V/V(phi(T)) is genuinely new and the main equivalence is plausible, but Proposition 1's proof as written has a sign error that breaks the safety guarantee. The stress-test note is right.

What's actually new: the paper gives a converse theorem that turns a strictly compatible CLF-CBF pair into a single smooth CLBF, with a PDE characterization and exact level sets {W<=1}=C. The formula W(x)=V(x)/V(phi(T(x),x)) and the boundary-only strict compatibility condition are stronger than what [16] and [18] offer, and the level-set verification in Theorem 1 is clean. The backward hitting time argument in Lemma 1 is a nice piece of work, aside from the O(1/|x|) typo for T(x) (the proof actually gives logarithmic growth).

The soft spot is proportionally large. In the proof of Proposition 1, Assumption 1(2) says for each x in partial C there is a u_x with phi(x,u_x)<0 and psi(x,u_x)>0. The proof instead states phi<0 and psi<0, then defines epsilon_x from both negative values and later imposes psi<=-epsilon/2 on a neighborhood. So the constructed controller satisfies psi<0 on partial C, meaning the closed-loop field points outward and C is not forward invariant. That is exactly the property Lemma 1 needs, so Theorem 1 has no valid controller to lean on. The same wrong inequality is used in both parts (a) and (b). This is a fixable sign flip, not a structural flaw -- flip the sign, set epsilon_x from -phi and psi, and the partition-of-unity argument plausibly works. But as written, the central implication is unsupported.

Two smaller issues: the smoothing step under Assumption 3 is sketched rather than demonstrated, and the example only shows that the unsmoothed W can fail to be C^1 at 0; it doesn't validate the actual smoothing construction. Neither is in the same league as the sign error. The citation to [15] for the integral identity is fine; it's a standard result, and there are no fitted parameters or hidden compatibility assumptions.

Who should read this: anyone working on CLF-CBF compatibility, safe stabilization, or converse Lyapunov theorems. It deserves a serious referee, with the explicit instruction to have the authors repair Proposition 1 and make the smoothing lemma explicit. I would not desk-reject it; I'd send it to review with the expectation of major revision.

Referee Report

3 major / 4 minor

Summary. The manuscript claims an iff characterization of joint safety and stability certificates: a strictly compatible CLF-CBF pair exists iff a single smooth CLBF exists with the safe set appearing exactly as a sublevel set. The proof route is constructive: Proposition 1 builds a safe stabilizing feedback from the compatible pair; Lemma 1 establishes C^1 regularity and gradient estimates for the boundary hitting time T(x); Theorem 1 then defines W(x)=V(x)/V(phi(T(x),x)), shows that W solves the PDE grad W . F = -omega_1, and verifies the level-set conditions {W<=1}=C, {W=1}=partial C. An example illustrates that without exponential stabilizability the unsmoothed W need not be C^1 at the origin, motivating a composition-based smoothing step.

Significance. If the proof can be repaired, this is a valuable converse theorem: it reduces the existence of a compatible CLF-CBF pair to the existence of a single Lyapunov-like function with a prescribed boundary level set, and it provides an explicit PDE characterization. The construction is parameter-free, the hitting-time regularity estimate is of independent interest, and the paper names the precise regularity trade-off through Assumptions 2 and 3. The current version, however, contains sign errors in two load-bearing places, so the main theorem is not proven as written.

major comments (3)

