Lyapunov Exponent as Physics-Informed Dense Reward: RL Discovery of Stabilization Beyond the Kapitza Pendulum

Reinforcement learning (RL) has become the de facto method for achieving locomotion on humanoid robots in practice, yet stability analysis of the corresponding control policies is lacking. Recent work has attempted to merge control theoretic ideas with reinforcement learning through control guided learning. A notable example of this is the use of a control Lyapunov function (CLF) to synthesize the reinforcement learning rewards, a technique known as CLF-RL, which has shown practical success. This paper investigates the stability properties of optimal controllers using CLF-RL with the goal of bridging experimentally observed stability with theoretical guarantees. The RL problem is viewed as an optimal control problem and exponential stability is proven in both continuous and discrete time using both core CLF reward terms and the additional terms used in practice. The theoretical bounds are numerically verified on systems such as the double integrator and cart-pole. Finally, the CLF guided rewards are implemented for a walking humanoid robot to generate stable periodic orbits.

Safe reinforcement learning (RL) typically asks $\textit{what}$ an agent should do. We ask $\textit{when}$ it needs to act, and show that a single policy can jointly learn control inputs and communication-efficient timing decisions under a pointwise Lyapunov safety shield. We focus on stabilization around a known equilibrium, where CARE-based LQR backups, Lyapunov certificates, and classical Lyapunov-STC are well defined, enabling clean comparison against analytical baselines. A run-time assurance (RTA) layer overrides the policy via a one-step-ahead Lyapunov prediction and a precomputed LQR backup, providing a strictly stronger guarantee than constrained MDP methods that enforce safety only in expectation. On an inverted pendulum, cart--pole, and planar quadrotor, the learned policy achieves $1.91\times$, $1.45\times$, and $3.51\times$ higher mean inter-sample interval (MSI) than a Lyapunov-triggered baseline; a fixed LQR controller at the same average rate is unstable on all three plants, showing that adaptive timing, not a lower average rate, makes sparsity safe. A CARE-derived Lyapunov reward transfers across environments without redesign, with a single weight $w_c$ controlling the stability--communication tradeoff; ablations confirm the RTA shield is essential, with its removal reducing MSI by $1.27$--$1.84\times$ and degrading state norms. A preference-conditioned extension recovers the full tradeoff frontier from one model at $\tfrac{2}{11}$ of training compute, and SAC experiments show the results are algorithm-agnostic across discrete and continuous domains. A 12-state 3D quadrotor case study extends the framework to higher-dimensional systems where classical STC is intractable, and robustness to $\pm30\%$ mass variation and disturbances shows graceful degradation, with the RTA absorbing what the learned policy cannot.

We propose KL-Coupled Policy Regularization (KCPR), a policy coordination framework for Reward-Punishment Reinforcement Learning (RPRL). Based on KCPR, we derive KL-Coupled Soft Optimality (KCSO) and develop its deep realization, klDMP. Unlike existing RPRL approaches that optimize reward-seeking and punishment-related policies largely independently, KCPR enables direct interactions between companion policies by treating each as a dynamically learned prior for the other. KCSO yields coupled soft-optimal policies and KL-regularized Bellman operators, allowing reward and punishment information to jointly influence value propagation. To improve learning stability, we introduce a companion-prior softening mechanism and evaluate separate replay-buffer designs for balancing reward- and punishment-related experience. Experiments in grid-world and Gazebo robotic navigation tasks demonstrate that klDMP improves safety and learning stability while maintaining competitive task performance compared with DQN, SQL and softDMP. These results suggest that policy-level coordination provides an effective mechanism for integrating multiple behavioral objectives and may serve as a useful design principle for reinforcement learning systems with interacting motivational processes.