Optimal-Decay Control Barrier Functions (OD-CBFs)

• Optimal-Decay Control Barrier Functions (OD-CBFs) are safety frameworks that dynamically scale the decay term to enforce constraints in control-affine systems.
• They reformulate standard barrier conditions by optimizing a decay multiplier online, thereby addressing infeasibility caused by actuator bounds and other constraints.
• Higher-order variants like OD-HOCBF and OD-ReCBF extend the approach to systems with vanishing relative degree, enhancing safety and control invariance in complex scenarios.

Searching arXiv for recent OD-CBF papers and adjacent CBF-decay work. arxiv_search query: "Optimal-Decay Control Barrier Functions control barrier function decay feasibility higher-order" Optimal-Decay Control Barrier Functions (OD-CBFs) are control barrier function formulations in which the decay term in the barrier inequality is not fixed a priori, but is scaled online by an optimized nonnegative factor. For a control-affine system $\dot x=f(x)+g(x)u$ and a candidate safe set $\mathcal C=\{x:h(x)\ge 0\}$, standard zeroing CBFs enforce $L_f h(x)+L_g h(x)u\ge -\alpha(h(x))$, whereas OD-CBFs replace this with $L_f h(x)+L_g h(x)u\ge -\omega \alpha(h(x))$, where $\omega$ is selected by the safety filter itself. In the named OD-CBF formulation, this modification is introduced to address infeasibility under additional constraints, especially input limits, and later formalized as a broader framework with results on validity, control invariance, forward invariance, feasibility, Lipschitz continuity, closed-form controllers, and higher-order extensions (Zeng et al., 2021, Ong et al., 17 Jul 2025).

1. Standard barrier framework and OD-CBF definition

OD-CBFs are built on the standard control-affine nonlinear system

$\dot x=f(x)+g(x)u,$

with a safe set expressed as a zero-superlevel set

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

In the zeroing-CBF framework, safety is enforced through the existence of an extended class-$\mathcal K$ or class-$\mathcal K^e$ function $\alpha$ such that

$\mathcal C=\{x:h(x)\ge 0\}$0

or, in controller form,

$\mathcal C=\{x:h(x)\ge 0\}$1

The OD-CBF modification replaces the fixed decay law by

$\mathcal C=\{x:h(x)\ge 0\}$2

with $\mathcal C=\{x:h(x)\ge 0\}$3 determined online by the safety filter (Zeng et al., 2021, Ong et al., 17 Jul 2025).

The 2021 formulation presents OD-CBFs as an online optimization over the decay-rate multiplier $\mathcal C=\{x:h(x)\ge 0\}$4 in order to guarantee point-wise feasibility when the state lies inside the safe set (Zeng et al., 2021). The 2025 theoretical treatment defines an OD-CBF through the condition

$\mathcal C=\{x:h(x)\ge 0\}$5

and interprets the method as formalizing the choice of $\mathcal C=\{x:h(x)\ge 0\}$6 by allowing the safety filter to choose the effective decay scaling online (Ong et al., 17 Jul 2025).

A central structural feature is that the OD scaling acts only away from the boundary. Because $\mathcal C=\{x:h(x)\ge 0\}$7, the OD-CBF condition reduces on $\mathcal C=\{x:h(x)\ge 0\}$8 to $\mathcal C=\{x:h(x)\ge 0\}$9. This preserves the usual Nagumo-type tangency logic at the boundary while modifying the admissible barrier dynamics in the interior. This suggests that OD-CBFs are not a replacement for boundary invariance conditions, but a mechanism for changing how aggressively safety is enforced away from the boundary.

2. Optimization-based safety filters and point-wise feasibility

The main practical motivation for OD-CBFs is the infeasibility of standard CBF-QPs and CLF-CBF-QPs once additional constraints, especially actuator bounds, are imposed. In the standard CBF-QP, one solves

$L_f h(x)+L_g h(x)u\ge -\alpha(h(x))$0

subject to the CBF constraint and admissible-input constraints $L_f h(x)+L_g h(x)u\ge -\alpha(h(x))$1. In the CLF-CBF-QP, a CLF relaxation variable $L_f h(x)+L_g h(x)u\ge -\alpha(h(x))$2 softens the stabilization objective, but the CBF inequality remains hard. If the CBF-admissible control set and the input-admissible set do not intersect, the QP is infeasible (Zeng et al., 2021).

