Zeroing Barrier Functions in Control Systems

Updated 7 January 2026

Zeroing Barrier Functions (ZBFs) are mathematical tools that enforce forward invariance by leveraging Lyapunov-like inequalities for safety in dynamical systems.
They extend to high-order, stochastic, and manifold settings, incorporating robust and data-driven methods to handle uncertainties and disturbances.
ZBFs underpin practical control synthesis via quadratic programs, ensuring safety in applications ranging from automotive safety to aerospace and robotics.

Zeroing Barrier Functions (ZBFs) provide a rigorous framework for certifying and enforcing forward invariance of safety-critical sets in control and dynamical systems. Originally formulated to express geometric safety conditions via Lyapunov-like inequalities and extended to data-driven, robust, high-order, and stochastic settings, ZBFs underpin a wide family of control barrier function methodologies for both deterministic and stochastic systems. The theory enables the synthesis of feedback controllers guaranteeing safety even in the presence of model uncertainties, input constraints, adversarial perturbations, and generalizes naturally to complex domains such as manifolds and Lie groups.

1. Mathematical Foundations of Zeroing Barrier Functions

Let $h: \mathbb{R}^n \to \mathbb{R}$ be a continuously differentiable function, and define the "safe set" $\mathcal{C} = \{x \in \mathbb{R}^n : h(x) \ge 0\}$ with boundary $\partial\mathcal{C} = \{x : h(x) = 0\}$ . An extended class- $\mathcal{K}$ function is any strictly increasing $\alpha: (-b, a) \rightarrow \mathbb{R}$ with $\alpha(0)=0$ . The ZBF condition for an autonomous system $\dot{x} = f(x)$ is

$L_f h(x) \ge -\alpha(h(x)), \qquad \forall x \in D \supseteq \mathcal{C}$

where $L_f h(x) = \nabla h(x)^\top f(x)$ .

If $h$ satisfies this inequality, then by Nagumo's theorem, $\mathcal{C}$ is forward invariant: any trajectory starting in $\mathcal{C}$ remains in $\mathcal{C}$ for all future time. On the boundary $\partial\mathcal{C}$ , $h(x)=0$ and so $L_f h(x) \ge 0$ ensures the vector field points into or is tangent to the set. This forward invariance property extends to control-affine systems $\dot{x} = f(x) + g(x)u$ by requiring that

$\sup_{u \in U} \left[ L_f h(x) + L_g h(x)u + \alpha(h(x)) \right] \ge 0, \qquad \forall x \in D$

where $L_g h(x) = \nabla h(x)^\top g(x)$ . Enforcing the affine constraint $L_f h(x) + L_g h(x)u + \alpha(h(x)) \ge 0$ in real time yields a zeroing control barrier function (ZCBF) (Ames et al., 2016).

High-order zeroing barrier functions (HO-ZBFs) address constraints of relative degree $r>1$ by defining a cascade of auxiliary functions $\psi_i$ through differentiated and "damped" barrier constraints, enforcing invariance of nested safe sets $\bigcap_{i=1}^r \{x : \psi_{i-1}(x) \ge 0\}$ (Tan et al., 2021).

2. Relationships to Reciprocal Barrier Functions and Robust Variants

Reciprocal barrier functions (RBFs) are defined on $\mathrm{Int}(\mathcal{C})$ via $B(x)$ such that $1/\alpha_1(h(x)) \leq B(x) \leq 1/\alpha_2(h(x))$ and $L_f B(x) \leq \alpha_3(h(x))$ for suitable extended class- $\mathcal{K}$ functions $\alpha_i$ . Forward invariance of $\mathrm{Int}(\mathcal{C})$ is obtained if an RBF exists. On compact sets, the existence of a ZBF is also necessary for invariance, and under a contractivity condition ( $L_f h > 0$ on $\partial\mathcal{C}$ ), the RBF and ZBF frameworks are equivalent for certifying invariance (Ames et al., 2016).

