Characterization of Safe Stabilization and Control Lyapunov-Barrier Functions via Zubov Equation Formulation

Abstract: Design and analysis of stabilizing controllers with safety guarantees for nonlinear systems have received considerable attention in recent years. Control Lyapunov-barrier functions (CLBFs) provide a powerful framework for simultaneously ensuring stability and safety; however, their construction for nonlinear systems remains challenging. To address this issue, we build on recent advances in PDE-based characterizations of control Lyapunov functions and Lyapunov-barrier functions for autonomous systems, and propose a succinct Zubov-HJB PDE formulation for safe stabilization of nonlinear control-affine systems under a common compatibility assumption. We further show that the viscosity solution of this PDE yields a maximal CLBF, enabling (not necessarily continuous) feedback synthesis with stability and safety guarantees. In light of recent advances in neural-network-based methods for solving Zubov-type PDEs, this theoretical framework also provides a natural interface to emerging numerical approaches.

• The paper introduces a PDE-based framework using a Zubov equation formulation to synthesize maximal, possibly nonsmooth, control Lyapunov-barrier functions for safe stabilization.
• It unifies stability and safety by bridging control Lyapunov and barrier functions, thereby easing compatibility constraints in nonlinear control-affine systems.
• The framework supports integration with neural PDE solvers for scalable verification and feedback synthesis in high-dimensional, safety-critical applications.

Characterization of Safe Stabilization and CLBFs via Zubov Equations

Introduction and Motivation

The paper addresses the synthesis and characterization of stabilizing controllers with formal safety guarantees for nonlinear control-affine systems. The primary mathematical tool for encoding both stability and safety is the control Lyapunov-barrier function (CLBF), which unifies control Lyapunov function (CLF) and control barrier function (CBF) properties. However, constructing a CLBF, especially for nonlinear or control-affine systems under state constraints, remains significantly challenging, primarily due to the restrictive differentiability and compatibility requirements imposed by conventional methods.

The paper overcomes these limitations by introducing a scalable and general partial differential equation (PDE)-based framework. This approach leverages the Zubov equation formulation to characterize the maximal domain of safe stabilization and constructs the associated maximal (possibly nonsmooth) CLBF, which need not admit a continuous feedback in general. Crucially, the paper establishes that the viscosity solution of this Zubov-HJB PDE subsumes the CLBF property, thus bridging Lyapunov and barrier notions in a single analytic framework.

Background: Safety, Stability, and Their Integration

In the context of safety-critical nonlinear systems, stabilization (to a desired setpoint, typically the origin) must be attained without violating safety constraints, formalized via a safe set $S$. CLFs certify asymptotic controllability, while CBFs enforce set invariance. Simultaneous use of CLF and CBF in optimization-based (e.g., QP) controllers raises feasibility issues due to possible incompatibility of objectives, motivating the quest for a unified CLBF. Classical approaches demand continuous differentiability and strong compatibility assumptions, which are sometimes violated due to topological or geometric obstructions or nonexistence of continuous stabilizing feedback.

The paper generalizes recent PDE-based converse Lyapunov theorems by lifting the need for smoothness and replacing pairwise CLF/CBF certificates with a single viscosity solution of a Zubov-HJB PDE. This generalizes the null-controllability domain to the safe domain $D$ from which stabilization can be achieved without leaving $S$, and encodes both objectives in a single value function.

PDE Characterization of Maximal Safe Domain and CLBF

Problem Formulation

The general continuous-time control-affine system is written as

$\dot{x} = f(x) + g(x)u$

with $x \in \mathbb{R}^n$, $u \in \mathbb{R}^m$, and locally Lipschitz $f,g$. The safe set $S$ is specified by a function $h(x) < 1$, and $U = S^c$ defines the unsafe set (obstacle). The null-controllability-with-safety domain $D$0 consists of all initial conditions from which there exists a control input such that the state evolves toward the origin without entering $D$1 at any time.

Main Theoretical Results

Value Function and Dynamic Programming

A value function $D$2 is defined as the infimum of the accumulated cost over all admissible control inputs:

$D$3

where $D$4 is the first exit time from the safe set $D$5, $D$6 is a suitable running cost (indicator) function, and $D$7 encodes the terminal cost and constraint violation.

Under sufficient regularity and a compatibility condition (which prevents pathological boundary behavior and ensures well-posed existence of a safe CLF), $D$8 is proved to possess the following properties:

• $D$9 iff $S$0;
• $S$1 diverges as $S$2 approaches $S$3 or $S$4;
• $S$5 is positive definite and continuous.