The OD-CBF-QP augments the decision variables with $L_f h(x)+L_g h(x)u\ge -\alpha(h(x))$3. In the 2021 formulation, the basic problem is

$L_f h(x)+L_g h(x)u\ge -\alpha(h(x))$4

subject to

$L_f h(x)+L_g h(x)u\ge -\alpha(h(x))$5

and the CLF-CBF version analogously adds $L_f h(x)+L_g h(x)u\ge -\alpha(h(x))$6 and the relaxed CLF constraint (Zeng et al., 2021). The resulting mechanism shifts the CBF half-space online until it intersects the admissible input set.

A key theorem states that when $L_f h(x)+L_g h(x)u\ge -\alpha(h(x))$7 is convex, the OD-CBF-QP and OD-CLF-CBF-QP are point-wise feasible for any $L_f h(x)+L_g h(x)u\ge -\alpha(h(x))$8 with $L_f h(x)+L_g h(x)u\ge -\alpha(h(x))$9 (Zeng et al., 2021). The proof uses the fact that inside the safe set one has $L_f h(x)+L_g h(x)u\ge -\omega \alpha(h(x))$0, so the term $L_f h(x)+L_g h(x)u\ge -\omega \alpha(h(x))$1 can be shifted over $L_f h(x)+L_g h(x)u\ge -\omega \alpha(h(x))$2 by varying $L_f h(x)+L_g h(x)u\ge -\omega \alpha(h(x))$3. The same paper is explicit that point-wise feasibility is weaker than persistent feasibility or forward invariance. It guarantees solvability of the optimization problem at a given interior state, not invariance of the full safe set under all trajectories.

The later theoretical treatment studies an OD-CBF-QP with unbounded inputs,

$L_f h(x)+L_g h(x)u\ge -\omega \alpha(h(x))$4

subject to

$L_f h(x)+L_g h(x)u\ge -\omega \alpha(h(x))$5

and proves feasibility on $L_f h(x)+L_g h(x)u\ge -\omega \alpha(h(x))$6, closed-form expressions for both $L_f h(x)+L_g h(x)u\ge -\omega \alpha(h(x))$7 and $L_f h(x)+L_g h(x)u\ge -\omega \alpha(h(x))$8, and local Lipschitz continuity under local Lipschitz assumptions on $L_f h(x)+L_g h(x)u\ge -\omega \alpha(h(x))$9, $\omega$0, and $\omega$1 (Ong et al., 17 Jul 2025). In that setting, the improved regularity comes from replacing the standard denominator $\omega$2 by a term of the form $\omega$3, which only vanishes on a smaller critical set.

3. Validity conditions, forward invariance, and control invariance

A major theoretical contribution of the 2025 analysis is an explicit characterization of OD-CBF validity. For ordinary CBFs, finding a compatible $\omega$4 can be difficult. For OD-CBFs, the paper proves that $\omega$5 is an OD-CBF on $\omega$6 if and only if

$\omega$7

This removes the need to search for a globally compatible fixed decay law when checking validity and reduces the analysis to the boundary points where control authority through $\omega$8 vanishes (Ong et al., 17 Jul 2025).

The same paper establishes a converse result for autonomous systems using optimal-decay barrier functions (OD-BFs): under local Lipschitzness of the vector field and regularity of the boundary, forward invariance of $\omega$9 is equivalent to the existence of an OD-BF inequality of the form

$\dot x=f(x)+g(x)u,$0

This links the OD formalism directly to Nagumo’s theorem and shows that the OD scaling can be interpreted as a way of parameterizing all forward-invariant barrier dynamics compatible with the boundary condition (Ong et al., 17 Jul 2025).

For controlled systems, the forward-invariance statement is sufficient rather than converse: if $\dot x=f(x)+g(x)u,$1 is an OD-CBF and a locally Lipschitz controller $\dot x=f(x)+g(x)u,$2 satisfies

$\dot x=f(x)+g(x)u,$3

on $\dot x=f(x)+g(x)u,$4, then $\dot x=f(x)+g(x)u,$5 is forward invariant (Ong et al., 17 Jul 2025). A corollary is that $\dot x=f(x)+g(x)u,$6 is control invariant, and if $\dot x=f(x)+g(x)u,$7 for a given safety constraint set $\dot x=f(x)+g(x)u,$8, then $\dot x=f(x)+g(x)u,$9 is a safe set.

Under additional assumptions, OD-CBF safety filters can also act as recovery mechanisms near the safe set. Specifically, if there exists a neighborhood $\mathcal C=\{x\in\mathbb R^n:h(x)\ge 0\}.$0 such that $\mathcal C=\{x\in\mathbb R^n:h(x)\ge 0\}.$1 for all $\mathcal C=\{x\in\mathbb R^n:h(x)\ge 0\}.$2, then the OD-CBF-QP is feasible on $\mathcal C=\{x\in\mathbb R^n:h(x)\ge 0\}.$3, and if $\mathcal C=\{x\in\mathbb R^n:h(x)\ge 0\}.$4 is compact and $\mathcal C=\{x\in\mathbb R^n:h(x)\ge 0\}.$5, the resulting controller makes $\mathcal C=\{x\in\mathbb R^n:h(x)\ge 0\}.$6 asymptotically stable (Ong et al., 17 Jul 2025). This suggests a local safe-set stabilization interpretation in addition to forward invariance.