Robust ZBF extensions handle modeling errors, bounded disturbances, or adversarial uncertainties. For an affine-in-control system with additive disturbance $d(t,x)$ and $\|d(t,x)\| \leq \delta$ , the robustified ZCBF derivative includes additional terms, and adapted constraints $L_f h(x) + L_g h(x) u + \alpha(h(x)) + l_h\delta \geq 0$ (with $l_h$ the Lipschitz constant of $\nabla h$ ) are enforced, certifying invariance under worst-case disturbances (Garg et al., 2022, Wang et al., 25 Jul 2025).

Reciprocal resistance-based barrier functions (RRBFs) further augment the ZBF inequality by introducing a term $\beta(1/h(x))$ , generating a buffer region that dominates near the set boundary and absorbs disturbance effects, guaranteeing invariance with a tunable safety margin (Wang et al., 25 Jul 2025).

3. Control Synthesis: Quadratic Programs and Mixed Objectives

Zeroing barrier function constraints admit natural integration with control Lyapunov function (CLF) objectives via real-time quadratic programs (QPs). The pointwise problem is formulated as

$\begin{aligned} & \min_{u,\,\delta} \quad \frac{1}{2} \begin{bmatrix} u \ \delta \end{bmatrix}^\top H(x)\begin{bmatrix} u \ \delta \end{bmatrix} + F(x)^\top \begin{bmatrix} u \ \delta \end{bmatrix} \ & \text{subject to:} \ & \qquad L_f V(x) + L_g V(x) u + c_3 V(x) - \delta \leq 0, \ & \qquad L_f h(x) + L_g h(x) u + \alpha(h(x)) \geq 0, \ & \qquad \delta \geq 0,\quad u \in U \end{aligned}$

where the CLF constraint is softened by slack $\delta$ and the ZBF constraint is imposed as a hard safety constraint (Ames et al., 2016). Under mild regularity assumptions, the resulting optimal control is locally Lipschitz in $x$ .

Controller blending and mixed-initiative frameworks have been developed to combine multiple Type-II ZCBFs, performance objectives, and input bounds, smoothly switching between nominal and safety-inducing behavior while retaining local Lipschitz continuity and enforcing all safety constraints (Cortez et al., 2022). For high-relative-degree constraints, tractable QP formulations rely on recasting the safety condition using a suitable HO-ZBF, potentially with state-dependent buffer corrections or using "braking" solutions (Tan et al., 2021, Breeden et al., 2021).

4. Extensions: Stochastic, Manifold, and Data-driven ZBFs

Stochastic Zeroing Barrier Functions

For Itô SDEs $dX_t = b(X_t) dt + \sum_{k=1}^m \sigma_k(X_t) dW_t^k$ , the infinitesimal generator $Lh(x)$ extends the classical Lie derivative, including both drift and diffusion terms: $Lh(x) = \nabla h(x)^\top b(x) + \frac{1}{2} \sum_{k=1}^m \|\sigma_k(x)^\top \nabla h(x)\|^2.$ A stochastic zeroing barrier function (SZBF) requires that

$L h(x) \ge -\alpha(h(x)), \qquad \sum_{k=1}^m \langle \nabla h(x), \sigma_k(x) \rangle = 0,$

with the orthogonality condition ensuring absence of a martingale term in $dh(X_t)$ , delivering almost sure forward invariance of the safe set (Tamba et al., 2020). More general stochastic ZCBFs incorporate probabilistic bounds on invariant set violation, with the control input appearing in the controlled drift term of the generator (Nishimura et al., 2022).

Zeroing Barrier Functions on Manifolds

ZBFs and ZCBFs generalize to smooth manifolds $M$ and structured domains such as matrix Lie groups $G$ , where the data-driven or geometric safe set is characterized as $S = \{x \in M : h(x) \geq 0\}$ . Energy-augmented zeroing CBFs encode kinematic and kinetic constraints for mechanical systems, notably in safety-critical rigid-body planning, yielding barrier constraints involving both configuration and energy variables, and guaranteeing invariance on $G \times \mathfrak{g}$ by enforcing affine input constraints inside QPs (Letti et al., 8 Dec 2025).