[III.A, Proposition 1 proof] The proof defines psi(x,u)=L_f h(x)+L_g h(x)u and then states that for x in partial C we have phi(x,u_x)<0 and psi(x,u_x)<0. This contradicts Assumption 1(2), Eq. (10), which requires psi(x,u_x)>0. The subsequent construction sets epsilon_x = min{-phi,-psi}>0 and requires psi(y,u_x) <= -epsilon/2 on a neighborhood, so the final feedback k satisfies grad h . F < 0 on partial C. This is the outward-pointing condition; Corollary 1, Eq. (6), requires the opposite sign for forward invariance. Since Proposition 1 supplies the vector field F used in Lemma 1 (which needs grad h . F > 0 on partial C) and in Theorem 1, the main result is not established as written. The error appears again in the summary of the construction in part (b). Replacing the sign so that psi>0 and setting epsilon_x = min{-phi, psi} seems to make the partition-of-unity argument go through, but this must be redone carefully.
[III.B, Theorem 1 proof, after Eq. (16)] For x in Int(C), T(x)<0, and the display V(phi(T(x),x)) = V(x) - integral_{T(x)}^0 omega(phi(t,x)) dt > V(x) has the wrong sign. Since omega is positive and the integration interval is [T(x),0] with T(x)<0, the integral is positive, so the right-hand side is less than V(x). The correct identity is V(phi(T(x),x)) = V(x) + integral_{T(x)}^0 omega(phi(t,x)) dt, which gives V(phi(T(x),x)) > V(x) and hence W(x)<1, as required. As written the displayed identity would give W(x)>1 in Int(C), contradicting condition (2) of Definition 4.
[Assumption 2] Assumption 2 defines P = grad^2 V(0), where V is only assumed continuously differentiable in Assumption 1. A C^1 function need not possess a Hessian at the origin, so the statement is not well-posed. Either Assumption 1 should be strengthened to require V to be C^2 in a neighborhood of 0, or Assumption 2 should be reformulated using stabilizability of (A,B) alone (e.g., taking P from the Lyapunov equation, as is actually done later in Lemma 1). As written, Proposition 1(a) and the part of Theorem 1 that relies on Assumption 2 rest on an ill-defined object.

minor comments (4)

[Lemma 1 proof] In the quadratic Lyapunov estimate, the term '-|x|^top' should read '-|x|^2'.
[Theorem 1 proof] The sentence 'T(x)=O(1/|x|)' should be 'grad T(x)=O(1/|x|)'; Lemma 1 proves |T(x)| = O(log(1/|x|)), not O(1/|x|).
[III.A, Proposition 1 proof] The notation 'D \(delta C \cup {0}\)' and 'delta C' appears to be a typo for 'D \setminus \partial C' and '\partial C'.
[Abstract and Section I] The abstract and introduction state an 'iff' characterization, but Theorem 1 only proves the direction from a strictly compatible pair to a CLBF. The converse direction follows easily from Definition 4 by taking h=1-W and V=W, but it should be stated explicitly for the claimed equivalence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CLBF is explicitly constructed from the assumed CLF-CBF pair and standard converse-Lyapunov results.

full rationale

The paper's central claim—that a strictly compatible CLF-CBF pair yields a single smooth Lyapunov function with the safe set as a level set—is proved by an explicit construction. W(x)=V(x)/V(φ(T(x),x)) is built from the given V and the hitting time T of the closed-loop system under a controller synthesized in Proposition 1. No parameters are fitted, and no empirical data are involved. The only citation to the authors' own prior work is [15, Prop. 2], used for the standard integral identity V(x)=∫ω(φ(t,x))dt and the PDE consequence. That proposition does not contain the target result (a CLBF with prescribed boundary conditions) and is a classical converse-Lyapunov identity; therefore the citation is independent support and does not create circularity. The derivation chain is self-contained from the stated assumptions.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof introduces no fitted constants and no new physical or mathematical entities. The CLBF concept is from [22], and the hitting time T is a defined quantity, not a new postulate. The assumptions are standard regularity, stabilizability, and strict compatibility conditions; the only questionable load-bearing assumption is the strict boundary inequality (10), which the proof text itself mishandles.

assumptions (6)

standard math Converse Lyapunov theorem: an asymptotically stable C^1 vector field admits a smooth Lyapunov function on its domain of attraction.
Invoked in Theorem 1 proof via '[10], [14], [27]' to obtain the smooth V used in the construction W=V/V(phi(T)).
standard math Implicit function theorem applied to psi(t,x)=h(phi(t,x)) gives smoothness of the hitting time T(x).
Used in Lemma 1; requires partial psi/partial t = grad h . F > 0 on partial C, which is exactly the strict boundary condition (10).
standard math Nagumo's theorem / strict inward pointing vector field on partial C guarantees controlled forward invariance.
Used in Corollary 1 and Remark 3 to ensure C is forward invariant and trajectories cross partial C at most once.
domain assumption Strict compatibility Assumption 1: on partial C there exists a single u making the CLF derivative negative and the CBF derivative positive; plus Assumption 2 or 3 for regularity at the origin.
This is the theorem's main hypothesis. The strictness of (10) is load-bearing for Lemma 1 and for W being C^1.
standard math Integral representation V(x)=int_0^inf omega(phi(t,x)) dt for a smooth Lyapunov function.
Used in Theorem 1, cited to [15, Proposition 2], a standard Lyapunov integral identity; the cited paper shares authors but the identity is not the target result.
standard math Smoothing lemma ([27, Lemma 17], [14, Theorem 4.3]): a class K_infinity function rho can be chosen so that rho o W is smooth while preserving the 1-level set.
Used only for the Assumption 3 case to smooth W at the origin; the paper does not reproduce the lemma, so its applicability is not fully demonstrated.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Converse Control Lyapunov Theorem for Joint Safety and Stability." pith.science (2026). https://pith.science/paper/DSMKJ7BD

