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arXiv · 2507.00695

A Test-Function Approach to Incremental Stability

Daniel Pfrommer·Max Simchowitz·Ali Jadbabaie
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Abstract

This paper presents a novel framework for analyzing Incremental-Input-to-State Stability ($\delta$ISS) based on the idea of using rewards as "test functions." Whereas control theory traditionally deals with Lyapunov functions that satisfy a time-decrease condition, reinforcement learning (RL) value functions are constructed by exponentially decaying a Lipschitz reward function that may be non-smooth and unbounded on both sides. Thus, these RL-style value functions cannot be directly understood as Lyapunov certificates. We develop a new equivalence between a variant of incremental input-to-state stability of a closed-loop system under given a policy, and the regularity of RL-style value functions under adversarial selection of a H\"older-continuous reward function. This result highlights that the regularity of value functions, and their connection to incremental stability, can be understood in a way that is distinct from the traditional Lyapunov-based approach to certifying stability in control theory.

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Daniel Pfrommer, Max Simchowitz, Ali Jadbabaie. 2025-07-01. A Test-Function Approach to Incremental Stability. https://arxiv.org/abs/2507.00695

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