Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions

Mohamed Maghenem; Ricardo G. Sanfelice

REVIEW 1 major objections 6 minor 1 cited by

Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions

T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read By checking infinitesimal inequalities near the boundary of each constraint, a vector-valued barrier function can certify that a closed set is forward pre-invariant or pre-contractive for a hybrid inclusion, without computing any solution.

desk verdict Solid extension of barrier-function certificates to hybrid inclusions, but the proof of the boundary-only invariance theorem has a gap in the domain of the flow map, and several proofs are deferred to the same arXiv preprint. read the letter →

arxiv 1908.03980 v4 pith:LLUWM24A submitted 2019-08-12 math.OC

Mohamed Maghenem , Ricardo G. Sanfelice This is my paper · ORCID

classification math.OC MSC 93C3093D3034A6093B03

keywords barrierfunctionsforwardinvariancecontractivityhybridinclusionssystemssafetyverificationuniquenessset

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 sufficient conditions, expressed as infinitesimal inequalities, under which a closed set is forward pre-invariant or pre-contractive for a hybrid inclusion. The set is defined as the common region where several scalar functions—the barrier function candidate—are nonpositive, so safety constraints with multiple simultaneous requirements can be certified without flattening them into one nonsmooth scalar. Pre-invariance means every maximal solution starting in the set stays in it; pre-contractivity means solutions starting on the boundary immediately move to the interior. The conditions separate into flow constraints, checked on an outer neighborhood of each constraint's zero level set using the contingent cone of admissible directions, and jump constraints ensuring jumps land back in the union of flow and jump sets. The payoff is a certificate that avoids computing solutions, even when solutions are nonunique or terminate prematurely.

What carries the argument

The central object is a vector-valued barrier function candidate $B=(B_1,\dots,B_m)$ defining $K=\{x\in C\cup D: B(x)\le 0\}$. The argument runs on the active zero-level sets $M_i=\{x\in\partial K: B_i(x)=0\}$: the flow condition is checked only on an outer neighborhood $(U(M_i)\setminus K_{e_i})\cap C$, intersected with the contingent cone $T_C(x)$ of directions along which solutions can flow while staying in $C$. For contractivity, strict inequality together with the Dubovitsky-Miliutin cone forces flows to enter $\operatorname{int}(K)$. The boundary-only variant in Theorem 2 replaces the neighborhood check by a condition on $\partial K$ using a transversality assumption and inequality (21), a one-sided Lipschitz-like condition involving a uniqueness function $\rho$.

What would settle it

Consider the differential inclusion with $C=\mathbb{R}^2$, $D=\emptyset$, $F(x)=\{[1,\sqrt{|x_2|}]^\top\}$, $B(x)=x_2$, and $K=\{x_2\le 0\}$. The boundary inequality $\langle\nabla B(x),F(x)\rangle=0$ holds on $\partial K$, yet the solution $x(t)=(t,\frac14 t^2)$ starts in $K$ and leaves it. This example shows that a boundary-only certificate must exclude such flows, and condition (21) is precisely the exclusion; a variant satisfying (21) while still admitting a leaving solution would falsify Theorem 2.

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Extended reading notes

Core claim

The central claim is Theorem 1: given a $C^1$ vector barrier function candidate $B$ defining $K=\{x\in C\cup D: B(x)\le 0\}$, the set $K$ is forward pre-invariant if, for each component $i$, every flow direction $\eta\in F(x)\cap T_C(x)$ satisfies $\langle\nabla B_i(x),\eta\rangle\le 0$ on an outer neighborhood $(U(M_i)\setminus K_{e_i})\cap C$, and if every jump from $x\in D\cap K$ satisfies $B(\eta)\le 0$ for $\eta\in G(x)$ together with $G(x)\subset C\cup D$. The paper extends this to locally Lipschitz barrier functions using Clarke generalized gradients, relaxes the flow inequality to $\langle\nabla B_i(x),\eta\rangle\le \rho(B_i(x))$ with $\rho$ a uniqueness function, and provides boundary-only sufficient conditions under a transversality assumption and a one-sided Lipschitz-like condition on the flow map. For contractivity, strict inequalities and a non-tangentiality condition force solutions from the boundary to enter the interior of $K$.

Load-bearing premise

The argument hinges on the assertion that infinitesimal inequalities checked just outside the set, or on its boundary under the extra regularity condition, rule out every possible way a solution could slip outside; if some direction of escape is not captured by the active barrier functions and the contingent cone, the certificate can fail.

