Lyapunov-Assisted Deep Reinforcement Learning

• Lyapunov-Assisted DRL is a method integrating classical Lyapunov stability theory with deep reinforcement learning to ensure safety and stability in dynamic systems.
• It employs reward shaping, constrained policy improvement, and formal verification techniques to overcome the instability and sample inefficiency challenges of standard DRL.
• Empirical studies in robotics, safe navigation, and multi-agent control demonstrate that these approaches deliver enhanced convergence, safety guarantees, and robust performance.

Lyapunov-Assisted Deep Reinforcement Learning (DRL) comprises a class of methodologies that integrate Lyapunov stability theory—classically used for certifying the stability of nonlinear dynamical systems—into deep reinforcement learning pipelines. These techniques are designed to overcome the inherent instability, lack of safety guarantees, and sample inefficiency of standard DRL, especially in robotic and control scenarios where well-grounded stability or constraint satisfaction is crucial. Lyapunov-based mechanisms serve to shape rewards, design constraints, guide exploration, or enable formal guarantees for the closed-loop system behavior.

1. Foundational Principles and Motivation

Lyapunov stability theory provides conditions under which trajectories of a dynamical system converge to a desired equilibrium (typically the origin). A Lyapunov function $V(x)$ satisfies $V(0)=0$, $V(x)>0$ for $x\neq 0$, and exhibits a negative definite decrease along trajectories, either via a continuous-time Lie derivative $L_f V<0$ or a discrete-time decrement $V(f(x,u))<V(x)$. Classical control synthesizes policies that directly enforce these properties to guarantee stability.

In DRL, direct stability guarantees are generally absent. Lyapunov-assisted DRL methodologies embed Lyapunov-theoretic elements by:

• Learning proxy Lyapunov functions from observed data or expert rollouts.
• Directly incorporating Lyapunov decrease penalties or constraints into the RL loss or reward.
• Using Lyapunov functions to define safe sets, restrict policy improvement, or guide exploration.
• Formal, model-based or model-free computation of Lyapunov certificates for the closed-loop system, even with high-dimensional policies parameterized by deep networks.

The result is a DRL agent whose behavior is both reward-seeking and provably stabilizing or safe with respect to system-theoretic criteria (Ganai et al., 2023, Westenbroek et al., 2022, Zhang et al., 2020, Wang et al., 2024).

2. Lyapunov Function Learning and Proxy Models

One family of approaches uses data-driven learning to obtain a Lyapunov-like function $V_\theta(x)$, parameterized as a neural network: - Given an expert dataset of state transitions $\tau_E = \{(s, s')\}$, a loss function is constructed to ensure that $V_\theta(0)=0$, $V_\theta(x)\geq 0$ (positivity), and $V_\theta$ decreases along expert rollouts:

$$\mathcal{L}_{\rm proxy}(\theta) = \mathbb{E}_{(s,s')\sim\tau_E} \Big[ V_\theta(0)^2 + \max(0,-V_\theta(s)) + \beta_1\big(c + \frac{V_\theta(s')-V_\theta(s)}{\Delta t}\big)^2 \Big]$$

• The finite-difference term over $\Delta t$ enforces a decrease rate akin to the Lie derivative (discrete approximation), capturing the expert's convergence rate (Ganai et al., 2023).

This proxy model can then be leveraged as a reward function:

$$r(s,s') = g(V_\theta(s)) + \beta_2\min\left(0, \frac{V_\theta(s)-V_\theta(s')}{\Delta t}\right)$$

where $g$ is a monotone decreasing function (e.g., $- \log(1-e^{-kx^2})$). This reward drives the agent to descend the learned "Lyapunov landscape" at the expert's rate, even with purely observational LfO data.

Neural Lyapunov models can also be explicitly trained to satisfy monotonicity and positivity over prescribed domains using structured architectures and mixed-integer optimization, enabling certified computation of region-of-attraction (ROA) estimates and controller–certificate pairs (Wang et al., 2024, Mehrjou et al., 2020).

