Control Lyapunov-Barrier Functions

• CLBF is a unified certificate that encodes both stability and safety by using a single scalar function with distinct level sets.
• It employs analytic and PDE-based constructions to directly synthesize controllers with explicit reach-avoid properties.
• CLBF frameworks integrate optimization and data-driven methods to address high-order, stochastic, and sampled-data control challenges.

A Control Lyapunov-Barrier Function (CLBF) is a certificate for safety-critical control that merges Lyapunov-type stabilization and barrier-type safety into a single object or, in some parts of the literature, into a single synthesis framework. In its scalar-certificate form, a CLBF is a function $V$ whose value decreases under a suitable feedback while its level sets separate safe and unsafe states; in reach-avoid formulations, this is often written by requiring $V=0$ on a goal set, $V\ge c$ on an unsafe set, $V<c$ on the safe region, and $$\langle \nabla V(x), f(x,u)\rangle+\lambda V(x)\le 0$$ under an admissible controller (Mukherjee et al., 2023). A closely related formulation uses a safe threshold $c$ and interprets $h(x):=c-V(x)$ as the barrier, so that safety is encoded by $V(x)\le c$ and stability by dissipation of $V$ (Cheng et al., 29 Sep 2025).

1. Core definitions and conceptual scope

The CLBF concept sits at the intersection of two classical certificates. A Control Lyapunov Function (CLF) establishes convergence toward an equilibrium or goal by enforcing a decay inequality such as

$L_f V(x) + L_g V(x)\,u \le -c\,V(x),$

or, more generally, $V=0$0. A Control Barrier Function (CBF) establishes forward invariance of a safe set $V=0$1 by enforcing

$V=0$2

A CLBF unifies these requirements by using one scalar quantity to encode both attraction and exclusion (Cheng et al., 29 Sep 2025).

One common continuous-time safe-stabilization formulation takes a state $V=0$3, a safe region $V=0$4, and a threshold $V=0$5, and requires

$V=0$6

together with

$V=0$7

and a dissipation condition

$V=0$8

In this view, $V=0$9 is simultaneously a Lyapunov function and a barrier through its sublevel sets (Cheng et al., 29 Sep 2025).

A reach-avoid variant specializes the definition to a compact domain $V\ge c$0, an open goal set $V\ge c$1, a closed unsafe set $V\ge c$2, and a safe region $V\ge c$3. The certificate must satisfy

$V\ge c$4

with a controller ensuring

$V\ge c$5

This formulation makes the “reach” and “avoid” parts explicit in the value conditions alone (Mukherjee et al., 2023).

There is also a discrete-time, data-driven analogue. In the constrained-MDP formulation used for model-free robot learning, a CLBF $V\ge c$6 satisfies $V\ge c$7, $V\ge c$8 away from the goal, $V\ge c$9 on unsafe or irrecoverable states, $V<c$0 on safe states, and an expected decrease condition

$V<c$1

The role of the barrier threshold is then played by $V<c$2, while reachability is encoded through expected decay under the policy (Du et al., 2023).

The term itself is not used uniformly. Some works reserve “CLBF” for a single scalar certificate $V<c$3. Others use the term more operationally for controller synthesis schemes that combine CLF and CBF constraints, especially in quadratic-programming pipelines. That distinction is explicit in work that states it does not define a single composite CLBF while still pursuing the same joint safety-and-stability objective (Gemert et al., 2024).

2. PDE and analytic constructions

A major line of work constructs CLBFs directly from partial differential equations, thereby avoiding trajectory-based training. In harmonic CLBFs, the certificate is defined on the safe region $V<c$4 as the solution of the Laplace equation

$V<c$5

with Dirichlet conditions

$V<c$6

Because harmonic functions satisfy strong and weak maximum principles, the solution has no interior local maxima or minima, so $V<c$7 throughout the interior of $V<c$8. This realizes the CLBF value conditions by construction, and the paper proves uniqueness of the harmonic solution for a given reach-avoid specification (Mukherjee et al., 2023).

The associated controller is chosen by aligning the dynamics with the steepest descent direction of $V<c$9. For general nonlinear dynamics $$\langle \nabla V(x), f(x,u)\rangle+\lambda V(x)\le 0$$0,

$$\langle \nabla V(x), f(x,u)\rangle+\lambda V(x)\le 0$$1

For control-affine dynamics $$\langle \nabla V(x), f(x,u)\rangle+\lambda V(x)\le 0$$2, this reduces to a linear optimization in $$\langle \nabla V(x), f(x,u)\rangle+\lambda V(x)\le 0$$3, and under box input constraints the solution saturates componentwise according to the sign of $$\langle \nabla V(x), f(x,u)\rangle+\lambda V(x)\le 0$$4. This yields a runtime controller that is analytically simple once the PDE has been solved offline (Mukherjee et al., 2023).