Editorial extensions

If this is right

If the conditions hold, a safety region for a hybrid inclusion is certified without simulating or computing solutions, even when flows are set-valued and solutions may stop after finite hybrid time.
Multiple constraints can be handled componentwise; the active zero-level sets $M_i$ are checked separately, avoiding the need for a single smooth scalarization of an intersection of constraints.
The relaxed flow condition with uniqueness functions allows non-Lipschitz flows, such as those with $\rho(\omega)=\omega\log\omega$, to be covered by the pre-invariance certificate.
For contractive sets, the same machinery yields certificates that boundary states evolve inward, which is the ingredient behind set-induced Lyapunov functions and safety-plus-convergence designs.
The results apply directly to hybrid automata by encoding the mode as a discrete state, so mode-dependent safety sets can be certified with one barrier candidate per mode.

Reading between the lines

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

An extension the authors leave implicit is that the componentwise flow check could be combined with control synthesis, so a hybrid controller can enforce safety online by selecting flow inputs that satisfy the inequalities in (12).
A testable next step is to turn the sufficient conditions into an algorithmic search over polynomial barrier candidates using convex optimization; if a certificate is found, the theorems give a formal safety proof.
The boundary-only Theorem 2 relies on condition (21) being applied to a projection $y_1$ that is only argued to lie in either $\partial K_e\cap C$ or an exceptional set; a reader should check whether that projection is always admissible, since a gap there would restrict the theorem's validity to cases where the projection falls on $\partial(K\cap C)$.
Because the paper only proposes sufficient conditions, the examples showing failure when extra assumptions are dropped suggest that necessity would require different, likely cone-based, conditions; a systematic comparison of conservatism between Theorem 1 and Theorem 2 remains open.

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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

You should know two things about this paper. First, it is a real and mostly careful extension of barrier-function tools to hybrid inclusions, with multiple barriers and a uniqueness-function relaxation that are genuinely new. Second, the proof of Theorem 2, the advertised boundary-only result, has a gap that needs repair before the theorem is trustworthy. I also found it annoying that several proofs are punted to reference [41], which is this same preprint.

The good parts are substantial. Theorem 1 gives clean infinitesimal flow and jump conditions for forward pre-invariance using multiple C^1 barriers, avoiding the single-scalar-function assumption common in the literature. Proposition 1 relaxes the flow inequality using uniqueness functions in a way I have not seen before for hybrid inclusions, and the examples—thermostat, bouncing ball, the non-convex flow set in Example 6—are illustrative and honestly discussed. The contractivity section extends the known C-set/Minkowski-functional approach to general closed sets and gives barrier-based sufficient conditions that look right. Most of the proofs I checked are carefully constructed, especially Lemma 3 and the cone-characterization lemmas.

The soft spot is Theorem 2. The idea is to replace the neighborhood flow condition by a one-sided Lipschitz-like condition (21) plus boundary/Lyapunov-type checks. In the proof, the projection y1 used under case (a) can land on (U(∂Ke∩∂C)∩∂Ke)\C, i.e., on the boundary of C but outside C. That point is still in ∂(K∩C), so the stress-test's claim that it lies outside ∂(K∩C) is not correct. But the real problem is that F(y1) appears in the right-hand side of (21), and F is only assumed to be defined on C. If y1∉C, F(y1) may not exist, and the inclusion (21) is meaningless. The proof applies (21) with y=y1 anyway, so the differential inequality for δ1 is unsupported. This is fixable—for example, by assuming F is defined on a neighborhood of C, or by strengthening the projection argument to show y1∈C under the hypotheses—but it is not a cosmetic issue.

Secondary issue: Theorem 4, Lemma 1, Proposition 2, and Theorem 6 all say “proof in [41],” and [41] is the arXiv version of this same paper. That is circular as a submission practice and should not be accepted in peer review.

Who this is for: people working on safety and invariance for hybrid systems, especially control barrier functions for switched or impulsive dynamics. It deserves a serious referee, not a desk reject, but the review should ask for a repaired Theorem 2 and complete proofs in the manuscript.