4. Higher-order constraints, vanishing relative degree, and analytically derived decay laws

OD-CBFs are particularly relevant when safety constraints have relative degree greater than one or when relative degree vanishes on subsets of the state space. The 2025 OD-CBF theory extends existing higher-order constructions by introducing OD-HOCBF and OD-ReCBF variants (Ong et al., 17 Jul 2025). For a relative-degree-2 constraint $\mathcal C=\{x\in\mathbb R^n:h(x)\ge 0\}.$7, the OD-HOCBF construction uses

$\mathcal C=\{x\in\mathbb R^n:h(x)\ge 0\}.$8

A sufficient condition for $\mathcal C=\{x\in\mathbb R^n:h(x)\ge 0\}.$9 to be an OD-HOCBF is that

$\mathcal K$0

on the relevant subset where $\mathcal K$1. Under an additional boundary-regularity condition on $\mathcal K$2, the resulting set $\mathcal K$3 is safe (Ong et al., 17 Jul 2025).

The same paper develops an OD-ReCBF construction in which a rectified term is applied to the higher-order barrier candidate,

$\mathcal K$4

and proves OD-CBF validity under a weaker implication involving $\mathcal K$5, $\mathcal K$6, and $\mathcal K$7 (Ong et al., 17 Jul 2025). In the reported simulations, OD-HOCBF and OD-ReCBF are both demonstrated on a satellite control problem, with OD-HOCBF producing a smoother and less aggressive control signal.

A closely related analytical line does not use the OD-CBF name but is strongly aligned with its feasibility-maximizing motivation. For first- and second-order constraints under bounded control, an optimal barrier law is derived to maximize the feasible action space, and for relative-degree-2 constraints the resulting control barrier function is

$\mathcal K$8

That law is shown to make the induced safe set equal to the largest recursively feasible set under the control bounds, and the paper interprets the construction as encoding a dynamical motion primitive with an implicit model for future trajectory evolution, including time-varying components (Beaver, 2024). Although not called an OD-CBF formulation, it exemplifies the same principle of deriving a nonlinear decay law from bounded recoverability rather than imposing $\mathcal K$9 by hand.

5. Related approaches to decay design, prediction, and adaptation

OD-CBFs sit within a broader family of methods that treat the class-$\mathcal K^e$0 term, barrier evolution, or horizon-awareness as design variables rather than fixed ingredients. These methods overlap in motivation but differ in formalism.

Approach	Representative paper	Relation to OD-CBFs
Rate-tunable CBFs	(Garg et al., 2023)	Online adaptation of class-$\mathcal K^e$1 parameters
Predictive CBF synthesis	(Wiltz et al., 22 Apr 2025)	Derives a ZCBF and explicitly characterizes admissible $\mathcal K^e$2 from a finite-horizon OCP
Learning-based adaptive decay	(Chriat et al., 2023)	Learns $\mathcal K^e$3 in $\mathcal K^e$4 with multi-step training
Data-driven adaptive-decay IO-DCBF	(Bajelani et al., 24 Feb 2025)	Optimizes discrete-time decay $\mathcal K^e$5 online in an input-output safety filter

Rate-Tunable CBFs (RT-CBFs) treat the coefficients inside linear class-$\mathcal K^e$6 functions $\mathcal K^e$7 as dynamic variables and adapt them online to make the controller less or more conservative without jeopardizing safety (Garg et al., 2023). In the same tutorial, Bird’s Eye CBFs (BECBFs) address the “pointwise-only optimal character” of standard CBF-QPs by building a barrier from worst-case predicted safety along a nominal future trajectory. RT-CBFs are the closest in-paper analogue to decay-rate tuning, while BECBFs address the same myopia critique by predictive barrier construction rather than decay scaling (Garg et al., 2023).

Predictive synthesis methods push the design question even further upstream. A finite-horizon optimal-control construction inspired by Hamilton–Jacobi reachability synthesizes a zeroing CBF from an OCP, explicitly characterizes the associated extended class-$\mathcal K^e$8 function $\mathcal K^e$9, and allows $\alpha$0 to be chosen as a design parameter in controller synthesis (Wiltz et al., 22 Apr 2025). The resulting barrier can account for time-varying constraints without recomputation when those variations are encoded predictively, and its values can be computed pointwise without solving a PDE over the entire state space (Wiltz et al., 22 Apr 2025). This suggests a predictive interpretation of decay-law design that is closely related to OD-CBF motivations, even though the method is not named as such.

