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To address PDEs in the feedback loop, a boundary control law and a re-design of the additive probing signal are mandatory. The latter, commonly called \u2018trajectory generation\u2019 or \u2018motion planning\u2019, involves designing perturbation signals that anticipate their propagation through PDEs. Specifically, this requires solving motion planning problems for systems governed by parabolic and hyperbolic PDEs. Physics-informed neural networks (PINNs) is a powerful tool for solving PDEs by embedding physical laws as constraints in the neural network\u2019s loss function, enabling efficient solutions for high-dimensional, non-linear and complex problems. This paper proposes a novel construction integrating PINN and ES, automating the motion planning process for specific PDE systems and eliminating the need for case-by-case analytical derivations. The proposed strategy efficiently extracts perturbation signals, optimizing the PDE system.<\/jats:p>","DOI":"10.1093\/imamci\/dnaf046","type":"journal-article","created":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T13:01:04Z","timestamp":1765544464000},"source":"Crossref","is-referenced-by-count":0,"title":["Extremum seeking boundary control for PDE systems using PINNs"],"prefix":"10.1093","volume":"43","author":[{"given":"Haojin","family":"Guo","sequence":"first","affiliation":[{"name":"Institute of Precision Guidance and Control, School of Astronautics, Northwestern Polytechnical University , Xi\u2019an 710072, Shaanxi ,","place":["China"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zongyi","family":"Guo","sequence":"additional","affiliation":[{"name":"Institute of Precision Guidance and Control, School of Astronautics, Northwestern Polytechnical University , Xi\u2019an 710072, Shaanxi ,","place":["China"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianguo","family":"Guo","sequence":"additional","affiliation":[{"name":"Institute of Precision Guidance and Control, School of Astronautics, Northwestern Polytechnical University , Xi\u2019an 710072, Shaanxi ,","place":["China"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tiago Roux","family":"Oliveira","sequence":"additional","affiliation":[{"name":"Department of Electronics and Telecommunication Engineering, State University of Rio de Janeiro , Rio de Janeiro, RJ 20550-900 \u00a0","place":["Brazil"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2026,1,10]]},"reference":[{"key":"2026031206213343500_ref1","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.automatica.2006.07.021","article-title":"Parameter convergence in adaptive extremum-seeking control","volume":"43","author":"Adetola","year":"2007","journal-title":"Automatica"},{"key":"2026031206213343500_ref2","doi-asserted-by":"crossref","first-page":"5310","DOI":"10.1109\/TAC.2023.3347499","article-title":"Neural operators for bypassing gain and control computations in PDE backstepping","volume":"69","author":"Bhan","year":"2024","journal-title":"IEEE Trans. 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