Learning and Data-driven Synthesis of ZBFs

When the explicit safe set is unknown, ZBFs can be synthesized from safe/unsafe data via supervised learning. Radial basis function (RBF) neural networks and polynomial feature lifts allow construction of a smooth $h_\alpha(x)$ separating safe and unsafe regions, with constraints on the learned function's value at sampled points. This translates into a tractable linear program whose solution $h_\alpha(x)$ induces a zeroing barrier function—after optional rescaling—for use in standard CBF-QP controllers (Abuaish et al., 2022). Human demonstration, kinesthetic teaching, and clustering further enable data-driven identification of polyhedral safe sets using affine barrier functions (Saveriano et al., 2020).

5. Practical Algorithms and Applications

Practical safety-critical controllers leveraging ZBFs are realized by solving, at each time step, a convex quadratic program that enforces the ZBF constraint, optionally unifying multiple safety barriers and performance objectives. The framework supports input constraints, actuator saturation, and robust or attack-resilient safety via buffer terms, robustified Lie derivatives, and two-player min-max verifications for sampled domains (Ames et al., 2016, Garg et al., 2022, Breeden et al., 2021).

Applications span adaptive cruise control (keeping time-headway), lane keeping (lateral-yaw models with actuator limits), mobile-robot navigation (partitioning workspace using neural ZBFs), robotic workspace limitation (data-driven, incrementally learned polyhedral ZBFs), spacecraft collision avoidance (high relative degree, polyhedral asteroid boundary), and resilience to cyber-physical attacks (hybrid ZCBF-based recovery) (Ames et al., 2016, Abuaish et al., 2022, Saveriano et al., 2020, Breeden et al., 2021, Garg et al., 2022).

Simulation and experimental studies consistently confirm that ZBF-based QP controllers maintain forward invariance of safety sets under model perturbations, exogenous disturbances, and actuator attacks, with safety always prevailing over softened performance objectives.

6. Limitations, Variants, and Research Directions

The main limitations are associated with feasibility and conservatism, especially in high-relative-degree constraints, tight or coupled actuator constraints, or high-dimensional systems. For stochastic systems, strong safety guarantees often require unbounded control near the boundary—a practical limitation remedied probabilistically by stochastic ZCBFs, at the expense of providing only high-probability, rather than almost sure, invariance (Nishimura et al., 2022, Tamba et al., 2020).

Type-II ZCBFs relax the strictness of the classical ZBF by enforcing safety constraints only in an annulus around the set, with $\alpha(h) \leq 0$ for $h<0$ , making them suitable for non-compact, passivity-based, or robust scenarios. Reciprocal resistance-based ZBFs leverage a buffer term that naturally adapts to disturbance magnitude and does not require explicit bound knowledge, which is advantageous for time-varying or unknown uncertainties (Wang et al., 25 Jul 2025, Cortez et al., 2022).

Research challenges remain in extending ZBF theory to hybrid, sampled-data, and high-dimensional stochastic systems, developing tractable controllers in high dimensionality, and achieving robustness under intermittent or adversarial disturbances. Open questions also include compositionality, decentralized implementation, and data-efficient learning of ZBFs for unknown or dynamic environments.

Summary Table: Key ZBF Classes and Guarantees

Class	Core Condition	Invariance Guarantee
Classical ZBF	$L_f h \ge -\alpha(h)$	Forward invariance
Zeroing CBF (control)	$L_f h + L_g h\,u + \alpha(h)\ge 0$	Forward invariance
High-Order ZBF / CBF	$\psi_r = L_f^r h + \cdots + \alpha_r(\psi_{r-1})\ge 0$	Nested set invariance
Stochastic ZBF	$Lh \ge -\alpha(h)$ , $\nabla h \perp \sigma_k$	a.s. invariance
Robust / RRBF	$L_f h + \alpha(h)\ge \beta(1/h) + D$	Disturbance-invariant
Data-driven (learned) ZBF	Trained $h_\alpha(x)$ from safe/unsafe samples	Empirical invariance