The same work also studies a superharmonic Poisson variant,

$$\langle \nabla V(x), f(x,u)\rangle+\lambda V(x)\le 0$$5

with the same boundary conditions. That variant can increase visual contrast between safe and unsafe regions, but it may introduce interior local minima, whereas the harmonic construction avoids interior minima by the maximum principle. In the reported 2D quadrotor landing problem, the harmonic CLBF achieved $$\langle \nabla V(x), f(x,u)\rangle+\lambda V(x)\le 0$$6 goal reachability and safety over $$\langle \nabla V(x), f(x,u)\rangle+\lambda V(x)\le 0$$7 random initial conditions, with mean time $$\langle \nabla V(x), f(x,u)\rangle+\lambda V(x)\le 0$$8 s and standard deviation $$\langle \nabla V(x), f(x,u)\rangle+\lambda V(x)\le 0$$9, outperforming the cited rCLBF-QP and Robust-MPC baselines on that task (Mukherjee et al., 2023).

A more global analytic characterization comes from a Zubov–Hamilton–Jacobi–Bellman formulation for safe stabilization. There, a value function $c$0 is defined by an exit-time optimal-control problem over the safe null-controllability domain $c$1, and a bounded CLBF is obtained through a Zubov transformation $c$2. The resulting viscosity solution satisfies

$c$3

with $c$4 and $c$5 on the boundary of the maximal safe stabilizable set. In that formulation, the sublevel condition $c$6 exactly characterizes the largest safe domain of attraction, so the solution is a maximal CLBF in the sense that no other CLBF can certify a larger safe stabilizable set (Meng et al., 1 Apr 2026).

These PDE-based constructions occupy different points on the design spectrum. Harmonic CLBFs encode reach-avoid geometry directly on a fixed domain and produce a unique certificate with strong geometric structure. Zubov-type formulations characterize the maximal safe stabilizable region but require solving a fully nonlinear PDE. Both make explicit that CLBF synthesis can be posed as a boundary-value or viscosity-solution problem rather than solely as online optimization or offline learning (Mukherjee et al., 2023, Meng et al., 1 Apr 2026).

3. Online synthesis, quadratic programs, and compatibility

In many implementations, CLBF objectives are enforced through optimization rather than by directly evaluating a closed-form feedback. A standard synthesis pattern embeds CLF decrease and CBF invariance as constraints in a quadratic program,

$c$7

subject to

$c$8

with $c$9 used to relax the CLF when necessary. This architecture is ubiquitous because it makes safety hard and stabilization soft, but it also exposes a basic issue: separate existence of a CLF and a CBF does not imply that a bounded input exists that satisfies both inequalities simultaneously (Cheng et al., 29 Sep 2025).

That feasibility question has become a central theme. One line of work derives exact necessary and sufficient compatibility conditions for a CLF $h(x):=c-V(x)$0 and a CBF $h(x):=c-V(x)$1 under polytopic input bounds $h(x):=c-V(x)$2, independent of any nominal controller. By writing the joint CLF-CBF-input constraints as a linear inequality system in $h(x):=c-V(x)$3, applying Farkas’ lemma, and then translating emptiness of the resulting semialgebraic set into Positivstellensatz and SOS conditions, compatibility can be verified or synthesized directly over a region $h(x):=c-V(x)$4 (Dai et al., 2024). A closely related bounded-control verification framework gives exact conditions for ball and box input sets and then constructs a single smooth CLBF by a patching formula

$h(x):=c-V(x)$5

where $h(x):=c-V(x)$6 is a smooth bump supported on a boundary band. In that framework the safe set is certified exactly by $h(x):=c-V(x)$7 (Liu, 13 Nov 2025).

Compatibility also motivates alternatives to the standard slack-weighted QP. One paper shows, via a literature example, that the usual CLF-CBF-QP can produce slow convergence and oscillatory inputs because the slack weight mediates an undesirable trade-off between stabilization and safety. It then replaces the CLF constraint by a cost that penalizes deviation from Sontag’s formula using

$h(x):=c-V(x)$8

and solves a QP with only CBF constraints. Under a control-sharing assumption, this yields local asymptotic stability and safety on an explicit domain of attraction $h(x):=c-V(x)$9 (Gemert et al., 2024).

A broader implementation lesson is that “CLBF-based control” often refers to an overview stack rather than to a unique analytic formula. Spacecraft attitude control under reaction-wheel constraints, for example, uses an optimal-decay CLF-CBF-QP with a decision variable $V(x)\le c$0 that scales the CLF decay rate online to mitigate chattering at low sampling frequencies, while zeroing CBFs enforce hard wheel-momentum bounds (Shahraki et al., 2024). High-order CLF-CBF-QP controllers for bicyclist safety in autonomous driving similarly treat safety as hard through HOCBF inequalities and stabilization as soft through a slackened HOCLF constraint (Chen et al., 14 Dec 2025). These are CLBF methodologies in the operational sense, even when they do not define a single scalar CLBF.

