Uniform Feasibility For Smoothed Backup Control Barrier Functions

Anil Alan; Bart De Schutter

arxiv: 2511.13499 · v2 · submitted 2025-11-17 · 🧮 math.OC · cs.SY· eess.SY

Uniform Feasibility For Smoothed Backup Control Barrier Functions

Anil Alan , Bart De Schutter This is my paper

Pith reviewed 2026-05-17 20:37 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords control barrier functionsbackup control barrier functionslog-sum-exp smoothingsafety filtersfeasibility guaranteesnonsmooth safe setsextended CBF

0 comments

The pith

Smoothing the pointwise minimum in backup control barrier functions yields a priori feasibility under explicit safety conditions on the backup set.

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

The paper shows that replacing the nonsmooth minimum in a safe-set definition with its log-sum-exp approximation turns the function into a valid control barrier function when a strict safety condition holds. For compact sets this produces an explicit lower bound on the smoothing parameter that keeps the safety constraint feasible. For unbounded sets, tail conditions on the functions ensure the smoothed version satisfies an extended barrier property uniformly. Applied to backup CBFs, the authors conclude that safety of a compact terminal backup set under its controller, plus a boundary-safety condition on the trajectories, is enough to guarantee feasibility without extra online checks.

Core claim

Safety of a compact terminal backup set under a backup controller, together with a condition ensuring safety of the backup trajectories on the relevant boundary of the safe set, is sufficient for constraint feasibility for BCBFs. The log-sum-exp smoothing of the defining minimum makes the resulting function a CBF (or extended CBF) for a range of the smoothing parameter when the strict safety condition holds, supplying explicit lower bounds for compact sets and uniform tail conditions for unbounded sets.

What carries the argument

Log-sum-exp smoothing of the pointwise minimum that defines the safe set, which converts the nonsmooth construction into a differentiable function that satisfies the CBF inequality under the strict safety condition.

If this is right

For compact safe sets an explicit lower bound on the smoothing parameter makes the smooth function a CBF.
For unbounded sets, tail conditions on the defining functions ensure the smoothed function satisfies an extended CBF condition uniformly.
These conditions together give a recipe for a priori feasibility guarantees on smooth inner approximations of nonsmooth safe sets.
For backup CBFs the combination of terminal-set safety and boundary safety is sufficient for the overall safety-filter constraint to remain feasible.

Where Pith is reading between the lines

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

The same smoothing argument could be tried on other nonsmooth safe-set constructions that arise from pointwise minima in control design.
Engineers could select the smoothing parameter offline once and for all, removing the need to monitor feasibility during real-time operation.
Numerical checks on simple unbounded examples could test whether the tail conditions are easy to verify in practice.

Load-bearing premise

The strict safety condition must hold for the backup controller and the backup trajectories must remain safe on the boundary.

What would settle it

A simulation or calculation showing that the safety filter becomes infeasible for a smoothing parameter above the derived lower bound, even though the compact backup set is safe under the backup controller and the boundary condition holds.

read the original abstract

We study feasibility guarantees for safety filters developed using Control Barrier Functions (CBFs) when a safe set is defined using the pointwise minimum of continuously differentiable functions, a construction that is common for the backup CBF (BCBF) method and typically nonsmooth. We replace the minimum by its log-sum-exp (soft-min) smoothing and show that, under a strict safety condition, the smooth function becomes a CBF (or extended CBF) for a range of the smoothing parameter. For compact safe sets, we derive an explicit lower bound on the smoothing parameter that makes the smooth function a CBF and hence renders the corresponding safety constraint feasible. For unbounded sets, we introduce tail conditions under which the smooth function satisfies an extended CBF condition uniformly. Finally, we apply these results to BCBFs. We show that safety of a compact (terminal) backup set under a backup controller, together with a condition ensuring safety of the backup trajectories on the relevant boundary of the safe set, is sufficient for constraint feasibility for BCBFs. These results provide a recipe for a priori feasibility guarantees for smooth inner approximations of nonsmooth safe sets without the need for additional online certification.

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 paper gives explicit lower bounds on the log-sum-exp smoothing parameter that make backup CBF safety filters feasible a priori for compact sets, but only after you verify an extra strict boundary condition that does not come for free from backup-set invariance.

