Barrier Certificates and Lyapunov Functions

• Barrier certificates and Lyapunov functions are fundamental tools that guarantee safety and asymptotic stability in nonlinear dynamical systems.
• They use differential inequalities to ensure that trajectories remain within safe sets and converge to desired equilibria even under disturbances.
• Modern research integrates these methods into a unified Lyapunov-barrier framework, enabling robust control, formal verification, and learning-based synthesis.

Barrier certificates and Lyapunov functions are fundamental mathematical tools for analyzing and certifying safety and stability of nonlinear dynamical systems, both in theory and in computational practice. Barrier certificates provide invariance guarantees that a trajectory remains within a prescribed safe set and avoids unsafe states, while Lyapunov functions certify asymptotic stability of a target set or equilibrium. Recent advances have unified these approaches, showing that under appropriate robust safety and stability conditions, there exists a single function—often termed a Lyapunov-barrier function or control Lyapunov-barrier function—that simultaneously certifies both safety and stability objectives. This synthesis underpins modern formal methods, learning-based control, and robust synthesis for safety-critical systems.

1. Fundamental Definitions and Theoretical Framework

Barrier Certificates. For a continuous-time system $\dot{x} = f(x)$ with $f: \mathbb{R}^n \to \mathbb{R}^n$ locally Lipschitz, initial safe set $W \subset \mathbb{R}^n$, unsafe set $U \subset \mathbb{R}^n$, and disturbance bound $\delta \ge 0$, a continuously differentiable function $B: \mathbb{R}^n \to \mathbb{R}$ is a $\delta$-robust barrier function for $(W, U)$ if:

• $B(x) \ge 0$ for all $x \in W$,
• $f: \mathbb{R}^n \to \mathbb{R}^n$0 for all $f: \mathbb{R}^n \to \mathbb{R}^n$1,
• $f: \mathbb{R}^n \to \mathbb{R}^n$2 for all $f: \mathbb{R}^n \to \mathbb{R}^n$3 with $f: \mathbb{R}^n \to \mathbb{R}^n$4 and $f: \mathbb{R}^n \to \mathbb{R}^n$5.

The superlevel set $f: \mathbb{R}^n \to \mathbb{R}^n$6 is robustly forward invariant under $f: \mathbb{R}^n \to \mathbb{R}^n$7-perturbations (Liu, 2020).

Lyapunov Functions. For a closed set $f: \mathbb{R}^n \to \mathbb{R}^n$8, a function $f: \mathbb{R}^n \to \mathbb{R}^n$9, $W \subset \mathbb{R}^n$0, is a $W \subset \mathbb{R}^n$1-robust Lyapunov function if there exist class-$W \subset \mathbb{R}^n$2 functions $W \subset \mathbb{R}^n$3 such that

• $W \subset \mathbb{R}^n$4 for all $W \subset \mathbb{R}^n$5,
• $W \subset \mathbb{R}^n$6 for all $W \subset \mathbb{R}^n$7, $W \subset \mathbb{R}^n$8.

This characterizes robust uniform asymptotic stability (RUAS) of $W \subset \mathbb{R}^n$9.

Joint Certificates and Converse Theorems. A central result is the converse barrier function theorem: Under robust safety conditions, any robustly safe set admits a robust barrier certificate derived from a Lyapunov function for its forward reachable set. Specifically, if $U \subset \mathbb{R}^n$0 is $U \subset \mathbb{R}^n$1-robustly safe with respect to $U \subset \mathbb{R}^n$2, $U \subset \mathbb{R}^n$3 is compact, and $U \subset \mathbb{R}^n$4, then for any $U \subset \mathbb{R}^n$5, there exists a smooth $U \subset \mathbb{R}^n$6 with robust Lyapunov properties for $U \subset \mathbb{R}^n$7, and

$U \subset \mathbb{R}^n$8

is a robust barrier function (Liu, 2020). This result links set invariance (barrier conditions) with attractivity (Lyapunov conditions), providing a unified certificate for both properties.

