Control Lyapunov Barrier Functions

Updated 20 February 2026

Control Lyapunov Barrier Functions (CLBFs) are a unified framework that combine stabilization and safety criteria into a single scalar certificate for nonlinear control-affine systems.
They utilize advanced methods such as convex SOS synthesis, neural network parameterizations, and patching techniques to address high-relative degree and stochastic constraints.
CLBFs are pivotal in applications like real-time safe robotics, formal verification, and safety-critical control, offering practical solutions for uncertain system dynamics.

Control Lyapunov Barrier Functions (CLBFs) provide a unified mathematical framework for synthesizing controllers that simultaneously guarantee asymptotic stabilization and safety (i.e., forward invariance of a prescribed safe set) for nonlinear control-affine systems, both in deterministic and stochastic settings. Unlike traditional methods that employ separate Control Lyapunov Functions (CLFs) and Control Barrier Functions (CBFs), often leading to conflicting constraints and potential infeasibility, CLBFs encode safety and stabilization objectives within a single function or certificate. Recent progress includes constructions for relative-degree- $m$ constraints, robust and high-order formulations, convex and sum-of-squares (SOS) synthesis, neural network parameterizations, and new verification and patching methodologies. CLBFs are central in real-time safe control, formal verification, learning-based control, high-assurance robotics, and safety-critical systems.

1. Mathematical Foundations and Definitions

The canonical context is a nonlinear control-affine system: $\dot{x} = f(x) + g(x) u, \;\; x \in \mathbb{R}^n, \;\; u \in \mathbb{R}^m$ The essence of a CLBF is to combine the Lyapunov stabilization and barrier invariance properties within a single scalar function $W(x)$ or $V(x)$ .

Control Lyapunov Function (CLF): $V(x)$ is positive definite and radially unbounded. There exists a control $u$ such that

$\inf_{u} \{ L_f V(x) + L_g V(x) u + \alpha(V(x)) \} \leq 0, \;\; \forall x \ne 0$

for some class- $\mathcal{K}$ function $\alpha$ .

Control Barrier Function (CBF): $h(x)$ defines the safe set $\mathcal{C} = \{ x : h(x) \geq 0 \}$ . $h$ is a CBF if, for some class- $\mathcal{K}$ function $\gamma$ ,

$\sup_{u} \{ L_f h(x) + L_g h(x) u + \gamma(h(x)) \} \geq 0, \;\; \forall x \in \mathcal{C}$

Control Lyapunov Barrier Function (CLBF): Let $W(x)$ $W (x)$ or $V(x)$ $V (x)$ satisfy, e.g.,
- $V(0) = 0, V(x) > 0$ for $x \ne 0$
- $V(x) \geq c$ on unsafe set $\mathcal{D}$ , $V(x) < c$ on safe set
- $\inf_{u} \{ L_f V(x) + L_g V(x) u + \lambda V(x) \} \leq 0, \;\; \forall x \ne 0$

Alternative formulations adjust the precise level-set semantics and may vary for reach-avoid, stochastic, or high-order cases (Dawson et al., 2021, Du et al., 2023, Mukherjee et al., 2023, Zheng et al., 2022, Xiao et al., 2021, Sarkar et al., 2020).

2. Construction Methodologies

CLBF construction has diversified beyond simple linear combinations of CLFs and CBFs, particularly to address high relative-degree constraints, conservative feasible sets, and computational scalability.

2.1 Nonlinear Scaling for High-Relative Degree Constraints

Pyon & Park's nonlinear scaling-based CLBF (Pyon et al., 18 Sep 2025) addresses relative-degree-2 constraints by defining: $W(x) = (1 + \theta\sigma(x_1)) V(x) - k, \qquad \sigma(x_1) = \frac{1}{1 + e^{l(x_1 - d - \delta/2)}}$ Here, $V(x)$ is a quadratic CLF, $\sigma(x_1)$ is a smooth sigmoid, and $d$ marks the unsafe boundary. Parameters $(l, \delta, \theta, k)$ are explicitly chosen so that $W(x)$ is large on/in the unsafe set and behaves as a Lyapunov function elsewhere, with sufficient conditions guaranteeing $W>0$ on the unsafe set and $W$ decreasing elsewhere (safe stabilization).

2.2 Patchwork and Patching of CLFs and CBFs

A strictly compatible pair $(V, h)$ —where $V$ is CLF and $h$ a CBF—can be fused via a $C^\infty$ convex patching function: $W(x) = (1-b(x))V(x) + b(x)h(x)$ where $b(x)$ is a smooth bump function supported near the safe set boundary. This ensures $W$ is a CLF in most of the region, a CBF near the boundary, and compatible in the patch region, providing a single smooth function that certifies both safety and asymptotic stability under bounded controls (Liu, 13 Nov 2025).

2.3 High-Order and Stochastic CLBFs

HOCLBFs generalize CLBFs for constraints of relative degree $m$ , embedding cascaded derivative conditions for guarantees on invariance and finite-time reachability (Xiao et al., 2021, Sarkar et al., 2020). The stochastic versions require conditions on the infinitesimal generator (Itô drift) of $V(x)$ or $W(x)$ , integrating diffusion and higher moments (Zheng et al., 2022, Sarkar et al., 2020).

2.4 Convex Optimization and SOS Synthesis

For polynomial systems, CLBFs can be constructed via convex (SOS-based) optimization. Sufficient conditions, often via Positivstellensatz, translate region-of-attraction and safety-set containment into tractable LMI constraints on the polynomial coefficients. Compatibility (existence of joint $u$ satisfying both decrease and invariance) can now be verified or synthesized without reference to a nominal controller (Dai et al., 2024, Schneeberger et al., 2023, Dai et al., 2022, Schneeberger et al., 2024).