read the letter

The main point is that log-sum-exp smoothing of the min in a backup CBF can be made to satisfy the CBF inequality with an explicit lower bound on the smoothing parameter, provided the backup trajectories meet a safety condition on the boundary of the safe set. For compact terminal backup sets this yields a concrete threshold you can compute ahead of time so the QP constraint stays feasible. For unbounded sets the same idea is extended with tail conditions that deliver uniform extended CBF behavior. The authors then apply the result directly to the BCBF construction and state that backup-set safety plus the boundary condition is enough to guarantee the smoothed version works without online feasibility checks. This is the concrete advance over prior smoothing work on CBFs. The derivations rest on standard Lie-derivative calculations and known properties of the soft-min, so the steps are traceable once the boundary assumption is granted. The explicit bound for the compact case and the tail conditions are the parts that were not already in the literature they cite. The practical payoff is a recipe that removes the need for runtime certification of the safety filter, which is a real pain point when the safe set is defined by a nonsmooth min. The soft spot is that the required strict inequality for the smoothed function does not follow automatically from plain invariance of the backup set. The weighting induced by the log-sum-exp can weaken the Lie derivative on the boundary, so the extra condition must still be checked or ensured separately. If that check turns out to be comparable in difficulty to the original feasibility problem, the a priori guarantee loses some of its bite. The tail conditions for unbounded sets look similarly non-trivial to verify in applications. This work is aimed at control researchers who already use backup CBFs for robotics or autonomous systems and want to pick the smoothing parameter with a guarantee rather than by trial and error. Readers who need the explicit bound or the uniform extended-CBF statement will get direct value. The claims are stated clearly enough and rest on established theory, so the paper deserves a serious referee to verify the bound derivations and assess how restrictive the boundary and tail conditions are in practice. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The paper studies feasibility of safety constraints for smoothed Backup Control Barrier Functions (BCBFs). It replaces the nonsmooth pointwise minimum of C1 functions defining the safe set with its log-sum-exp (soft-min) approximation. Under a strict safety condition, the smoothed function is shown to be a CBF (or extended CBF) for a range of the smoothing parameter. Explicit lower bounds on the parameter are derived for compact safe sets; tail conditions are introduced for unbounded sets to obtain uniform extended CBF properties. The results are applied to BCBFs: safety of a compact terminal backup set under a backup controller, together with a boundary condition on backup trajectories, is claimed to be sufficient for a priori feasibility of the smoothed BCBF constraints without online certification.

Significance. If the central sufficiency statements hold, the work supplies a concrete recipe for obtaining uniform feasibility guarantees for inner approximations of nonsmooth safe sets arising in the BCBF method. The explicit parameter bounds for the compact case and the uniform tail conditions for unbounded domains are potentially useful strengths for practical deployment in safety-critical control. The approach avoids online feasibility checks by shifting verification to offline conditions on the backup set and trajectories.

major comments (2)

[§4, Theorem 4.2] §4 (application to BCBFs) and the statement following Theorem 4.2: the central sufficiency claim asserts that safety of the compact terminal backup set under the backup controller plus 'a condition ensuring safety of the backup trajectories on the relevant boundary' is enough for the smoothed BCBF constraint to be feasible. However, the strict Lie-derivative inequality required for the log-sum-exp function to be a CBF (after accounting for the soft-min gradient weighting) is stronger than plain invariance of the backup set. The manuscript treats this boundary condition as an assumption rather than deriving that it implies a uniform positive margin on the smoothed Lie derivative; without an explicit reduction or certificate showing the margin remains positive and finite for the chosen smoothing parameter, the lower bound on the smoothing parameter may be infinite or nonexistent.
[§3.3] §3.3 (unbounded sets): the tail conditions are introduced to obtain a uniform extended CBF property, but the proof sketch does not quantify how the tail decay interacts with the log-sum-exp smoothing to preserve a uniform lower bound on the CBF inequality at infinity. If the tail conditions only guarantee invariance of the original nonsmooth set, the same gap as in the compact case appears for the smoothed version.

minor comments (2)

[Throughout] Notation for the smoothing parameter (denoted variously as ε or μ) should be unified across sections and theorems for clarity.
[Abstract and §1] The abstract and introduction would benefit from a brief remark on how the boundary condition can be verified in practice for common backup controllers (e.g., via Lyapunov functions or reachability).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important points regarding the transfer of strict invariance margins from the nonsmooth to the smoothed setting in both the compact BCBF application and the unbounded case. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§4, Theorem 4.2] §4 (application to BCBFs) and the statement following Theorem 4.2: the central sufficiency claim asserts that safety of the compact terminal backup set under the backup controller plus 'a condition ensuring safety of the backup trajectories on the relevant boundary' is enough for the smoothed BCBF constraint to be feasible. However, the strict Lie-derivative inequality required for the log-sum-exp function to be a CBF (after accounting for the soft-min gradient weighting) is stronger than plain invariance of the backup set. The manuscript treats this boundary condition as an assumption rather than deriving that it implies a uniform positive margin on the smoothed Lie derivative; without an explicit reduction or certificate showing the margin remains positive and finite for the chosen smoothing parameter, the lower bound on the smoothing parameter may be infinite or non