@misc{pith2026250912182,
  author       = {Pith},
  title        = {Pith review of: A Converse Control Lyapunov Theorem for Joint Safety and Stability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DSMKJ7BD}},
  note         = {Machine review of arXiv:2509.12182}
}

read the original abstract

We show that the existence of a strictly compatible pair of control Lyapunov and control barrier functions is equivalent to the existence of a single smooth Lyapunov function that certifies both asymptotic stability and safety. This characterization complements existing literature on converse Lyapunov functions by establishing a partial differential equation (PDE) characterization with prescribed boundary conditions on the safe set, ensuring that the safe set is exactly certified by this Lyapunov function. The result also implies that if a safety and stability specification cannot be certified by a single Lyapunov function, then any pair of control Lyapunov and control barrier functions necessarily leads to a conflict and cannot be satisfied simultaneously in a robust sense.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references

[16]

Smooth converse Lyapunov-barrier theorems for asymptotic stability with safety constraints and reach-avoid-stay specifications.Automatica, 144:110478, 2022

Yiming Meng, Yinan Li, Maxwell Fitzsimmons, and Jun Liu. Smooth converse Lyapunov-barrier theorems for asymptotic stability with safety constraints and reach-avoid-stay specifications.Automatica, 144:110478, 2022

2022
[18]

Converse theorems for certificates of safety and stability, 2025

Pol Mestres and Jorge Cort ´es. Converse theorems for certificates of safety and stability, 2025

2025
[15]

Physics-informed neural network Lyapunov functions: PDE charac- terization, learning, and verification.Automatica, 175:112193, 2025

Jun Liu, Yiming Meng, Maxwell Fitzsimmons, and Ruikun Zhou. Physics-informed neural network Lyapunov functions: PDE charac- terization, learning, and verification.Automatica, 175:112193, 2025

2025
[1]

Control barrier func- tions: Theory and applications

Aaron D Ames, Samuel Coogan, Magnus Egerstedt, Gennaro No- tomista, Koushil Sreenath, and Paulo Tabuada. Control barrier func- tions: Theory and applications. InProc. of ECC, pages 3420–3431. IEEE, 2019

2019
[2]

Control barrier function based quadratic programs for safety critical systems.IEEE Transactions on Automatic Control, 62(8):3861–3876, 2017

Aaron D Ames, Xiangru Xu, Jessy W Grizzle, and Paulo Tabuada. Control barrier function based quadratic programs for safety critical systems.IEEE Transactions on Automatic Control, 62(8):3861–3876, 2017

2017
[3]

Stabilization with relaxed controls.Nonlinear Analysis: Theory, Methods & Applications, 7(11):1163–1173, 1983

Zvi Artstein. Stabilization with relaxed controls.Nonlinear Analysis: Theory, Methods & Applications, 7(11):1163–1173, 1983

1983
[4]

On (the existence of) control Lyapunov barrier functions

Philipp Braun and Christopher M Kellett. On (the existence of) control Lyapunov barrier functions. 2017

2017
[5]

stabilization with guaranteed safety using control Lyapunov–barrier function

Philipp Braun and Christopher M Kellett. Comment on “stabilization with guaranteed safety using control Lyapunov–barrier function”. Automatica, 122:109225, 2020

2020

Show all 31 references

[6]

Crowl and J.F

D.A. Crowl and J.F. Louvar.Chemical Process Safety: Fundamentals with Applications. Prentice Hall, 1990

1990
[7]

Verification and synthesis of compatible control lyapunov and control barrier functions

