MSACL: Multi-Step Actor-Critic Learning with Lyapunov Certificates for Exponentially Stabilizing Control

Achieving provable stability in model-free reinforcement learning (RL) remains a challenge, particularly in balancing exploration with rigorous safety. This article introduces MSACL, a framework that integrates exponential stability theory with maximum entropy RL through multi-step Lyapunov certificate learning. Unlike methods relying on complex reward engineering, MSACL utilizes off-policy multi-step data to learn Lyapunov certificates satisfying theoretical stability conditions. By introducing Exponential Stability Labels (ESL) and a $λ$-weighted aggregation mechanism, the framework effectively balances the bias-variance trade-off in multi-step learning. Policy optimization is guided by a stability-aware advantage function, ensuring the learned policy promotes rapid Lyapunov descent. We evaluate MSACL across six benchmarks, including stabilization and nonlinear tracking tasks, demonstrating its superiority over state-of-the-art Lyapunov-based RL algorithms. MSACL achieves exponential stability and rapid convergence under simple rewards, while exhibiting significant robustness to uncertainties and generalization to unseen trajectories. Sensitivity analysis establishes the multi-step horizon $n=20$ as a robust default across diverse systems. By linking Lyapunov theory with off-policy actor-critic frameworks, MSACL provides a foundation for verifiably safe learning-based control. Source code and benchmark environments will be made publicly available.