Authors: We agree that an explicit derivation is needed to confirm that the boundary safety condition on backup trajectories produces a uniform positive margin for the weighted Lie derivative of the log-sum-exp function. The condition is formulated so that each active boundary function satisfies a strict decrease under the backup controller; because the soft-min gradient is a convex combination of the individual gradients, this strict inequality is inherited by the smoothed function with a margin that depends continuously on the smoothing parameter. In the revision we will insert a supporting lemma that extracts an explicit positive lower bound on the smoothed Lie derivative directly from the boundary condition, thereby guaranteeing that the lower bound on the smoothing parameter remains finite and can be computed from the same data used to verify the original nonsmooth invariance. revision: yes
Referee: [§3.3] §3.3 (unbounded sets): the tail conditions are introduced to obtain a uniform extended CBF property, but the proof sketch does not quantify how the tail decay interacts with the log-sum-exp smoothing to preserve a uniform lower bound on the CBF inequality at infinity. If the tail conditions only guarantee invariance of the original nonsmooth set, the same gap as in the compact case appears for the smoothed version.

Authors: The tail conditions are designed to make the contribution of functions that become active only at large distances decay faster than any polynomial growth in the state, so that their effect on both the value and the gradient of the log-sum-exp remains uniformly small. We acknowledge that the current sketch does not spell out the quantitative interaction with the smoothing parameter. In the revision we will expand the proof of the uniform extended-CBF property to include explicit estimates: for any prescribed margin, a sufficiently rapid tail decay (as stated in the assumption) permits a finite upper bound on the smoothing parameter that keeps the smoothed Lie derivative bounded away from zero uniformly outside a compact set. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on standard CBF and log-sum-exp properties

full rationale

The paper claims that safety of a compact terminal backup set under a backup controller, together with an explicit boundary safety condition on backup trajectories, suffices to make the log-sum-exp smoothed function a CBF (or extended CBF) for a range of the smoothing parameter, thereby guaranteeing feasibility. This chain is built from the definition of CBFs, the explicit lower-bound derivation for the smoothing parameter in the compact case, and stated tail conditions for unbounded sets. No step reduces the target feasibility result to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified. The boundary condition is introduced as an additional assumption rather than derived from the backup-set invariance alone, but this is an explicit premise, not a circular reduction. The overall argument remains self-contained against external CBF theory and does not rename known results or smuggle ansatzes via citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard mathematical properties of the log-sum-exp approximation and conventional CBF definitions; no new free parameters are fitted and no new entities are postulated.

axioms (2)

domain assumption The safe set is the sublevel set of the pointwise minimum of continuously differentiable functions.
This is the standard nonsmooth construction for backup CBFs invoked throughout the abstract.
standard math Log-sum-exp provides a differentiable approximation to the minimum that converges uniformly on compact sets under suitable conditions.
Invoked to replace the nonsmooth min while preserving CBF properties for a range of the smoothing parameter.

pith-pipeline@v0.9.0 · 5500 in / 1369 out tokens · 37869 ms · 2026-05-17T20:37:31.443411+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Control barrier functions: Theory and applications,

A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, “Control barrier functions: Theory and applications,” in2019 18th European Control Conference (ECC), 2019, pp. 3420– 3431

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On the necessity of barrier certificates,

S. Prajna and A. Rantzer, “On the necessity of barrier certificates,” IFAC Proceedings Volumes, vol. 38, no. 1, pp. 526–531, 2005, 16th IFAC World Congress

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Converse barrier certificate theorems,

R. Wisniewski and C. Sloth, “Converse barrier certificate theorems,” IEEE Transactions on Automatic Control, vol. 61, no. 5, pp. 1356– 1361, 2016

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Converse theorems for safety and barrier certificates,

S. Ratschan, “Converse theorems for safety and barrier certificates,” IEEE Transactions on Automatic Control, vol. 63, no. 8, pp. 2628– 2632, 2018

work page 2018

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Converse barrier functions via Lyapunov functions,

J. Liu, “Converse barrier functions via Lyapunov functions,”IEEE Transactions on Automatic Control, vol. 67, no. 1, pp. 497–503, 2022

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Towards a framework for realizable safety critical control through active set invariance,

T. Gurriet, A. Singletary, J. Reher, L. Ciarletta, E. Feron, and A. Ames, “Towards a framework for realizable safety critical control through active set invariance,” inInternational Conference on Cyber-Physical Systems (ICCPS), 2018, pp. 98–106

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Mestres and J

P. Mestres and J. Cort ´es, “Converse theorems for certificates of safety and stability,”arXiv preprint arXiv:2406.14823, 2024

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I. Mitchell, A. Bayen, and C. Tomlin, “A time-dependent Hamilton- Jacobi formulation of reachable sets for continuous dynamic games,” IEEE Transactions on Automatic Control, vol. 50, no. 7, pp. 947–957, 2005

