{"ID":2899620,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.00695","arxiv_id":"2507.00695","title":"A Test-Function Approach to Incremental Stability","abstract":"This paper presents a novel framework for analyzing Incremental-Input-to-State Stability ($δ$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 Hölder-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.","short_abstract":"This paper presents a novel framework for analyzing Incremental-Input-to-State Stability ($δ$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...","url_abs":"https://arxiv.org/abs/2507.00695","url_pdf":"https://arxiv.org/pdf/2507.00695v2","authors":"[\"Daniel Pfrommer\",\"Max Simchowitz\",\"Ali Jadbabaie\"]","published":"2025-07-01T11:46:52Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"eess.SY\"]","methods":"[\"Reinforcement Learning\"]","has_code":false}