Barrier Sufficiency in Safety Verification

• Barrier sufficiency is the use of certificate functions with differential, difference, or variational inequalities to ensure system safety and invariance without solving full reachability problems.
• It applies to both continuous and discrete systems through techniques like Lie derivatives and exponential control barrier functions, enabling safe stabilization and forward invariance.
• Recent advances incorporate stochastic, compositional, and parameterized barrier functions to reduce conservatism and certify safety under uncertainty and resource constraints.

Barrier sufficiency is the use of a barrier or barrier-like function as a certificate whose existence, together with prescribed differential, difference, generator, or variational inequalities, is enough to establish a target property such as forward invariance, safety, reach-avoid satisfaction, robustness, or a quantified resource margin. In most of the literature the result is one-sided—sufficiency without necessity—but recent stochastic verification work explicitly studies both directions: one 2024 contribution examines necessary and sufficient barrier-like conditions for infinite-horizon safety verification and reach-avoid verification of stochastic discrete-time systems, derived through a relaxation of Bellman equations, with a necessary and sufficient barrier-like condition for safety verification and two such conditions for reach-avoid verification under certain assumptions (Xue, 2024).

1. Core formulation

In its classical state-invariant form, barrier sufficiency is the implication

$$\text{certificate inequalities} \;\Longrightarrow\; \text{safety or invariance}.$$

For continuous systems, a canonical pattern is to find a scalar function whose sign separates initial and unsafe sets and whose Lie derivative is constrained so that the separating sign cannot be lost along trajectories. A general formulation appears in "Barrier Certificates Revisited", where analytic functions $\varphi$ and $\psi$ satisfy $\varphi(x)\le 0$ on the initial set, $\varphi(x)>0$ on the unsafe set, and

$\mathcal{L}_f \varphi(x)-\psi(\varphi(x))\le 0$

on the domain, together with a one-dimensional comparison property for $\dot\theta=\psi(\theta)$; these conditions imply safety (Dai et al., 2013).

In discrete time, the same logic appears in exponential control barrier form. For a safe set

$\mathcal{C}=\{x\in\mathbb{R}^n\mid h(x)\ge 0\},$

the discrete-time exponential control barrier condition requires a control satisfying

$\Delta h(x_k,u_k)+\lambda h(x_k)\ge 0,$

with $\lambda\in[0,1]$. Any feedback $\varphi$0, where $\varphi$1 is the corresponding admissible control set, renders $\varphi$2 forward invariant (Squires et al., 2021).

This establishes the basic meaning of barrier sufficiency: the barrier need not solve the full reachability problem explicitly; it only needs to satisfy a closed family of inequalities that is strong enough to force the desired property.

2. Deterministic invariance and safe stabilization

In safety-critical control, non-strict zeroing conditions remain the standard sufficiency mechanism. For a time-varying safe set

$\varphi$3

the inequality

$\varphi$4

is sufficient for forward invariance, hence for safety. At the same time, this non-strictness does not prevent trajectories from remaining on the boundary for arbitrarily long time intervals, so sufficiency for invariance is weaker than sufficiency for boundary liveness (Han et al., 17 Mar 2026).

Safe stabilization extends this idea by combining barrier inequalities with Lyapunov decrease. "Permissive Barrier Certificates for Safe Stabilization Using Sum-of-squares" constructs barrier certificates that explicitly maximize the region where the system can be stabilized without violating safety constraints. The barrier certified region is allowed to take any arbitrary shape and is proved to be strictly larger than safe regions generated with Lyapunov sublevel set based methods; the construction also unites a Lyapunov function with multiple barrier functions that might not be compatible with each other (Wang et al., 2018).

For disturbed affine nonlinear systems, reciprocal-resistance constructions strengthen sufficiency near the boundary. A reciprocal resistance-based barrier function introduces a term of the form

$\varphi$5

so that the associated RRBF, RRCBF, and HO-RRCBF inequalities are sufficient for forward invariance of $\varphi$6 or $\varphi$7, while also generating an inner buffer set $\varphi$8 where invariance is strongest (Wang et al., 25 Jul 2025). The same paper emphasizes that robustness is achieved without requiring explicit disturbance bounds in the design inequalities.

Barrier states theory reframes sufficiency at the system level. There the barrier value $\varphi$9 is embedded into an auxiliary dynamical state, and boundedness of the barrier value is equivalent to safety. The main theorem states that the original control system is safely stabilizable at the origin if and only if the safety embedded system is stabilizable at the origin, so stabilization of the augmented system becomes sufficient for safe stabilization of the original one (Almubarak et al., 2023).