3. Integration into Deep RL Algorithms

Lyapunov-assisted DRL pipelines modify standard RL training in several ways:

• Reward and Cost Shaping: The one-step reward is augmented with a Lyapunov-decrement term, as in

$$r(x, u) = -\ell(x, u) + \alpha \big[ W(F(x, u)) - W(x) \big]$$

where $W$ is a Control Lyapunov Function and $\alpha$ tunes the trade-off between nominal reward and stability (Westenbroek et al., 2022).

• Constrained Policy Improvement: Actor loss functions integrate Lyapunov decrease constraints via Lagrangian multipliers, e.g., in multi-agent decentralized MASAC:

$$J_\pi^L(\phi) = \mathbb{E}[... + \beta \Delta L_\phi(...)]$$

where $\Delta L_\phi$ represents the Lyapunov drift (Zhang et al., 2020).

• Safety Shields and Sets: Lyapunov functions are used to define or approximate safe subsets of state space, and policy improvement is restricted to maintain monotonic decrease within those sets (Huh et al., 2020, Zhang et al., 2020).
• Hierarchical or Hybrid Schemes: For system classes with mixed discrete–continuous controls, hierarchical DRL architectures split action selection and subtask optimization, with Lyapunov-type virtual queues providing constraint satisfaction (Bi et al., 2021, Long et al., 2024, Younesi et al., 29 Dec 2025).
• Residual Control: In model-based control fusion, a stable controller is designed (e.g., by LMI solutions) to guarantee Lyapunov decrease; DRL only supplies a "residual" correction, thus maintaining stability by construction (Cao et al., 2023).

4. Theoretical Guarantees and Stability Analysis

The integration of Lyapunov arguments provides various formal assurances:

• Closed-Loop Stability: Provided $V(0)=0$, $V(x)>0$, and $V$ decreases along system trajectories, the origin is asymptotically stable by classical Lyapunov stability theorems (Westenbroek et al., 2022, Cao et al., 2023).
• Safety Constraints: When Lyapunov decrease is enforced in a prescribed set, sublevel sets are forward invariant—trajectories remain safe even under model uncertainty or high-dimensional policies (Xiong et al., 2022, Mandal et al., 2024).
• Sample Complexity Improvements: The Lyapunov term allows choice of small discount factor $\gamma$, reducing variance and stabilizing learning, especially in off-policy RL (Westenbroek et al., 2022).
• Formal Verification: For discrete-time policies parameterized by NNs, compositional Lyapunov-barrier certificates can be jointly trained and formally verified via neural network verification engines, furnishing both safety and liveness guarantees over large initial regions (Mandal et al., 2024).

Key analytic tools include:

• Drift–plus–penalty bounds (for queuing and constraint systems) (Bi et al., 2021, Bae et al., 2020, Xu et al., 4 Jun 2025, Younesi et al., 29 Dec 2025).
• Monotonicity and envelope arguments in region-of-attraction estimation (Mehrjou et al., 2020, Wang et al., 2024).
• Lagrangian relaxation and primal-dual updates to enforce Lyapunov constraints in high-dimensional actor–critic methods (Huh et al., 2020, Xiong et al., 2022).

5. Empirical Results and Application Domains

Lyapunov-assisted DRL has been empirically validated in:

• Continuous stabilization and robotics: LSO-LLPM (Lyapunov-proxy landscape) and CLF-augmented rewards enabled sample-efficient stabilization of acrobot, quadrotor, hopper, walker, and path-tracking robots, often matching or exceeding previous LfO or imitation-learned methods with as few as 10 expert trajectories (Ganai et al., 2023, Westenbroek et al., 2022).
• Safe navigation: Co-learning of control policy and twin neural Lyapunov functions delivered near-zero collision rates and high reachability in high-dimensional simulated navigation, surpassing reward-constraint and penalty-only baselines (Xiong et al., 2022).
• Decentralized multi-agent control: Incorporating Lyapunov decrement constraints into multi-agent Soft Actor-Critic drastically improved policy reliability and cooperative convergence without access to analytic dynamics (Zhang et al., 2020).
• Queuing and resource allocation: Drift–plus–penalty rewards driven by Lyapunov functions facilitated stable queue management in edge computing and routing, outperforming classical DPP and pure RL schemes, including in split-edge-cloud systems serving LLMs (Bi et al., 2021, Younesi et al., 29 Dec 2025, Xu et al., 4 Jun 2025).
• Robustness and formal verification: Lyapunov exponent regularization in Dreamer V3 reduced closed-loop chaos, yielding greater robustness to observation noise and adversarial perturbations (Young et al., 2024). Formal neural Lyapunov-barrier certificates enabled scalable formal safety and liveness certification of DRL policies for spacecraft control (Mandal et al., 2024).

