{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T16:22:22Z","timestamp":1776183742589,"version":"3.50.1"},"reference-count":31,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2022,1,22]],"date-time":"2022-01-22T00:00:00Z","timestamp":1642809600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,3,2]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this work, an observer-based output feedback controller for a one-dimensional wave equation with van der Pol type nonlinear boundary conditions is considered. The uncontrolled system with the energy injection and the cubic velocity nonlinearity reveals different kinds of dynamical behaviours, for example, chaotic acoustic vibration, period-doubling bifurcation, square wave and some other dynamical patterns. Firstly, an observer is designed, and next, we propose an observer-based output feedback controller using the backstepping approach. We prove well-posedness and globally asymptotic stability of the observer error and corresponding closed-loop system. To this purpose, pre-compactness of the trajectories is shown for the large-time analysis. The validity of the theoretical results is illustrated by some numerical simulations.<\/jats:p>","DOI":"10.1093\/imamci\/dnab045","type":"journal-article","created":{"date-parts":[[2021,12,4]],"date-time":"2021-12-04T12:09:00Z","timestamp":1638619740000},"page":"295-321","source":"Crossref","is-referenced-by-count":6,"title":["Observer-based global output feedback controller design for an unstable wave PDE with nonlinear boundary condition"],"prefix":"10.1093","volume":"39","author":[{"given":"Najmeh","family":"Ghaderi","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 41938-33697, Rasht, Iran"}]},{"given":"Mohammad","family":"Keyanpour","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 41938-33697, Rasht, Iran"},{"name":"Center of Excellence for Mathematical Modeling, Optimization and Combinatorial Computing (MMOCC), University of Guilan, P.O. Box 41938-33697, Rasht, Iran"}]}],"member":"286","published-online":{"date-parts":[[2022,1,22]]},"reference":[{"key":"2022030218251512100_ref1","volume-title":"Introduction to the Theory of Fourier\u2019s Series and Integrals","author":"Carslaw","year":"1930"},{"key":"2022030218251512100_ref2","doi-asserted-by":"crossref","first-page":"4265","DOI":"10.1090\/S0002-9947-98-02022-4","article-title":"Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part I: Controlled hysteresis","volume":"350","author":"Chen","year":"1998","journal-title":"Trans. Amer. Math. 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