$S$6 is further shown to be the unique maximal viscosity solution of the HJB equation:

$S$7

Zubov Equation Formulation

To handle unboundedness and facilitate computation over a larger domain, a nonlinear transformation of $S$8 is used, $S$9, with $\dot{x} = f(x) + g(x)u$0 a strictly increasing function ($\dot{x} = f(x) + g(x)u$1, $\dot{x} = f(x) + g(x)u$2 as $\dot{x} = f(x) + g(x)u$3, e.g., $\dot{x} = f(x) + g(x)u$4).

$\dot{x} = f(x) + g(x)u$5 is then shown to be the unique viscosity solution of a generalized Zubov equation:

$\dot{x} = f(x) + g(x)u$6

with $\dot{x} = f(x) + g(x)u$7, $\dot{x} = f(x) + g(x)u$8 as $\dot{x} = f(x) + g(x)u$9. The set $x \in \mathbb{R}^n$0 coincides with $x \in \mathbb{R}^n$1.

Feedback Synthesis

The viscosity solution $x \in \mathbb{R}^n$2 is a maximal (possibly nonsmooth) CLBF; thus, one can synthesize state feedback (not necessarily continuous) that certifies both invariance with respect to $x \in \mathbb{R}^n$3 and stabilization to the origin. Sontag-type formulas and relaxed control constructions can be applied in this nonsmooth context.

Interface with Neural Solver Paradigms

The PDE characterization—particularly the Zubov equation—serves as a natural interface for physics-informed neural network (PINN) methods and other modern mesh- or sample-based solvers. This aligns with advances in the computation and verification of maximal Lyapunov/barrier functions using deep learning frameworks (Liu et al., 2023, 2604.00941).

Comparison with Prior Work and Theoretical Implications

The proposed characterization generalizes and subsumes several previous approaches:

• Prior smooth-converse results require both differentiability and explicit construction of CLF/CBF pairs or separate PDEs, whereas the present framework yields a single maximal CLBF via a single PDE, covering nonsmooth and continuous cases alike [camilli2008control], [meng2025towards], [quartz2025converse].
• The compatibility condition employed here is less restrictive than existing versions, eliminates extraneous obstructions, and results in maximal domains by construction.
• The transformed Zubov PDE is defined globally (or over enlarged regions of interest), which is crucial for practical numerical computation (e.g., PINNs, SMT-based verifiers).

Theoretically, this work unifies stability and safety certificates, linking classical Lyapunov theory, integer-constrained reachability, and modern viscosity solutions of HJB/Zubov equations in a compact analytic framework. The nonsmooth CLBFs derived in this manner are maximal, providing exhaustive certification of the stabilizable safe region $x \in \mathbb{R}^n$4.

Practical Implications and Future Directions

This framework enables direct use of state-of-the-art neural PDE solvers for the synthesis and verification of safe stabilizing feedback in complex nonlinear systems. It provides the foundation for formally-correct policy learning, neural control certification, and scalable computational synthesis for high-dimensional, safety-critical autonomous systems.

Strong numerical evidence and simulation results in related recent studies (Liu et al., 2023), [liu2025physics], [liu2025formally] suggest that this PDE-based approach delivers improved performance and less conservative safe regions compared to SOS or QP-based methods, and is compatible with formal verification engines (e.g., SMT solvers).

Open theoretical challenges include relaxing or removing the compatibility assumption—which, while standard and never known to be violated, may still exclude highly pathological cases—and extending these results to underactuated, hybrid, or uncertainty-affected systems.

Conclusion

This paper presents a rigorous Zubov-HJB PDE-based characterization of safe stabilization for nonlinear control-affine systems. It provides explicit analytic and computational machinery for constructing maximal (potentially nonsmooth) CLBFs, synthesizing safe stabilizing controls even in the absence of smoothness and continuous feedback. This work unifies CLF/CBF notions, extends the arsenal of nonsmooth control methods, and opens significant pathways toward neural PDE-based verification and controller synthesis for safety-critical nonlinear systems.

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Key references:

• "Control Lyapunov functions and Zubov’s method" [camilli2008control]
• "Physics-informed neural network Lyapunov functions: PDE characterization, learning, and verification" (Liu et al., 2023)
• "Formally verified physics-informed neural control Lyapunov functions" [liu2025formally]
• "Towards Learning and Verifying Maximal Lyapunov-Barrier Functions with a Zubov PDE Formulation" [meng2025towards]
• "A Converse Control Lyapunov Theorem for Joint Safety and Stability" [quartz2025converse]
• "Smooth converse Lyapunov-barrier theorems for asymptotic stability with safety constraints..." [meng2022smooth]