Referee Report

1 major / 6 minor

Summary. The paper studies sufficient conditions for forward pre-invariance and pre-contractivity of closed sets for hybrid inclusions modeled as in (1). The set of interest is described by the intersection of zero-sublevel sets of a vector-valued barrier function candidate, and the results give infinitesimal conditions on the flow and jump maps that avoid computing solutions. Several variants are presented: C^1 barrier functions (Theorem 1), boundary-only flow checks under a one-sided Lipschitz condition on the flow map (Theorem 2), cone-based conditions (Theorem 3), locally Lipschitz barrier functions (Theorems 4 and 6), relaxed flow conditions using uniqueness functions (Proposition 1), and contractivity results for general closed sets and for C-sets. Examples include a thermostat, a bouncing ball, and a two-dimensional nonconvex example.

Significance. If the results hold, the paper extends barrier-function certificates from differential equations and hybrid automata to the broader setting of hybrid inclusions, addressing the complications of nonunique solutions and solutions that terminate prematurely. The explicit treatment of multiple barrier functions and the use of uniqueness functions to relax the flow inequalities are potentially valuable for safety verification in hybrid systems. The examples are illustrative and the overall structure is mostly careful. However, the main boundary-only relaxation, Theorem 2, currently contains a derivation gap that affects the advertised relaxation of Theorem 1, so the significance is conditional on repairing that gap.

major comments (1)

[Section 3.2, Theorem 2 (proof, conditions (21), (26), (27))] The proof applies the one-sided Lipschitz condition (21) to the point y1, defined as a projection of x(t,0) onto Ke. The text explicitly allows y1 to lie in (U(∂Ke∩∂C)∩∂Ke)\C, i.e., outside C. Since condition (21) is assumed only for y∈∂(K∩C), and y1∉C implies y1∉K∩C, the inequality used to pass from (26) to (27) is not available for such y1. This is a genuine derivation gap in the advertised boundary-only relaxation of Theorem 1. The proof must be repaired, for example by strengthening (21) to hold for all y∈Ke in the relevant neighborhood, or by providing a separate argument for the case y1∉C.

minor comments (6)

[Throughout (e.g., before Theorem 4, Proposition 2, Lemma 1, Theorem 6)] Several statements contain the sentence 'The proof is in [41]' where [41] is the same arXiv preprint; the proofs actually appear in the manuscript, so these sentences should be deleted or corrected to avoid a circular-reference appearance.
[References] References [27] and [29] are the same conference paper; they should be merged or renumbered to avoid duplication.
[Example 1] The definition of G(x) is written as 'G(x) :=[0,x 2]× [0,|x1|]', which appears malformed; it should be a set such as {0}×[0,|x_1|] or equivalent.
[Theorem 5] Condition (41) is stated as 'for each i ={1,2,...,m}' instead of 'for each i∈{1,2,...,m}'.
[Throughout] There are several typos, e.g., 'there exits' instead of 'there exists' in multiple places, and 'diﬀerent forward invariance' in the abstract should be 'different forward invariance'.
[Proof of Theorem 2] The projections y1 and y2 are not necessarily unique unless the sets are convex; the argument should clarify that any projection possessing the stated properties is selected, since the distance equality |x−y_i| = |x|_Ke or |x|_K∩C holds for any projection.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: main theorems are derived from standard viability and nonsmooth analysis; self-citations are pointers and not load-bearing.

full rationale

Walking the derivation chain, Theorem 1 proves forward pre-invariance by contradiction, integrating B_i along a hypothetical escaping flow and using condition (12) on the exit neighborhood; the jump conditions (13)-(14) handle escapes by jumps. This is a direct sufficient-condition argument, not an equivalence with the conclusion. Proposition 1 replaces the sign condition by rho(B_i(x)) and invokes the definition of a uniqueness function: the differential inequality obtained for B_i(x(t)) matches (19), so the comparison principle gives delta identically zero. Theorem 2 adapts the external proof of Redheffer [39], adding the one-sided Lipschitz condition (21); Theorem 3 is Aubin's external-cone condition [9]; Theorems 4-6 use Clarke gradients and the same cone machinery. No fitted parameter is renamed as a prediction, and no barrier candidate is defined in terms of the invariance it certifies. The only self-citations are [27,28] (preliminary conference versions) and [41] (the same arXiv preprint used as a proof pointer); the proofs actually appear in the text or rest on external references [9,14,31,39], so these self-citations are not load-bearing. The proof gap in Theorem 2 - applying (21) to a projection y1 that need not lie in d(K cap C) - is a correctness or regularity concern, not a circularity, and is therefore not scored as circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The paper's assumptions are regularity and transversality conditions on the hybrid inclusion data, not fitted values.