2. Robustness, Discrete-Time Analogues, and Practical Implications

The robust generalization of barrier and Lyapunov functions ensures the invariance and stability properties hold even under bounded perturbations in the dynamics. In discrete-time, a $U \subset \mathbb{R}^n$9-robust barrier function $\delta \ge 0$0 for $\delta \ge 0$1 satisfies:

• $\delta \ge 0$2 on $\delta \ge 0$3,
• $\delta \ge 0$4 on $\delta \ge 0$5,
• $\delta \ge 0$6 for all $\delta \ge 0$7 with $\delta \ge 0$8, $\delta \ge 0$9.

The continuous-time and discrete-time Lyapunov and barrier conditions translate via the Lie derivative and difference operator, ensuring uniform treatments across system classes (Liu, 2020).

Unification of Safety and Stability. The Lyapunov-barrier certificate $B: \mathbb{R}^n \to \mathbb{R}$0 verifies:

• Asymptotic attraction to the safe set $B: \mathbb{R}^n \to \mathbb{R}$1 (stability property),
• Forward invariance of the initial safe set $B: \mathbb{R}^n \to \mathbb{R}$2 (safety property).

This function subsumes classical one-sided barrier conditions (which can be conservative) and accommodates extended conditions used in control applications, such as the extended (Ames–Xu) conditions and reciprocal barrier functions.

Exemplary Application. For the scalar system $B: \mathbb{R}^n \to \mathbb{R}$3 with $B: \mathbb{R}^n \to \mathbb{R}$4, $B: \mathbb{R}^n \to \mathbb{R}$5, and $B: \mathbb{R}^n \to \mathbb{R}$6, one computes $B: \mathbb{R}^n \to \mathbb{R}$7, a Lyapunov function $B: \mathbb{R}^n \to \mathbb{R}$8, and $B: \mathbb{R}^n \to \mathbb{R}$9; this $\delta$0 is positive on $\delta$1, negative on $\delta$2, and satisfies the strict barrier-derivative condition (Liu, 2020).

3. Connections to Other Certificate Frameworks and Extensions

Barrier and Lyapunov functions are at the core of multiple methodologies, including:

• Sum-of-squares (SOS) certificate synthesis: Convex relaxations can be used to compute polynomial barrier and Lyapunov functions via SDP, although with some conservatism due to degree and domain restrictions (Wang et al., 2018, Schneeberger et al., 2023).
• Learning-based certificates: Neural network parameterizations allow for certificate learning in high dimensions, with loss functions encoding the required invariance and descent properties on sampled points (Dawson et al., 2021, Wang et al., 5 Feb 2026, Dawson et al., 2022).
• High-order and temporal-logic certificates: High-order control Lyapunov-barrier functions (HOCLBFs) extend the framework to constraints with arbitrary relative degree, enabling satisfaction of signal temporal logic specifications (Xiao et al., 2021).
• Control-system and robust control extensions: Control Lyapunov-barrier functions (CLBFs) unify safety and asymptotic stability for control-affine systems with bounded inputs, often with explicit patching constructions (e.g., smooth bump interpolation between barrier and Lyapunov regions) (Liu, 13 Nov 2025, Liu et al., 2 Oct 2025).

These frameworks generalize the basic theory to constrained, stochastic, hybrid, and learning-enabled control systems.

4. Formal Synthesis, Verification, and Learning

Modern research emphasizes formal certificate synthesis and verification under uncertainty:

• Neural Lyapunov-barrier certificates can be trained with adversarial, Lipschitz, and neighborhood losses to ensure robustness to state and model perturbations. Certification can be performed via adversarial attacks or formal verification tools (SMT/MILP) (Wang et al., 5 Feb 2026, Mandal et al., 2024).
• Loss functions for neural training enforce initialization (low energy for initial states), progress (Lyapunov/barrier decrease), and separation (unsafe region avoidance). Global Lipschitz regularization is crucial for certifying robustness to perturbations.
• Compositional and filtered certificates facilitate formal verification on large or complex state spaces by dividing the problem into smaller certified domains, each with its own neural certificate, and switching policies as needed (Mandal et al., 2024).
• SMT-verifiable constructions use Farkas' lemma and universal quantification checks for strict compatibility of CBF–CLF pairs, enabling formal guarantees for smooth patched certificates (Liu, 13 Nov 2025, Liu et al., 2 Oct 2025).

The resulting controllers and certificates guarantee both safety and convergence under rich classes of uncertainty and in challenging, high-dimensional domains.