Learning-based and data-driven variants replace analytical derivation by adaptation from trajectories or measurements. Adaptive Multi-step CBFs (AM-CBFs) parameterize the class-$\alpha$1 function $\alpha$2 by a neural network, train it jointly with a reinforcement learning policy, and use a multi-step training/single-step execution paradigm so that the runtime safety filter remains a standard convex QP (Chriat et al., 2023). Data-driven input-output CBFs for discrete-time LTI systems derive a barrier from the maximal invariant set in an input-output lifted state and then optimize the decay parameter $\alpha$3 online in the safety filter

$\alpha$4

thereby trading decay rate against feasibility and recursive feasibility (Bajelani et al., 24 Feb 2025).

HJB-based safe optimal control is another adjacent line. One paper embeds safety into an infinite-horizon value function through a barrier-like running cost rather than optimizing a decay parameter (Cohen et al., 2020), while another solves a constrained HJB problem with exact CBF constraints and derives a closed-form optimal safe controller via KKT conditions (Almubarak et al., 2021). These methods support the same critique of myopic one-step filtering, but the optimized object is the value function or Hamiltonian controller rather than the barrier decay law.

6. Applications, misconceptions, and limitations

The named OD-CBF formulation is demonstrated on adaptive cruise control, where the standard CLF-CBF-QP becomes infeasible for sufficiently high initial speed because the barrier constraint conflicts with acceleration bounds, while the OD-CLF-CBF-QP remains feasible by adjusting $\alpha$5 (Zeng et al., 2021). The later OD-CBF theory demonstrates higher-order variants on a satellite control problem involving a radial safety annulus, with OD-HOCBF and OD-ReCBF both maintaining safety despite vanishing-relative-degree issues (Ong et al., 17 Jul 2025). Closely related methods extend the same design philosophy to first- and second-order Dubins car systems (Chriat et al., 2023), unknown time-delay systems in an input-output formulation (Bajelani et al., 24 Feb 2025), and predictive handling of time-varying constraints (Wiltz et al., 22 Apr 2025).

A common misconception is that OD-CBFs solve all feasibility and invariance issues once $\alpha$6 is optimized. The 2021 paper is explicit that point-wise feasibility is not persistent feasibility or forward invariance of the full safe set (Zeng et al., 2021). Another misconception is that OD-CBFs are merely slack-variable relaxations of safety constraints. In the named formulation, the safety inequality remains structurally a barrier inequality; the change is that the decay multiplier is optimized online rather than fixed by the designer (Zeng et al., 2021). A further source of confusion is the tendency to group all tunable or predictive barrier methods under the OD-CBF label. Rate-tunable CBFs, predictive finite-horizon synthesis, adaptive learning of $\alpha$7, and exact-time barrier tracking all target related limitations, but they optimize different objects: class-$\alpha$8 coefficients, value-function-induced barriers, learned decay maps, or reference barrier trajectories rather than the specific OD-CBF multiplier $\alpha$9 (Garg et al., 2023, Wiltz et al., 22 Apr 2025, Chen et al., 19 Mar 2026).

The main technical limitations remain tied to model assumptions and constraint classes. The 2025 OD-CBF theory works primarily with unbounded inputs and identifies bounded-input extensions as future work (Ong et al., 17 Jul 2025). The 2021 feasibility theorem applies only on the interior of the safe set; at the boundary $\mathcal C=\{x:h(x)\ge 0\}$00, $\mathcal C=\{x:h(x)\ge 0\}$01, so optimizing $\mathcal C=\{x:h(x)\ge 0\}$02 does not change the barrier condition there (Zeng et al., 2021). Higher-order and predictive variants can improve feasibility and reduce conservatism, but they introduce additional structure: finite-horizon OCPs, learned function approximators, lifted input-output representations, or auxiliary controller dynamics (Wiltz et al., 22 Apr 2025, Chriat et al., 2023, Bajelani et al., 24 Feb 2025, Rabiee et al., 2024).

Within the broader CBF literature, OD-CBFs therefore occupy a specific position: they preserve the standard affine-in-control barrier inequality, keep the runtime filter QP-compatible, and move the decay-rate decision from offline hand tuning into the online optimization loop. That repositioning is narrow in form but consequential in effect, because it changes feasibility, conditioning, and the interpretation of the class-$\mathcal C=\{x:h(x)\ge 0\}$03 term from a fixed design choice to a dynamically allocated safety resource (Zeng et al., 2021, Ong et al., 17 Jul 2025).