Hongkai Dai, Chuanrui Jiang, Hongchao Zhang, and Andrew Clark. Verification and synthesis of compatible control lyapunov and control barrier functions. InProc. of CDC, pages 8178–8185. IEEE, 2024

2024
[8]

Safe nonlinear control using robust neural Lyapunov-barrier functions

Charles Dawson, Zengyi Qin, Sicun Gao, and Chuchu Fan. Safe nonlinear control using robust neural Lyapunov-barrier functions. In Conference on Robot Learning, pages 1724–1735. PMLR, 2022

2022
[9]

Challenges in autonomous vehi- cle testing and validation.SAE International Journal of Transportation Safety, 4(1):15–24, 2016

Philip Koopman and Michael Wagner. Challenges in autonomous vehi- cle testing and validation.SAE International Journal of Transportation Safety, 4(1):15–24, 2016

2016
[10]

On the inversion of Lyapunov’s second theorem on stability of motion.Amer

Jaroslav Kurzweil. On the inversion of Lyapunov’s second theorem on stability of motion.Amer. Math. Soc. Transl, 24(2):19–77, 1963

1963
[11]

Lee.Introduction to Smooth Manifolds

John M. Lee.Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Springer, 2003

2003
[12]

Haoqi Li, Jiangping Hu, Xiaoming Hu, and Bijoy K Ghosh. Stabiliza- tion of nonlinear safety-critical systems by relaxed converse lyapunov- barrier approach and its applications in robotic systems.Autonomous Intelligent Systems, 4(1):1–8, 2024

2024
[13]

A graphical interpretation and universal formula for safe stabilization

Ming Li and Zhiyong Sun. A graphical interpretation and universal formula for safe stabilization. In2023 American Control Conference (ACC), pages 3012–3017. IEEE, 2023

2023
[14]

A smooth converse Lyapunov theorem for robust stability.SIAM Journal on Control and Optimization, 34(1):124–160, 1996

Yuandan Lin, Eduardo D Sontag, and Yuan Wang. A smooth converse Lyapunov theorem for robust stability.SIAM Journal on Control and Optimization, 34(1):124–160, 1996

1996
[17]

Optimization-based safe stabilizing feedback with guaranteed region of attraction.IEEE Control Systems Letters, 7:367–372, 2022

Pol Mestres and Jorge Cort ´es. Optimization-based safe stabilizing feedback with guaranteed region of attraction.IEEE Control Systems Letters, 7:367–372, 2022

2022
[19]

¨Uber die lage der integralkurven gew ¨ohnlicher differentialgleichungen.Proceedings of the Physico-Mathematical Society of Japan, 24:551–559, 1942

Mitio Nagumo. ¨Uber die lage der integralkurven gew ¨ohnlicher differentialgleichungen.Proceedings of the Physico-Mathematical Society of Japan, 24:551–559, 1942

1942
[20]

Ong.Uniting and Balancing Control Objectives: Safety, Stability, Smoothness, and Resource Conservation

P. Ong.Uniting and Balancing Control Objectives: Safety, Stability, Smoothness, and Resource Conservation. Ph.d. dissertation, University of California, San Diego, San Diego, CA, 2022

2022
[21]

Universal formula for smooth safe stabilization

Pio Ong and Jorge Cort ´es. Universal formula for smooth safe stabilization. In2019 IEEE 58th Conference on Decision and Control (CDC), pages 2373–2378. IEEE, 2019

2019
[22]

Stabiliza- tion with guaranteed safety using control Lyapunov–barrier function

Muhammad Zakiyullah Romdlony and Bayu Jayawardhana. Stabiliza- tion with guaranteed safety using control Lyapunov–barrier function. Automatica, 66:39–47, 2016

2016
[23]

A Lyapunov-like characterization of asymptotic controllability.SIAM journal on control and optimization, 21(3):462– 471, 1983

Eduardo D Sontag. A Lyapunov-like characterization of asymptotic controllability.SIAM journal on control and optimization, 21(3):462– 471, 1983

1983
[24]

A ‘universal’construction of Artstein’s theorem on nonlinear stabilization.Systems & Control Letters, 13(2):117–123, 1989

Eduardo D Sontag. A ‘universal’construction of Artstein’s theorem on nonlinear stabilization.Systems & Control Letters, 13(2):117–123, 1989

1989
[25]