3. Robustness, boundary liveness, and resource margins

Under uncertainty, barrier sufficiency often becomes parametric. Parameterized barrier functions introduce

$\psi$0

so that safety can be certified not only for $\psi$1 but for any superlevel set $\psi$2. In this framework, $\psi$3 quantifies safety degradation, $\psi$4 recovers the nominal safe set, and $\psi$5 quantifies conservativeness. The paper further shows that input-to-state safety is a special case of this parameterized construction (Alan et al., 2023).

Forward invariance alone is not enough when one also wants trajectories to leave boundary layers in finite time. A Matrosov-type auxiliary function framework addresses exactly this gap. If $\psi$6 is bounded by $\psi$7 on a forward-invariant compact set and satisfies

$\psi$8

throughout a boundary layer $\psi$9, then any continuous residence interval in that layer has uniformly bounded length,

$\varphi(x)\le 0$0

while forward invariance is preserved (Han et al., 17 Mar 2026). This makes barrier sufficiency strictly stronger than invariance-only certification.

Resource-aware formulations replace geometric safety by budget sufficiency. In energy-constrained robotics, the barrier

$\varphi(x)\le 0$1

compares remaining allocable energy with the estimated energy-to-go along a homing path. Within the resulting CBF-QP layer, the proved guarantee is that

$\varphi(x)\le 0$2

so full budget usage can occur only within the charging region (Fouad et al., 2023). This is barrier sufficiency in a viability sense: the state remains inside a set from which safe recovery is still feasible.

4. Stochastic safety and reach-avoid

In stochastic systems, barrier sufficiency is tied to Bellman relaxations, supermartingale or subsolution inequalities, and probability bounds. The 2024 infinite-horizon result is explicit about this structure: safety verification concerns the probability that the system, starting from a specified initial state, remains within a safe set always and exceeds a prescribed lower bound, while reach-avoid verification concerns the probability of eventually reaching a target set while remaining within the safe set until the first hit of the target. The paper formulates one necessary and sufficient barrier-like condition for safety verification and two necessary and sufficient barrier-like conditions for reach-avoid verification under certain assumptions (Xue, 2024).

Finite-time stochastic barrier sufficiency has also been refined by removing boundedness assumptions that previously restricted auxiliary functions on unbounded state spaces. For discrete-time systems, a single barrier-like function $\varphi(x)\le 0$3 satisfying

$\varphi(x)\le 0$4

on the safe set, together with a terminal-side inequality on $\varphi(x)\le 0$5, is sufficient to derive an upper bound on finite-time safety probabilities (Xue et al., 23 Sep 2025). For continuous-time systems, a function $\varphi(x)\le 0$6 satisfying

$\varphi(x)\le 0$7

on $\varphi(x)\le 0$8, together with boundary conditions on $\varphi(x)\le 0$9 and $\varphi(x)>0$0, is sufficient to derive lower bounds on finite-time reach-avoid probabilities (Xue et al., 23 Sep 2025). The removal of boundedness assumptions is important because it enlarges the admissible certificate class, especially on unbounded state spaces, and facilitates semidefinite programming with polynomial functions.

5. Compositional, geometric, and transition-based generalizations

Barrier sufficiency becomes compositional when a large system is verified through local certificates. For interconnected switched impulsive systems, the compositional construction uses local pseudo barrier functions $\varphi(x)>0$1 satisfying local sign conditions and coupled flow inequalities of the form

$\varphi(x)>0$2

together with local jump inequalities

$\varphi(x)>0$3

The paper presents sufficient conditions under which these local barrier functions, rather than a single global one, guarantee safety of the interconnected switched impulsive system (Bieker et al., 2024).

A geometric reinterpretation appears in control barrier corridors. Given a goal-parametrized feedback law $\varphi(x)>0$4, the control barrier corridor

$\varphi(x)>0$5

is the set of goals whose immediate closed-loop motion satisfies the CBF inequality. For convex barrier functions, and provided the control convergence rate matches the barrier decay rate, individual state safety extends locally over these control barrier corridors, yielding safely reachable persistent goal selection as the robot moves (Arslan et al., 6 Mar 2026). Here sufficiency is no longer only about the current state; it becomes a statement about safe local goal regions.

For temporal logic and recurrence properties, state invariants are insufficiently expressive. Closure certificates extend barrier certificates from state invariants to transition invariants. A closure certificate $\varphi(x)>0$6, or in the product with an automaton $\varphi(x)>0$7, over-approximates the transitive closure of the transition relation and combines this with a decrease condition by at least $\varphi(x)>0$8 on recurrent or accepting regions. This is sufficient for safety, persistence, and LTL/ω-regular verification, and the paper gives both SOS- and SMT-based characterizations for automated synthesis (Murali et al., 2023).

