{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T15:28:25Z","timestamp":1785511705215,"version":"3.56.0"},"reference-count":48,"publisher":"Optica Publishing Group","issue":"1","license":[{"start":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T00:00:00Z","timestamp":1767312000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/doi.org\/10.1364\/OA_License_v2#VOR-OA"},{"start":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T00:00:00Z","timestamp":1767312000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/opg.optica.org\/policies\/opg-tdm-policy.json"}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"short-container-title":["Opt. Express"],"published-print":{"date-parts":[[2026,1,12]]},"abstract":"<jats:p>Diffractive optical networks (DONs), characterized by light-speed computation and low power consumption, represent a promising platform for artificial intelligence (AI) implementation, with current applications primarily focused on object recognition and image classification. However, their potential for continuous, provably stable nonlinear control remains largely unexplored. This work introduces a Lyapunov-constrained reinforcement learning diffractive-optical network (LC-RLDON) framework designed for stability control of continuous nonlinear dynamical systems that challenge conventional control strategies. The proposed approach integrates reinforcement learning with differentiable Lyapunov conditions to iteratively optimize the DON toward an optimal Bellman policy, guaranteeing closed-loop stability while overcoming the cumulative drift problems inherent in behavior cloning methods. During deployment, the system requires only a passive DON and a lightweight electronic linear layer to form the optical Actor, enabling highly efficient real-time inference. Unlike previous DON-based controllers limited to binary tasks, LC-RLDON directly handles continuous-variable control scenarios. Experimental results on underactuated rotary inverted pendulums demonstrate LC-RLDON's superior performance by achieving stable equilibrium in 2.8 seconds and recovering within 2.1 seconds from a 1.1 N disturbance, while behavior cloning consistently fails. These findings establish DONs as capable of delivering real-time, formally safe control, paving the way for practical implementation in low-power, high-performance intelligent systems for agile robotics and autonomous vehicles.<\/jats:p>","DOI":"10.1364\/oe.574487","type":"journal-article","created":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T07:00:09Z","timestamp":1765782009000},"page":"15","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Verifiably stable nonlinear control with reinforcement-learned diffractive optical networks"],"prefix":"10.1364","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-4968-9556","authenticated-orcid":true,"given":"Mingliang","family":"Xie","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04qr3zq92","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6488-1682","authenticated-orcid":true,"given":"Xiren","family":"Zhang","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04qr3zq92","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinghui","family":"Cai","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04qr3zq92","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yisong","family":"Yue","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04qr3zq92","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"285","published-online":{"date-parts":[[2026,1,2]]},"reference":[{"key":"oe-34-1-15-R1","doi-asserted-by":"publisher","first-page":"eade4538","DOI":"10.1126\/scirobotics.ade4538","type":"journal-article","volume":"8","author":"Chen","year":"2023","journal-title":"Sci. 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