Springer Science & Business Media, 1998

Eduardo D Sontag.Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer Science & Business Media, 1998

1998
[26]

Stability and stabilization: discontinuities and the effect of disturbances

Eduardo D Sontag. Stability and stabilization: discontinuities and the effect of disturbances. InNonlinear analysis, differential equations and control, pages 551–598. Springer, 1999

1999
[27]

A smooth Lyapunov function from a class-KLestimate involving two positive semidefinite functions

Andrew R Teel and Laurent Praly. A smooth Lyapunov function from a class-KLestimate involving two positive semidefinite functions. ESAIM: Control, Optimisation and Calculus of Variations, 5:313–367, 2000

2000
[28]

Constructive safety using control barrier functions.IFAC Proceedings Volumes, 40(12):462–467, 2007

Peter Wieland and Frank Allg ¨ower. Constructive safety using control barrier functions.IFAC Proceedings Volumes, 40(12):462–467, 2007

2007
[29]

Control Lyapunov-barrier function-based model predictive control of nonlinear systems.Au- tomatica, 109:108508, 2019

Zhe Wu, Fahad Albalawi, Zhihao Zhang, Junfeng Zhang, Helen Durand, and Panagiotis D Christofides. Control Lyapunov-barrier function-based model predictive control of nonlinear systems.Au- tomatica, 109:108508, 2019

2019
[30]

Constrained control of input–output linearizable systems using control sharing barrier functions.Automatica, 87:195–201, 2018

Xiangru Xu. Constrained control of input–output linearizable systems using control sharing barrier functions.Automatica, 87:195–201, 2018

2018
[31]

Robustness of control barrier functions for safety critical control

Xiangru Xu, Paulo Tabuada, Jessy W Grizzle, and Aaron D Ames. Robustness of control barrier functions for safety critical control. IFAC-PapersOnLine, 48(27):54–61, 2015

2015

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [16]

Smooth converse Lyapunov-barrier theorems for asymptotic stability with safety constraints and reach-avoid-stay specifications.Automatica, 144:110478, 2022

Yiming Meng, Yinan Li, Maxwell Fitzsimmons, and Jun Liu. Smooth converse Lyapunov-barrier theorems for asymptotic stability with safety constraints and reach-avoid-stay specifications.Automatica, 144:110478, 2022

2022

[2] [18]

Converse theorems for certificates of safety and stability, 2025

Pol Mestres and Jorge Cort ´es. Converse theorems for certificates of safety and stability, 2025

2025

[3] [15]

Physics-informed neural network Lyapunov functions: PDE charac- terization, learning, and verification.Automatica, 175:112193, 2025

Jun Liu, Yiming Meng, Maxwell Fitzsimmons, and Ruikun Zhou. Physics-informed neural network Lyapunov functions: PDE charac- terization, learning, and verification.Automatica, 175:112193, 2025

2025

[4] [1]

Control barrier func- tions: Theory and applications

Aaron D Ames, Samuel Coogan, Magnus Egerstedt, Gennaro No- tomista, Koushil Sreenath, and Paulo Tabuada. Control barrier func- tions: Theory and applications. InProc. of ECC, pages 3420–3431. IEEE, 2019

2019

[5] [2]

Control barrier function based quadratic programs for safety critical systems.IEEE Transactions on Automatic Control, 62(8):3861–3876, 2017

Aaron D Ames, Xiangru Xu, Jessy W Grizzle, and Paulo Tabuada. Control barrier function based quadratic programs for safety critical systems.IEEE Transactions on Automatic Control, 62(8):3861–3876, 2017

2017

[6] [3]

Stabilization with relaxed controls.Nonlinear Analysis: Theory, Methods & Applications, 7(11):1163–1173, 1983

Zvi Artstein. Stabilization with relaxed controls.Nonlinear Analysis: Theory, Methods & Applications, 7(11):1163–1173, 1983

1983

[7] [4]

On (the existence of) control Lyapunov barrier functions

Philipp Braun and Christopher M Kellett. On (the existence of) control Lyapunov barrier functions. 2017

2017

[8] [5]

stabilization with guaranteed safety using control Lyapunov–barrier function

Philipp Braun and Christopher M Kellett. Comment on “stabilization with guaranteed safety using control Lyapunov–barrier function”. Automatica, 122:109225, 2020

2020