6. Beyond control: convex analysis, finite-extensibility, and literal barriers

Barrier sufficiency also appears outside control design. In subdifferential theory, barrier functions are used as a sufficient technical device to handle lower semicontinuous functions in Banach spaces. The construction uses the Minkowski functional $\varphi(x)>0$9 of a bounded open convex neighborhood $\mathcal{L}_f \varphi(x)-\psi(\varphi(x))\le 0$0 and the barrier

$\mathcal{L}_f \varphi(x)-\psi(\varphi(x))\le 0$1

which blows up at $\mathcal{L}_f \varphi(x)-\psi(\varphi(x))\le 0$2. This barrier framework is sufficient to prove a barrier-based version of the Correa–Jofré–Thibault theorem: if a feasible subdifferential is monotone, then the underlying proper lower semicontinuous function is convex (Ivanov et al., 2019).

In numerical rheology, barrier sufficiency becomes an admissibility-and-entropy question. For diffusive FENE flows, positive definiteness of the conformation tensor is insufficient because the model also requires the finite-extensibility constraint $\mathcal{L}_f \varphi(x)-\psi(\varphi(x))\le 0$3. The relevant free energy contains a trace barrier,

$\mathcal{L}_f \varphi(x)-\psi(\varphi(x))\le 0$4

and the paper develops a barrier-preserving entropy-compatible discretization that proves finite-extensibility preservation at entropy quadrature points, existence and bisection computability of the maximal entropy-admissible reconstruction parameter, a fully discrete free-energy inequality with relaxation and molecular-diffusion barrier dissipation, a quantitative AP stress closure, and a fixed-discretization Newtonian limit (Peng, 26 May 2026). In this setting, positivity alone is not sufficient; compatibility with the barrier free energy is also required.

A different, literal use of the term appears in opinion dynamics with mobile agents separated by a physical barrier. There the control parameter is the opening size $\mathcal{L}_f \varphi(x)-\psi(\varphi(x))\le 0$5, and the paper defines the critical opening size $\mathcal{L}_f \varphi(x)-\psi(\varphi(x))\le 0$6 as the largest value of $\mathcal{L}_f \varphi(x)-\psi(\varphi(x))\le 0$7 that still gives $\mathcal{L}_f \varphi(x)-\psi(\varphi(x))\le 0$8. Accordingly,

$\mathcal{L}_f \varphi(x)-\psi(\varphi(x))\le 0$9

means the barrier is sufficient to maintain stalemate, while $\dot\theta=\psi(\theta)$0 makes the barrier insufficient and consensus becomes likely. On the consensus side, the relaxation time diverges as

$\dot\theta=\psi(\theta)$1

when $\dot\theta=\psi(\theta)$2 (Xiao et al., 6 May 2025). This usage is conceptually distinct from certificate-based barriers, but it preserves the same threshold logic: a barrier is sufficient when it is strong enough to force a qualitative property.

7. Conservatism, limitations, and ongoing directions

A recurring limitation is that sufficiency is rarely necessity. Model-based barrier functions built from specific backup maneuvers can be needlessly restrictive: in fixed-wing collision avoidance, the paper constructs cases in which using a barrier function makes two aircraft come closer to colliding than if there were no barrier function at all, and cases in which the barrier function labels the system as unsafe even when the vehicles start arbitrarily far apart (Squires et al., 2021). This is not a failure of soundness; it is a manifestation of conservatism.

Expressiveness also trades off against synthesis complexity. "Barrier Certificates Revisited" explicitly notes that a stronger condition on barrier certificates means that less expressive barrier certificates can be synthesized, whereas synthesizing more expressive barrier certificates often means higher complexity. The same work therefore develops weaker generalized and combined barrier conditions while still keeping convexity, and it emphasizes symbolic checking to avoid unsoundness caused by numeric error in SDP-based synthesis (Dai et al., 2013).

Across stochastic verification, recurrence reasoning, and boundary liveness, recent work has targeted exactly these conservatism gaps. Earlier stochastic barrier methods often relied on bounded auxiliary functions, which limited applicability on unbounded state spaces; refined finite-time conditions remove this assumption (Xue et al., 23 Sep 2025). State-triplet barrier tactics for ω-regular refutation are conservative because recurrence requires reasoning about the well-foundedness of the transitive closure of the transition relation, motivating closure certificates (Murali et al., 2023). Non-strict barrier conditions guarantee forward invariance but not escape from boundary layers, motivating Matrosov-type auxiliary-function refinements (Han et al., 17 Mar 2026). This suggests a broad trend: barrier sufficiency is evolving from bare invariance certificates toward richer certificate architectures that preserve soundness while reducing conservatism, enlarging admissible template classes, and, in some cases, integrating necessity with sufficiency (Xue, 2024).