Barrier Lyapunov Functions (BLF)

Updated 9 March 2026

Barrier Lyapunov Functions are analytical tools that enforce state and output constraints in nonlinear control systems by integrating Lyapunov decay with barrier penalties.
They have evolved into progressive, fractional, and robust variants that offer smooth corrective actions, finite tracking, and resilience to uncertainties.
BLF-based designs are widely applied in robotics, power electronics, and learning-based control, delivering rigorous stability and safety guarantees in practical implementations.

Barrier Lyapunov Functions (BLF) provide a rigorous framework for enforcing state and output constraints in nonlinear control systems by fusing Lyapunov-based stability guarantees with barrier constraints that preclude boundary violations. BLFs have been extended across deterministic, stochastic, robust, data-driven, and learning architectures, with proven sufficiency and necessity in both smooth and nonsmooth formulations. Advanced BLF variants support optimization-based control, learning-based certificates, and robust, high-relative-degree problems, with verified stability, constraint satisfaction, and computational tractability.

1. Fundamental Principles and Definitions

A Barrier Lyapunov Function for a system $\dot x = f(x)$ , $x\in D\subset\mathbb R^n$ , is a continuously differentiable, positive-definite $V:D\to\mathbb R_+$ such that:

$V(x)\to+\infty$ as $x\rightarrow\partial D$
Along system trajectories, $\dot V(x)\le0$

This construction certifies forward invariance of $D$ : for all $t\ge0$ , $x(t)\in D$ if $x(0)\in D$ (Nohooji et al., 2024, Sarkar et al., 2020).

Classical BLFs often employ the logarithmic form $V(x) = \frac{1}{2}\ln\left(\frac{k^2}{k^2-x^2}\right)$ for enforcing $|x|<k$ ; as $|x|\to k$ , $V(x)\to\infty$ . The BLF is distinguished from generic barrier certificates in that it also enforces Lyapunov-style decay, yielding a direct synthesis procedure for constraint-aware stabilizing controllers.

Extended definitions encompass Control Lyapunov Barrier Functions (CLBFs) in feedback systems—where a single scalar function $V(x)$ is used to enforce both stability (Lyapunov) and forward invariance (barrier), see e.g. (Mukherjee et al., 2023, Meng et al., 2020)—as well as nonsmooth and maximal (Zubov-PDE-based) BLFs (Meng et al., 12 Nov 2025, Lan et al., 2024).

2. BLF Synthesis in Nonlinear Control: Classical and Progressive Forms

Classical BLFs

For single-state or output-constrained systems, classical BLFs induce high corrective action as the constraint boundary is approached, potentially applying substantial (and sometimes unnecessary) actuation even far from the boundary. This often produces chattering or overly aggressive responses (Nohooji et al., 2024).

Progressive (Smooth-Zone) BLFs

"Progressive BLFs" (p-BLFs) modulate the rate of increase in the penalty near the constraint, providing a free zone of minimal control and a smooth, progressive growth in corrective gain towards the boundary. Two canonical forms:

Logarithmic-based p-BLF:

$V_{\rm log}(x) = \frac{1}{2\,\beta}\, \ln\left(\frac{k^2}{k^2 - x^2}\right)$

Rational-based p-BLF:

$V_{\rm rat}(x) = \frac{x^2}{2\,(k^2-x^2)\,(1+\beta x^2)}$

where $\beta>0$ defines the width of the "free" region (small $V$ for $|x|\ll k$ ), and $k$ the constraint gap (Nohooji et al., 2024).

When $|x|\ll k$ , both barrier terms grow slowly, yielding negligible control effort; as $|x|\to k$ , control effort increases automatically and smoothly, repelling the state from the constraint—removing discontinuities and chattering.

BLF-Based Constrained Control Synthesis

For strict-feedback systems: $\dot x_1 = f_1(x_1) + g_1(x_1)x_2 \ \dot x_2 = f_2(x_1, x_2) + g_2(x_1)u$ tracking errors $z_1 = x_1 - x_d$ , $z_2 = x_2 - \alpha_1$ , the p-BLF-based controller uses backstepping to sequentially select virtual controls and the control input $u$ to guarantee $\dot V \le 0$ with the desired constraint satisfaction (Nohooji et al., 2024).

Generalizations to full-state constraints concatenate BLFs for each constrained state, yielding composite Lyapunov-barrier functions with analogous guarantees.

3. Beyond Classical BLFs: Fractional, Harmonic, and Maximal Variants

Fractional BLFs (FBLFs) replace transcendental penalties with rational forms, such as $f_{FI}(V, b_\nu) = V/(b_\nu - V)$ , which exhibit stronger gradient near the boundary and improved finite-interval tracking behavior in iterative learning schemes (Sun, 2023).