4. Data-driven and learned CLBFs

A second major direction learns the certificate from data. In robust neural CLBFs, the system is modeled as

$V(x)\le c$1

with parametric uncertainty $V(x)\le c$2. A neural certificate $V(x)\le c$3 is trained so that $V(x)\le c$4, $V(x)\le c$5 on the safe set, $V(x)\le c$6 on the unsafe set, and

$V(x)\le c$7

holds across all scenario vertices. The runtime controller solves a small robust QP over these scenario inequalities. In simulation, this approach matched or exceeded robust MPC while reducing computational costs by an order of magnitude across tasks including car tracking, obstacle avoidance, satellite rendezvous, and flight control with a learned ground-effect model (Dawson et al., 2021).

A model-free alternative appears in constrained reinforcement learning. The Lyapunov Barrier Actor-Critic (LBAC) treats the CLBF as a critic $V(x)\le c$8, defines $V(x)\le c$9, and enforces a discrete expected-decrease condition from replay-buffer data without an explicit dynamics model. The method uses terminal-cost shaping, Lagrangian penalties, and a theorem that links a sufficiently large unsafe terminal cost $V$0 to the threshold property $V$1 on unsafe states and $V$2 on safe states. In simulation and on a real Crazyflie 2.0 platform, the learned CLBF reduced safety violations relative to model-free RL baselines and avoided the CLF-CBF conflicts observed with a model-based QP controller (Du et al., 2023).

More recently, Safe and Stable Diffusion ($V$3Diff) recasts CLBF enforcement as trajectory sampling from a distribution

$V$4

where safety is encoded by the hard condition $V$5, and stability is encoded by a soft penalty on

$V$6

The paper follows Romdlony–Jayawardhana’s unified CLBF view, but replaces pointwise QP enforcement by diffusion-guided trajectory generation and CLBF learning. Its theory is phrased in Almost Lyapunov terms: dissipation may fail on a small set $V$7, yet one still obtains almost-sure exponential decay with an additive buffer. Empirically, over eight environments $V$8Diff reports average safety rate $V$9, average normalized terminal error $L_f V(x) + L_g V(x)\,u \le -c\,V(x),$0, and average inference time $L_f V(x) + L_g V(x)\,u \le -c\,V(x),$1 ms, compared with $L_f V(x) + L_g V(x)\,u \le -c\,V(x),$2, $L_f V(x) + L_g V(x)\,u \le -c\,V(x),$3, and $L_f V(x) + L_g V(x)\,u \le -c\,V(x),$4 ms for rCLBF-QP, and $L_f V(x) + L_g V(x)\,u \le -c\,V(x),$5, $L_f V(x) + L_g V(x)\,u \le -c\,V(x),$6, and $L_f V(x) + L_g V(x)\,u \le -c\,V(x),$7 ms for MPC (Cheng et al., 29 Sep 2025).

Across these learning-based formulations, the CLBF serves three related roles: as a certificate to constrain learning, as a shaping function that defines safe sublevel sets, and as a surrogate objective whose derivative is easier to supervise than long-horizon reward alone. What changes is the enforcement mechanism—robust QP, actor-critic constraint learning, or diffusion-based trajectory guidance (Dawson et al., 2021, Du et al., 2023, Cheng et al., 29 Sep 2025).

5. High-order, stochastic, and sampled-data extensions

Relative degree is a persistent obstacle for CLBF design. When the safety output does not depend on the control at first order, standard CBF inequalities cease to be directly usable. One response is to use high-order barrier constructions. Another is to redesign the CLBF itself. For scalar second-order control-affine systems with an unsafe half-space $L_f V(x) + L_g V(x)\,u \le -c\,V(x),$8, a nonlinear scaling-based design defines

$L_f V(x) + L_g V(x)\,u \le -c\,V(x),$9

where $V=0$00 is a quadratic CLF and $V=0$01 is a logistic function centered near the safety boundary. The paper derives explicit parameter conditions, including

$V=0$02

to ensure $V=0$03 on the unsafe set and $V=0$04 on the set where $V=0$05. Sontag’s formula then yields a safe stabilizer, and the same construction is used as an add-on safety term for feedback-linearized task-space control of a planar robot manipulator (Pyon et al., 18 Sep 2025).

