Lyapunov Exponents as Rewards: RL Revisits Inverted Pendulum Stabilization

Introduction

The inverted pendulum is one of the most familiar testbeds in control theory and reinforcement learning. Its appeal is simple: the upright position is naturally unstable, so even small perturbations can quickly lead to failure. The arXiv paper “Lyapunov Exponent as Physics-Informed Dense Reward” revisits this benchmark in a more physically structured setting: stabilizing an inverted pendulum through vertical motion.

The main idea is not merely to train an agent that keeps the pendulum up. Instead, the paper proposes using the Lyapunov characteristic exponent, or LCE, as a dense reward signal. Since Lyapunov exponents describe how sensitive a dynamical trajectory is to small perturbations, they are closely tied to the notion of stability that the control problem is trying to achieve.

Key points

• A physics-informed reward signal: Many reinforcement learning control tasks depend on hand-designed rewards, such as penalties for angular deviation or bonuses for staying alive. This work suggests that LCE can provide a more principled feedback signal because it reflects the stability properties of the system itself.
• A connection to the Kapitza pendulum: The target system relates to the famous Kapitza pendulum, where rapid vertical oscillation can stabilize the otherwise unstable inverted position. This phenomenon is a classic example of counterintuitive stabilization in nonlinear dynamics.
• Discovery and extension of known behavior: According to the abstract, the RL agent successfully found the oscillatory motion associated with the Kapitza pendulum. More notably, it also damped the pivoting motion, leaving the pendulum in a strictly upright position.
• Dense feedback for difficult exploration: Sparse rewards often make control learning brittle because the agent receives useful information only after success or failure. A continuous stability-related measure may provide richer guidance during exploration.

Why it matters

The paper points toward a broader theme: reinforcement learning for physical systems can benefit from reward functions grounded in dynamical systems theory. Rather than encoding only surface-level objectives, researchers can incorporate quantities that directly measure stability, sensitivity, or energy structure.

For robotics and AI control, this is potentially valuable. Real-world reward design is difficult, and unsafe exploration can be expensive. If a stability metric such as the Lyapunov characteristic exponent can serve as an effective learning signal, it may help agents discover more interpretable and physically meaningful control strategies.

That said, the available material is limited to the abstract and arXiv metadata. It does not provide enough detail to assess the exact algorithm, the LCE computation procedure, the training setup, or the robustness of the results. The claim is therefore best read as an intriguing research direction rather than a fully evaluated general solution.

Source: arXiv