Harmonic CLBFs solve the Dirichlet-Laplace boundary value problem for constrained optimal control—constructing a $C^2$ function $B_h$ satisfying Laplace’s equation in the safe set, prescribed values on target/unsafe boundaries. This guarantees the maximum principle, optimal safety/reachability geometry, and unique controller synthesis by gradient steepest descent (Mukherjee et al., 2023).

Maximal BLFs (Zubov-PDE approach) transform the problem of computing tight safe domains of attraction with stability/safety guarantees into a PDE with Dirichlet boundary conditions. Neural approximation and formal verification (via SMT solvers) yield a neural Lyapunov-barrier function $W_\theta(x)$ that is provably valid over a large portion of the theoretically maximal safe set (Meng et al., 12 Nov 2025).

4. Converse, Robust, and Stochastic BLF Theory

Converse Lyapunov-Barrier Theorems

It is proven that if a system enjoys a stability-with-safety or reach-avoid-stay property, then there exists a single $C^\infty$ Lyapunov-barrier function encoding both objectives (possibly at a reduced level of modeled disturbance). This closes a theoretical gap regarding sufficiency and necessity for existence, supporting learning or computational synthesis strategies (Meng et al., 2020, Liu, 2020).

Robust and Stochastic BLFs

Robust BLFs are constructed to handle parametric uncertainties or bounded disturbances, using worst-case Lyapunov and barrier decay inequalities over all admissible uncertainties (Dawson et al., 2021, Meng et al., 2022).

In stochastic systems (Itô diffusions), Stochastic Lyapunov-Barrier Functions (SLBFs) jointly enforce expected-value decay and supermartingale properties (via the extended generator), yielding probabilistic guarantees for reach-avoid-stay specifications. High-relative-degree generalizations embed sequential barrier-Lyapunov terms for nested derivative constraints, enforced via chained QP conditions (Sarkar et al., 2020, Meng et al., 2022).

5. Model Predictive, Optimization-Based, and Learning Approaches

BLFs in Predictive and Optimization-Based Control

Barrier Lyapunov Functions have been incorporated as explicit soft or hard constraints in Model Predictive Control (MPC), ensuring state and output constraint satisfaction at each prediction step. The BLF constraint is expressed as an affine or convex inequality in the control input, straightforwardly integrated within the QP-based MPC formulation (Mundheda et al., 2022). This operationalizes BLFs in high-fidelity, real-time safety-critical systems.

Neural and Data-Driven BLF Synthesis

Learning-based frameworks parameterize Lyapunov-barrier certificates and controllers by neural networks, trained with loss terms enforcing positive-definiteness, constraint satisfaction, and (robust) decay conditions at sampled states. “Proof” policies warm-start the associated inner QP solve. The neural approach accommodates arbitrary dynamics and parametric uncertainty, and can optionally be verified on large sampled or certified domains (Meng et al., 12 Nov 2025, Dawson et al., 2021).

Formal Verification

Learned BLFs can be verified by SMT interval methods (e.g., dReal, Z3) to certify Lyapunov decay and the barrier property uniformly over prescribed sublevel sets, closing the loop between data-driven synthesis and rigorous certification (Meng et al., 12 Nov 2025).

6. Nonsmooth, Nonexistence, and Advanced Topics

For certain architectures and nonconvex unsafe sets, smooth CLBFs cannot exist due to gradient cancellation or topological obstructions. Nonsmooth (max-combination) CLBFs, as in $V(x) = \max\{L(x), B(x)\}$ , always exist for bounded obstacles and guarantee barrier separation and Lyapunov convergence; controller synthesis reduces to region-specific feedback with provable forward invariance and $\mathcal{KL}$ -stability (Lan et al., 2024).

Fractional BLFs, robust chattering-suppressing modifications, and piecewise continuous feedback are developed to optimize performance and extend universal existence.

7. Applications and Practical Guidelines

BLFs are now deployed in domains including nonlinear process control, aerial manipulation, mobile robotics, power electronics, and learning-based/iterative control. Empirical results consistently report:

Hard constraint enforcement for all $t$ (no violations)
Smooth, chattering-free control action (especially by p-BLFs and FBLFs)
Reduced average actuation relative to classical (nonstratified) BLFs when far from constraints
Asymptotic target tracking and boundedness of all closed-loop signals (Nohooji et al., 2024, Mukherjee et al., 2023, Mundheda et al., 2022, Sun, 2023)

A typical guideline is to tune the aggressiveness parameter (e.g., $\beta$ in p-BLFs) to widen the unconstrained region; select gains (e.g., $\kappa_i$ ) to adjust convergence rate; and ensure the BLF construction matches the relative degree and smoothness properties of the constraints.

Simulation and hardware-oriented studies demonstrate the scalability of these methods with formal certification, real-time QP tractability, and implementation in the presence of uncertainty and disturbances. The theoretical foundations are now mature, supporting continued methodological innovation and deployment in increasingly challenging safety-critical control scenarios.

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