5. Hybrid, Stochastic, and Generalized System Classes

Lyapunov–barrier characterizations extend to systems with hybrid (flow-and-jump), stochastic, and set-valued (differential inclusion) dynamics:

• Hybrid systems: For systems with flows and jumps, a Lyapunov–barrier function must satisfy both continuous flow decrease and monotonicity conditions across discrete jumps. Under arc-topology arguments, necessity and sufficiency of the certificate can be established for perturbed hybrid systems (Meng et al., 2022).
• Stochastic systems: Stochastic Lyapunov-barrier functions provide sufficient (and, under technical assumptions, necessary) conditions for probabilistic reach–avoid–stay properties under Itô-diffusion dynamics. Reciprocal-type barrier functions are useful for encoding invariant sets in the presence of random noise (Meng et al., 2022, Sarkar et al., 2020).
• Set-valued differential inclusions: In settings without smooth right-hand sides, the appropriate notion is a time-varying or nonautonomous barrier function (potentially nonsmooth), constructed via marginal functions of reachable sets. Such functions are both necessary and sufficient for safety in general classes, sometimes requiring time dependence (Maghenem et al., 2022).

These extensions push the methodology beyond deterministic ODEs to encompass a broad spectrum of real-world dynamical phenomena.

6. Synthesis Algorithms, Compatibility, and Practical Considerations

The effectiveness of barrier/Lyapunov certificate synthesis depends on the compatibility between safety and stability objectives:

• Strict compatibility: A control barrier function and a control Lyapunov function are strictly compatible if, on the boundary of the safe set, there exists a control input making both $\delta$3 (stability) and $\delta$4 (safety). This is both necessary and sufficient for the existence of a single smooth Lyapunov-barrier certificate whose sublevel set equals the prescribed safe set (Quartz et al., 15 Sep 2025, Liu, 13 Nov 2025, Liu et al., 2 Oct 2025).
• Constructive patching techniques: Explicit smooth bump-patching can be used to interpolate between a Lyapunov function in the interior and a barrier function near the boundary of the safe set, resulting in a smooth unified certificate (Liu, 13 Nov 2025, Liu et al., 2 Oct 2025).
• Limitations and topological obstructions: In some cases, such as unbounded safe sets or when the complement of the safe set has a bounded component, no compatible CLF–CBF pair exists. This restriction is fundamental and constrains the class of systems for which unified certificates can be constructed (Mestres et al., 2024).
• Computational methods: Sum-of-squares relaxations, convex programming, neural network training, and SMT-based verification are all used in practice, each with trade-offs in scalability, conservatism, and formal guarantees (Schneeberger et al., 2023, Wang et al., 2018, Meng et al., 12 Nov 2025).

7. Impact, Applications, and Ongoing Research Directions

Barrier certificates and Lyapunov functions—and their synthesis, verification, and learning—are central in the design of safety-critical controllers in robotics, aerospace, cyberphysical systems, and autonomous vehicles. Their adoption has enabled:

• Data-driven and reinforcement learning controllers with formal safety/liveness certification (Wang et al., 5 Feb 2026, Dawson et al., 2021, Dawson et al., 2022).
• Real-time robust safe control under uncertainty, via QP-based online control using neural or polynomial certificates (Liu, 13 Nov 2025, Liu et al., 2 Oct 2025, Dawson et al., 2021).
• Modular safe control architectures, such as the Proxy-CBF approach for strict-feedback systems and hierarchical barrier/Lyapunov policies (Wang et al., 7 Jan 2025).
• Extension to high-order, temporal logic, stochastic, and hybrid dynamical systems (Xiao et al., 2021, Meng et al., 2022, Meng et al., 2022).

Active research addresses further scalability for high dimensions, less conservative synthesis (e.g., via PDE or neural approaches (Meng et al., 12 Nov 2025)), and synthesis under partial model knowledge or using reachability analysis (Mestres et al., 2024, Meng et al., 2022).

Barrier certificates and Lyapunov functions exemplify the mature intersection of control theory, formal methods, and learning, offering a principled framework for safe and stable system design. The converse theorems and robust synthesis methodologies ensure that, when safe robust operation is possible, these certificates exist and can, in principle, be found via computational means.