Lyapunov-Stable Neural Network Control

• Lyapunov-stable neural network control is the synthesis of neural controllers that use Lyapunov functions to ensure closed-loop stability and quantify regions of attraction.
• It integrates various methodologies including direct Lyapunov enforcement, optimization-based verification, and certified training with MILP/SDP to guarantee asymptotic performance.
• Recent approaches yield substantially larger regions of attraction and faster verification times compared to classical controllers, enhancing safety in critical applications.

Lyapunov-stable neural-network control is a research domain focused on synthesizing neural-network controllers that guarantee closed-loop stability via the verification of Lyapunov functions, providing explicit regions of attraction (ROA) in nonlinear, uncertain, or high-dimensional systems. The field encompasses a broad array of methodologies, including direct Lyapunov inequality enforcement, optimization-based learning, certified training with mixed-integer or semialgebraic programming, and model-free or model-based approaches. These methods deliver mathematically rigorous certificates ensuring that trajectories of the controlled system converge asymptotically—often exponentially—to a desired equilibrium, overcoming limitations of traditional control designs and reinforcement learning that lack formal guarantees.

1. Fundamental Concepts of Lyapunov Stability in Neural Control

The central concept is the use of a Lyapunov function $V(x)$—often parameterized as a neural network—that is positive definite and strictly decreases along closed-loop trajectories:

$$V(0) = 0, \qquad V(x) > 0 \:\forall x \neq 0, \qquad \dot{V}(x) < 0 \:\forall x \neq 0.$$

For discrete-time systems, the increment is

$V(f(x, \pi(x))) - V(x) < -\epsilon V(x).$

The control policy $\pi(x)$ itself may be a neural network, and the nonlinearities in the closed-loop system, plant, and neural controller introduce significant verification challenges.

Key Lyapunov stability notions such as incremental input-to-state stability (Basu et al., 25 Apr 2025), almost Lyapunov functions accommodating measure-zero violation pockets (Ke et al., 23 Sep 2025, Chang et al., 2021), and sublevel-set invariance underpin most methodologies for neural-controller certification.

2. Direct Lyapunov Inequality Enforcement via Data-driven or Model-based Synthesis

Several works invert the classical controller-learning loop by constructing datasets that by design satisfy Lyapunov inequalities and then fit neural-network policies to these points, ensuring an "out of the box" stable controller. For example, in multicopter interception (Ke et al., 23 Sep 2025), quadratic Lyapunov functions over 2D image error are paired with an image-based Jacobian model:

$$V(x) = \tfrac{1}{2}\bar{p}_x^2 + \tfrac{1}{2}\bar{p}_y^2,$$

with the velocity commands $u = ({^g}v_x,{^g}v_y)$ chosen—by a constrained minimization—for each sampled state to strictly enforce $D_x(x,u)<0$ and $D_y(x,u)<0$. Networks trained on these datasets inherit stability by construction. "Almost Lyapunov" conditions [LiuAlmostLyapunovfunctions2020] are verified empirically by dense sampling, with certified large ROAs and empirical convergence in real flights at high velocity.

3. Optimization-based Learning and Verification Schemes

A rich family of approaches employ convex or mixed-integer programming to certify Lyapunov conditions and maximize ROA size for neural-policy/feedback systems. Typical workflow (Dai et al., 2021, Wang et al., 15 Mar 2024, Wu et al., 2023):

• Represent $V(x)$ and $\pi(x)$ as ReLU or monotonic neural networks.
• Encode the decrease and positivity conditions as mixed-integer linear programs (MILP) or semidefinite programs (SDP), which check:

$$\forall x \in \mathcal{B} : V(x) \ge \epsilon_1 \|R(x-x^*)\|_1,$$

$V(f(x, \pi(x))) - V(x) + \epsilon_2 V(x) \le 0.$

• Use exact MILP solution for verification. When violations are found, counterexamples are added to the training set to guide refinement (counterexample-guided inductive synthesis).
• ROA is computed as the largest invariant sublevel set $S(\rho) = \{x \mid V(x) \le \rho\}$ with $S(\rho)$ contained in the verified region.

Innovative architectures (e.g., monotonic multi-layer star-convex Lyapunov NNs (Wang et al., 15 Mar 2024)) further accelerate convergence and enlarge certified ROA, outperforming fixed-size ReLU certificates.

