LC-SAC: Lyapunov-Constrained Soft Actor-Critic via Koopman Operator Theory for Trajectory Tracking and Stabilization

Reinforcement Learning (RL) has achieved remarkable success in solving complex sequential decision-making problems. However, its application to safety-critical physical systems remains constrained by the lack of stability guarantees. Standard RL algorithms prioritize reward maximization, often yielding policies that may induce oscillations or unbounded state divergence. In this work we propose a Lyapunov-Constrained Soft Actor-Critic (LC-SAC) algorithm using Koopman operator theory. We learn a linear lifted surrogate of the error dynamics via Extended Dynamic Mode Decomposition (EDMD) and solve the Discrete Algebraic Riccati Equation (DARE) to obtain a closed-form quadratic candidate Control Lyapunov Function (CLF). This CLF is incorporated into the SAC actor update as a Lagrangian penalty that aggregates the worst-case tail of violations via a Conditional Value-at-Risk (CVaR) objective, concentrating constraint pressure on rare but severe instability events. We further introduce three structural EDMD refinements spectral-radius normalization of the lifted A-matrix prior to the DARE solve, a physically meaningful LQR state cost, and a value-bias anchor enforcing V(0)=0 that make the closed-form CLF well-posed for higher-dimensional lifted models such as the cartpole and 3D quadrotor. The ablation study shows that a hard Lagrangian constraint is essential, replacing it with reward shaping (Lyap-RS-SAC) destabilizes learning and collapses return on quadrotor tasks.