assumptions (5)

domain assumption Standing assumptions: F outer semicontinuous, locally bounded, nonempty convex images on C; G(x) nonempty on D; K closed.
Section 2.3 states these as the baseline regularity for all results.
domain assumption Barrier function candidate B is C^1 (Theorems 1,2,5, Corollaries) or locally Lipschitz (Theorems 4,6).
Needed for gradient or Clarke generalized gradient computations in the proofs.
ad hoc to paper Transversality condition (Assumption 1): for each x∈∂Ke∩C there exists v with ∇B_i(x)⊤v<0 for all active i.
Used in Theorem 2 and Lemma 3 to characterize the contingent cone of Ke at the boundary.
ad hoc to paper One-sided Lipschitz-like condition (21) with a uniqueness function ρ in Theorem 2.
This is the key regularity assumption enabling boundary-only flow conditions; it holds automatically for locally Lipschitz F, but not in general.
standard math Uniqueness functions and Osgood-type criteria from ODE theory (Definition 7, Remark 9).
Borrowed from Redheffer and Agarwal-Lakshmikantham; used in Proposition 1 and Theorem 2.

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Cite this review

Pith. "Pith review of Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions." pith.science (2026). https://pith.science/paper/LLUWM24A

@misc{pith2026190803980,
  author       = {Pith},
  title        = {Pith review of: Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LLUWM24A}},
  note         = {Machine review of arXiv:1908.03980}
}

read the original abstract

This paper studies set invariance and contractivity in hybrid systems modeled by hybrid inclusions using barrier functions. After introducing the notion of a multiple barrier functions, we investigate the tightest possible sufficient conditions to guarantee different forward invariance and contractivity notions of a closed set for hybrid systems with nonuniqueness of solutions and solutions terminating prematurely. More precisely, we consider forward (pre-)invariance of sets, which guarantees solutions to stay in a set, and (pre-)contractivity, which further requires solutions that reach the boundary of the set to evolve (continuously or discretely) towards its interior. Our conditions for forward invariance and contractivity involve infinitesimal conditions in terms of multiple barrier functions. Examples illustrate the results. Keywords: Forward invariance, contractivity, barrier functions, hybrid dynamical systems.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Characterizing Safety: Minimal Barrier Functions from Scalar Comparison Systems
eess.SY 2019-08 conditional novelty 7.0 of 10

Set invariance is characterized by minimal barrier functions, and the minimal comparison functions are fully characterized by four verifiable conditions.

Reference graph

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Moreover, let a sequence{tn}n∈N⊂ (0,t′ 2−t) such that tn→ 0

such that ˙x(t,j ) exists thus ˙x(t,j )∈F (x(t,j )). Moreover, let a sequence{tn}n∈N⊂ (0,t′ 2−t) such that tn→ 0. That is, for vn(t) := (x(tn,j )−x(t,j ))/tn, we have limnvn(t) = ˙x(t,j ) and at the same timex(t,j )+tnvn(t) =x(tn,j )∈C. Hence, using (3), we conclude that ˙x(t,...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [41]

Maghenem and R

M. Maghenem and R. G. Sanfelice. Suﬃcient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions. arXiv preprint:1908.03980, 2019

arXiv 1908

[2] [1]

Wieland and F

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

work page 2007

[3] [2]

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work page 1974

[4] [3]

Taly and A

A. Taly and A. Tiwari. Deductive veriﬁcation of continuous dynamical systems. In Proceedings of the LIPIcs-Leibniz International Proceedings in Informatics , volume 4. Schloss Dagstuhl-Leibniz- Zentrum f¨ ur Informatik, 2009

work page 2009

[5] [4]

S. Prajna. Optimization-based methods for nonlinear and hybrid systems veriﬁcation . PhD thesis, California Institute of Technology, 2005

work page 2005

[6] [5]

R. G. Sanfelice, R. Goebel, and A. R. Teel. Invariance principles for hybrid systems with connections to detectability and asymptotic stability. IEEE Transactions on Automatic Control , 52(12):2282– 2297, 2007

work page 2007

[7] [6]

R. G. Sanfelice and A. R. Teel. Asymptotic stability in hybrid systems via nested matrosov functions. IEEE Transactions on Automatic Control , 54(7):1569–1574, 2009

work page 2009

[8] [7]

G. S. Ladde and S. Leela. Analysis of invariant sets. Annali di Matematica Pura ed Applicata , 94 (1):283–289, 1972

work page 1972