2.5 Learning-Based and Data-Driven CLBFs

Neural network parameterization of CLBFs, with losses encoding sub/superlevel set properties and decrease/invariance constraints, enables data-driven synthesis and online control with theoretical guarantees analogous to classic function-based CLBFs (Dawson et al., 2021, Du et al., 2023). Similarly, reinforcement learning approaches enforce the CLBF drift condition in the expected Bellman residual sense.

3. Parameter Selection, Verification, and Compatibility

Effective CLBF synthesis requires attention to:

Parameter selection: Ensuring shape and weighting are such that the function is positive on unsafe regions, decrease conditions are met wherever needed, and no spurious equilibria arise (Pyon et al., 18 Sep 2025, Liu, 13 Nov 2025).
Compatibility: The feasible set in which both CLF and CBF inequalities admit a simultaneous control $u$ can be strictly characterized via Farkas' lemma and stated as SOS-feasibility problems. This enables expansion of the compatibility region beyond nominal controller-based parameterizations (Dai et al., 2024).
Verification: SMT methods (e.g., dReal) and SOS checkers serve for formal guarantees over the compact certified region (Liu, 13 Nov 2025, Dai et al., 2022).

4. Implementation in Real-Time Control and Learning

CLBFs form the foundation of a wide variety of controller architectures:

CLBF-QP and CLBF-QCQP: At each control step, a (small) QP is solved to find $u$ minimizing control effort while enforcing CLBF-derived inequalities (decrease condition for stabilization, invariance for safety), including potential relaxations for feasibility (slack variables) (Xu et al., 2016, Schneeberger et al., 2024, Shahraki et al., 2024).
Feedback Linearization Add-On: In high relative-degree or multi-constraint settings, a safe feedback-linearization baseline controller is extended by CLBF-based add-on laws. Synthesis is decoupled for multi-constraint decoupled subsystems (Pyon et al., 18 Sep 2025).
Patchwork and Sontag’s Universal Formula: Explicit, globally defined feedbacks can be constructed using the universal formula for systems where $L_g W(x)\neq 0$ (Liu, 13 Nov 2025).
Learning Control via Neural CLBFs/LBAC: Neural networks parameterize both critics ( $V$ functions) and policies, optimized to satisfy CLBF conditions empirically or in expectation. Reach-avoid and stabilization are cast as a single RL objective, yielding improved safety and sample efficiency (Dawson et al., 2021, Du et al., 2023).
Reinforcement Learning for Reach-Avoid: Data-driven CLBFs unify reachability and safety by encoding a single value function, used for actor-critic synthesis or policy search (Du et al., 2023).

5. Applications and Computational Aspects

CLBFs play a pivotal role in:

Safe Robotics: Task-space control of manipulators, quadrotor/vehicle navigation, real-time obstacle avoidance, and safety filter integration into legacy architectures.
Stochastic MPC: In the presence of diffusion or jump uncertainties, CLBFs provide sampled-data safety/stability certificates suitable for constraint-encoded MPC and event-triggered updates (Zheng et al., 2022).
Spacecraft Attitude Control: CLBF-based QPs enable real-time, constraint-enforcing, low-chattering, and asymptotically stabilizing control under torque/momentum and reaction wheel constraints (Shahraki et al., 2024).
Formal Verification: CLBF synthesis with SMT or SOS tools enables certified initialization sets, safety regions, and formal region-of-attraction guarantees with less conservatism than traditional SOS designs (Liu, 13 Nov 2025, Dai et al., 2024).

The computational complexity is determined primarily by the underlying QP (or SOS) solve; in typical robotics or embedded implementations, real-time operation at kHz rates is possible for low-dimensional systems (Dawson et al., 2021, Xu et al., 2016). In SOS-based methods, polynomial degree and dimension are limiting parameters; however, convexification and alternation schemes have enabled tractable syntheses for moderate-size systems (Dai et al., 2024, Schneeberger et al., 2023, Dai et al., 2022).

6. Limitations, Extensions, and Future Directions

While CLBFs provide a powerful method for unifying safety and stabilization, key limitations and research challenges include:

Dimension and Scalability: Harmonic/PDE-based approaches guarantee boundary conditions, but mesh complexity limits dimension (Mukherjee et al., 2023).
Nonexistence and Conservatism: For some systems/constraints, no CLBF, or only very conservative ones, may exist, especially under tight input bounds or highly nonconvex safe sets (Liu, 13 Nov 2025, Zheng et al., 2022).
Spurious Equilibria: Standard CLF–CBF QPs can admit undesired equilibria at safe set boundaries; these can be mitigated by structural augmentations such as rotated Lyapunov functions and additional barrier terms (Reis et al., 2020).
High-Relative-Degree and Temporal Specifications: Generalizations (HOCLBFs, AVCLBFs) address high-order specifications and spatio-temporal constraint satisfaction, improving feasibility via auxiliary variable adaptation (Xiao et al., 2021, Liu et al., 2024).
Learning and Adaptation: Learning-based CLBFs (neural/LBAC) avoid model dependence but currently may lack formal guarantees outside sampled regions or under severe domain shift (Dawson et al., 2021, Du et al., 2023).

Extensions involve combining physics-informed neural nets (PINNs) to analytically satisfy PDE-based CLBFs (Mukherjee et al., 2023), adaptive mesh or spectral numerical solvers for high-dimensional harmonic CLBFs, and extensions to stochastic and hybrid/delayed systems. The open problem of synthesis and verification in highly nonlinear, constrained, and uncertain systems remains at the frontier.

References: