Explainable Stability Certification of Reinforcement Learning Control Policies Using Sparse Dynamics Identification

Reinforcement learning (RL) is a promising alternative to classical guidance and control methods; however, the black-box nature of deep neural network policies and the lack of interpretable stability evidence remain barriers to real-world aerospace adoption. This paper presents an a posteriori methodology using Sparse Identification of Nonlinear Dynamics (SINDy) to recover sparse analytical representations of the closed-loop dynamics induced by a trained RL controller, with the goal of certifying its stability. When the uncontrolled dynamics and input map are known, the identified model provides an explicit analytical approximation of the state-feedback control law, offering functional explainability through interpretable state couplings and nonlinear terms, as well as a lightweight surrogate for real-time deployment. In parallel, an analytical approximation of the positive cost to go is obtained as a candidate Lyapunov function. The Lyapunov conditions are evaluated for both the original RL policy and the reconstructed analytical controller over a bounded operating domain, while a Probably Approximately Correct (PAC) bound quantifies confidence in finite-sample verification. Demonstrations on a spring-mass oscillator, spacecraft attitude control, and asteroid hovering show that sparse analytical laws can reproduce and explain trained RL controllers, while PAC-supported Lyapunov analysis provides a principled framework for their systematic stability certification.