6. Limitations and Open Directions

Despite significant progress, Lyapunov-assisted DRL faces several challenges:

• Function Approximation: Guaranteeing Lyapunov conditions globally on the state space with neural function approximators remains nontrivial, especially in the presence of function mismatch or unmodeled dynamics (Ganai et al., 2023, Wang et al., 2024, Mandal et al., 2024).
• Region of Attraction Estimation: Efficiently enlarging and certifying the domain of attraction is computationally intensive for high-dimensional systems, though methods involving monotonic-layer NNs and MILP verification provide scalable pathways (Wang et al., 2024, Mehrjou et al., 2020).
• Policy Explorativity vs. Safety: There is a trade-off between reward maximization and conservatism induced by Lyapunov constraints. Excessive Lyapunov penalty can hamper exploration; insufficient weight can lead to instability (Westenbroek et al., 2022, Zhang et al., 2020).
• Compositionality and Modular Guarantees: Techniques exploiting compositional certificates or graph-based controller assembly suggest promising directions for safe motion planning and large-region feedback synthesis (Ghanbarzadeh et al., 2023, Mandal et al., 2024).
• Integration with Model-Free DRL: Not all methodologies seamlessly accommodate arbitrary model-free DRL frameworks; some require hybrid, hierarchical, or residual-control architectures to balance practicality and provable properties (Cao et al., 2023, Younesi et al., 29 Dec 2025).

7. Representative Algorithms and Empirical Comparisons

Below is a table summarizing principal methodologies, their Lyapunov mechanisms, and application domains.

Approach	Lyapunov Mechanism	Application Domain
LSO-LLPM (Ganai et al., 2023)	Learned V_θ proxy, reward shaping	Observation-based robotic stabilization
SAC+CLF (Westenbroek et al., 2022)	Control Lyapunov in cost	Cartpole/quadruped/biped real robots
MASAC+Lyap (Zhang et al., 2020)	Drift penalty in SAC actor loss	Multi-agent navigation, decentralized MARL
LyDROO (Bi et al., 2021)	Virtual-queue Lyapunov drift	Mobile-edge computation offloading
Model-Free Neural Lyapunov (Xiong et al., 2022)	Co-learned TNLF, region monitor	High-dimensional safe navigation
Monotonic Lyapunov NN (Wang et al., 2024)	Structured V(x), MILP-based certif.	Globally certified nonlinear stabilization
Lyapunov Barrier Cert (Mandal et al., 2024)	RWA+Barrier function, formal SMT	DRL-controlled safety/liveness verification
Splitwise (Younesi et al., 29 Dec 2025)	Drift-plus-penalty in reward	Adaptive LLM edge-cloud partitioning

These techniques consistently report enhanced stability, safety, constraint satisfaction, or sample efficiency compared to reward-only or unconstrained DRL, and in several robotics and resource allocation benchmarks, outperform the best prior baselines both in speed of convergence and ultimate performance.

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References:

(Ganai et al., 2023, Westenbroek et al., 2022, Zhang et al., 2020, Bi et al., 2021, Xiong et al., 2022, Wang et al., 2024, Mandal et al., 2024, Ghanbarzadeh et al., 2023, Younesi et al., 29 Dec 2025, Cao et al., 2023, Young et al., 2024, Xu et al., 4 Jun 2025, Bae et al., 2020, Mehrjou et al., 2020)