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[10] [10]

An online approach to active set invariance,

T. Gurriet, M. Mote, A. D. Ames, and E. Feron, “An online approach to active set invariance,” inIEEE Conference on Decision and Control (CDC), 2018, pp. 3592–3599

work page 2018

[11] [11]

Backup control barrier functions: Formulation and comparative study,

Y . Chen, M. Jankovic, M. Santillo, and A. D. Ames, “Backup control barrier functions: Formulation and comparative study,” inIEEE Conference on Decision and Control (CDC), 2021, pp. 6835–6841

work page 2021

[12] [12]

A predictive safety filter for learning-based control of constrained nonlinear dynamical systems,

K. P. Wabersich and M. N. Zeilinger, “A predictive safety filter for learning-based control of constrained nonlinear dynamical systems,” Automatica, vol. 129, p. 109597, 2021

work page 2021

[13] [13]

Predictive control barrier functions for online safety critical control,

J. Breeden and D. Panagou, “Predictive control barrier functions for online safety critical control,” in2022 IEEE 61st Conference on Decision and Control (CDC), 2022, pp. 924–931

work page 2022

[14] [14]

Inner approximations of the region of attraction for polynomial dynamical systems,

M. Korda, D. Henrion, and C. N. Jones, “Inner approximations of the region of attraction for polynomial dynamical systems,”IFAC Proceedings Volumes, vol. 46, no. 23, pp. 534–539, 2013, 9th IFAC Symposium on Nonlinear Control Systems

work page 2013

[15] [15]

Data- driven safety-critical control: Synthesizing control barrier functions with Koopman operators,

C. Folkestad, Y . Chen, A. D. Ames, and J. W. Burdick, “Data- driven safety-critical control: Synthesizing control barrier functions with Koopman operators,”IEEE Control Systems Letters, vol. 5, no. 6, pp. 2012–2017, 2020

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Safety-critical control with bounded inputs via reduced order models,

T. G. Molnar and A. D. Ames, “Safety-critical control with bounded inputs via reduced order models,” in2023 American Control Confer- ence (ACC), 2023, pp. 1414–1421

work page 2023

[17] [17]

Disturbance-robust backup control barrier functions: Safety under uncertain dynamics,

D. E. J. van Wijk, S. Coogan, T. G. Molnar, M. Majji, and K. L. Hobbs, “Disturbance-robust backup control barrier functions: Safety under uncertain dynamics,”IEEE Control Systems Letters, vol. 8, pp. 2817–2822, 2024

work page 2024

[18] [18]

A learning-based framework for safe human-robot collaboration with multiple backup control barrier functions,

N. C. Janwani, E. Das ¸, T. Touma, S. X. Wei, T. G. Molnar, and J. W. Burdick, “A learning-based framework for safe human-robot collaboration with multiple backup control barrier functions,” in2024 IEEE International Conference on Robotics and Automation (ICRA), 2024, pp. 11 676–11 682

work page 2024

[19] [19]

A scalable safety critical control framework for nonlinear systems,

T. Gurriet, M. Mote, A. Singletary, P. Nilsson, E. Feron, and A. D. Ames, “A scalable safety critical control framework for nonlinear systems,”IEEE Access, vol. 8, pp. 187 249–187 275, 2020

work page 2020

[20] [20]

Forward and control invariance analysis of backup control barrier function induced safe sets for online safety of nonlinear systems,

P. M. Rivera, “Forward and control invariance analysis of backup control barrier function induced safe sets for online safety of nonlinear systems,” in2024 IEEE 63rd Conference on Decision and Control (CDC), 2024, pp. 8150–8157

work page 2024

[21] [21]

Soft-minimum and soft-maximum barrier functions for safety with actuation constraints,

P. Rabiee and J. B. Hoagg, “Soft-minimum and soft-maximum barrier functions for safety with actuation constraints,”Automatica, vol. 171, p. 111921, 2025

work page 2025

[22] [22]

Predictive control barrier functions for piecewise affine systems with non-smooth constraints,

K. He, A. Alan, S. Shi, T. van den Boom, and B. De Schutter, “Predictive control barrier functions for piecewise affine systems with non-smooth constraints,”arXiv preprint arXiv:2510.21321, 2025

work page arXiv 2025

[23] [23]

F. H. Clarke, Y . S. Ledyaev, R. J. Stern, and R. Wolenski,Nonsmooth analysis and control theory, ser. Graduate Texts in Mathematics. New York, NY: Springer, 1998

work page 1998

[24] [24]

J. M. Lee,Introduction to smooth manifolds, 3rd ed., ser. Graduate Texts in Mathematics. New York, NY: Springer, 2003

work page 2003