4. Certified Training and Verification-friendly Controller Construction

Recent advances integrate certified training frameworks to facilitate verification and maximize ROA. The branch-and-bound (BaB) method dynamically partitions the input space during neural policy/Lyapunov training, adapting regions where verification bounds are tightest (Shi et al., 27 Nov 2024).

Framework summary:

• Define a single scalar violation function $g_\theta(x)$ for Lyapunov and invariance properties.
• At each training iteration, for a box $[\underline{x}, \overline{x}]$, compute differentiable verified bounds $\bar{g}$ (e.g. via CROWN+IBP linear relaxations), and adversarial PGD-maximized $g^A$.
• Split the hardest (least certified) boxes along dimensions that most reduce cumulative loss.
• Train NNs to minimize $L_\mathrm{box}$; after training, use αβ-CROWN for global verification.
• Empirical results on 2D quadrotor and pendulum tracking systems show certified ROAs up to $16\times$ larger and verification speeds $5\times$ faster than CEGIS baselines.

CT-BaB enables the construction of neural controllers that are simultaneously high-performing and verification-friendly across large input regions.

5. Advanced Methods: Semialgebraic, SOS, SMT and Incremental Stability

To overcome conservatism and accommodate more complex nonlinear systems or activation functions, high-dimensional approaches utilize:

• Semialgebraic input-output neural modeling (Detailleur et al., 28 Oct 2025), introducing polynomial graph constraints for each network neuron, enabling direct SOS/SDP search for Lyapunov functions and certified ROA for feedforward/Recurrent Equilibrium Networks (RENs), including custom models for softplus/tanh activations.
• Higher-order sum-of-squares multipliers and robust SDP-based certificates accommodate plant/model uncertainty and yield enlarged ROA compared to prior sector/IQC bounds (Newton et al., 2022).
• Physics-informed PINN approaches solve Zubov-HJB PDEs, yielding Lyapunov-CRFs that are formally verified by SMT (e.g. dReal, Z3) with explicit null-controllability region characterizations (Liu et al., 30 Sep 2024).
• For model-free or black-box systems, incremental input-to-state stability ($\delta$-ISS-CLF) networks validate global asymptotic convergence robustness via Lipschitz continuity and scenario convex programming (Basu et al., 25 Apr 2025).
• Data-driven stable dynamics identification (CoILS (Min et al., 2023)) and learning-based methods for switched systems (Debauche et al., 2023) demonstrate that neural architectures can match or outperform polynomial/SDP certificates—subject to sufficient sampling and careful regularization.

6. Experimental Results and Quantitative Performance

Representative benchmarks consistently show numerically validated, certified regions of attraction that substantially exceed those of classical LQR or sector-bounded designs:

System	Neural Region (ROA)	LQR / SDP / SOS	Speedup / Benefit
Inverted pendulum	3x–6x larger	ROA ≈ 0.2	MILP <10s; CT-BaB 5x faster
Path tracking	Order-of-mag.	ROA ≈ 0.9	ROA ≈ 9 (DITL); ≈16x (CT-BaB)
Quadrotor (2D,3D)	$>$10–100x	NA/unstable	All tested initial conditions stabilize
Van der Pol / Unicycle	Larger ROA vs LQR	–	Neural Lyapunov matches/exceeds SOS
Multicopter VB Interc.	$15\,\mathrm{m/s}$	– (no LQR)	Stability certified via "almost Lyap."

Control policies maintain stability and robustness despite plant uncertainties, actuator limits, or noisy measurements. Training times for Lyapunov NN certification are competitive (minutes to hours for 6–12D systems), with sampling, verification, and min–max optimization dominating cost.

7. Limitations, Scalability, and Prospects

Despite substantial advances, several practical limitations remain:

• Scalability to very high-dimensional ($d>12$) systems is limited by exponential partitioning and verification costs.
• Sampling-based enforcement may miss rare violation pockets; global guarantees require dense covering and Lipschitz/regularization bounds.
• Nonconvex optimization and counterexample-mining introduce runtime variance; tight integration of SDP/MILP/SMT solvers with neural architectures is evolving.
• Extensions to output-feedback, time-delay, and partially observed systems are active research areas. Incorporation of recurrent/temporal NNs and richer physical models is ongoing.

The field continues to evolve toward fully data-driven, model-free design pipelines that produce neural controllers with explicit, formally verified Lyapunov stability certificates and maximal regions of attraction—enabling deployment in safety-critical, performance-driven applications.