Stochastic generalizations replace deterministic Lie derivatives by the infinitesimal generator. A high-relative-degree stochastic CLF/CBF framework constructs recursive barrier quantities through

$V=0$06

and enforces the final CLF and barrier inequalities in a QP with slack on the Lyapunov side. Safety is obtained almost surely, while stability is in probability. The paper treats relative-degree-2 car navigation and a relative-degree-4 two-link pendulum with elastic actuator, making explicit that diffusion enters through second-order terms in both the Lyapunov and barrier recursions (Sarkar et al., 2020).

A direct stochastic CLBF construction appears in model predictive control for nonlinear affine systems. There, a stochastic CLBF

$V=0$07

combines an unconstrained stochastic CLF $V=0$08, obtained through dynamic feedback linearization, with multiple stochastic CBFs $V=0$09. The resulting sampled-data MPC enforces generator-based CLBF drift constraints and integrates event-triggering to improve performance during sampling intervals. The safety statement is forward invariance with probability $V=0$10, while the sampled-data closed loop is shown to be mean-square exponentially stable for sufficiently small intersample times (Zheng et al., 2022).

Not all work in this area is supportive of existing CLBF recovery formulations. A recent finite-time CBF paper explicitly identifies two limitations of CLBF-based recovery controllers for initially unsafe states: chattering near the safety boundary and lack of explicit control-bound handling. It proposes a strengthened constraint $V=0$11 and a linear inequality

$V=0$12

that guarantees finite-time return to the safe set under control bounds when a reachability-based comparison condition is satisfied. That result is presented as an alternative to CLBF-based finite-time recovery, not as a CLBF itself (Li et al., 23 Mar 2026).

Sampled-data execution is another extension. Event- and self-triggered CLBF control co-designs the input and the next execution time so as to maximize margins to the CLF and CBF boundaries at each update. The greedy formulation introduces margins $V=0$13 in the inequalities and computes a control that allows longer inter-execution intervals, with explicit lower bounds on dwell time under Lipschitz assumptions (Kishida, 2023).

6. Applications, limitations, and terminology

CLBF methods have been used in a wide range of domains. Reach-avoid navigation with harmonic CLBFs has been demonstrated on Roomba, differential-drive, car-like, and quadrotor systems, including a 2D landing problem with obstacle avoidance (Mukherjee et al., 2023). Learned and robust CLBF variants have been evaluated on inverted pendulum, Segway, neural lander, 2D and 3D quadrotor, spacecraft rendezvous, and F-16 aircraft models (Dawson et al., 2021, Cheng et al., 29 Sep 2025). CLF-CBF methodologies closely tied to the CLBF objective have also been applied to tumor dynamics with positivity constraints, autonomous-driving scenarios involving bicyclists, spacecraft attitude control with reaction-wheel limitations, and ac/dc power converters (Gemert et al., 2024, Chen et al., 14 Dec 2025, Shahraki et al., 2024, Schneeberger et al., 2024).

The main strengths of CLBF formulations are structural unification and explicit certificates. A single thresholded function can encode reachability and safety simultaneously; this removes gradient conflicts that arise when CLF and CBF objectives are designed independently, a point emphasized by the model-free RL work and by QP-based analyses of CLF-CBF conflict (Du et al., 2023, Gemert et al., 2024). In QP realizations, the same unification appears operationally: safety is imposed as a barrier constraint, stabilization as a Lyapunov inequality, and the optimizer mediates the trade-off under input bounds (Shahraki et al., 2024).

The limitations are equally consistent across the literature. PDE-based constructions can suffer from the curse of dimensionality, since finite-element or viscosity-solution computations scale poorly with state dimension (Mukherjee et al., 2023, Meng et al., 1 Apr 2026). SOS-based verification and synthesis require polynomial models or polynomial approximations and can become conservative or computationally expensive as degree and state dimension grow (Dai et al., 2022, Dai et al., 2024, Liu, 13 Nov 2025). Learning-based methods depend on model accuracy, data coverage, or differentiable surrogates, and their guarantees are often scenario-based, almost-sure, or “almost Lyapunov” rather than pointwise everywhere (Dawson et al., 2021, Du et al., 2023, Cheng et al., 29 Sep 2025). High-order and sampled-data settings introduce additional feasibility and numerical-conditioning issues, especially under hard control bounds (Zheng et al., 2022, Li et al., 23 Mar 2026).

A final source of confusion is terminological rather than technical. In one usage, a CLBF is literally a single scalar certificate $V=0$14 or $V=0$15 whose level sets simultaneously encode safety and stabilization. In another, “CLBF-based control” denotes any synthesis framework that enforces CLF and CBF conditions together, typically through a QP, even if no single analytic composite function is written down (Gemert et al., 2024, Shahraki et al., 2024). Both usages are established in current research. The first emphasizes certificate construction; the second emphasizes controller realization.