{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:28:23Z","timestamp":1787318903448,"version":"3.56.0"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"name":"I-Form"},{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001602","name":"Science Foundation Ireland","doi-asserted-by":"publisher","award":["16\/RC\/3872"],"award-info":[{"award-number":["16\/RC\/3872"]}],"id":[{"id":"10.13039\/501100001602","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2022,2]]},"abstract":"<jats:p>This paper is concerned with the proportional integral (PI) regulation control of the left Neumann trace of a one-dimensional semilinear wave equation. The control input is selected as the right Neumann trace. The control design goes as follows. First, a preliminary (classical) velocity feedback is applied in order to shift all but a finite number of the eivenvalues of the underlying unbounded operator into the open left half-plane. We then leverage the projection of the system trajectories into an adequate Riesz basis to obtain a truncated model of the system capturing the remaining unstable modes. The controller is computed by applying a classical PI control design scheme to this truncated model. Local stability of the resulting closed-loop infinite-dimensional system is obtained through the study of an adequate Lyapunov function. Finally, an estimate assessing the set point tracking performance of the left Neumann trace is derived.<\/jats:p>","DOI":"10.1137\/20m1346857","type":"journal-article","created":{"date-parts":[[2022,1,4]],"date-time":"2022-01-04T10:23:24Z","timestamp":1641291804000},"page":"1-21","source":"Crossref","is-referenced-by-count":11,"title":["Proportional Integral Regulation Control of a One-Dimensional Semilinear Wave Equation"],"prefix":"10.1137","volume":"60","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2782-4208","authenticated-orcid":true,"given":"Hugo","family":"Lhachemi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4456-2019","authenticated-orcid":true,"given":"Christophe","family":"Prieur","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0656-7003","authenticated-orcid":true,"given":"Emmanuel","family":"Tr\u00e9lat","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,1,4]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1982.1103023"},{"key":"atypb2","unstructured":"K. J. \\AAstr\u00f6m and T. H\u00e4gglund,\n                      PID Controllers: Theory, Design, and Tuning\n                      , 2nd ed., Instrument Society of America, Research Triangle Park, NC, 1995."},{"key":"atypb3","unstructured":"K. J. \\AAstr\u00f6m and R. M. Murray,\n                      Feedback Systems: An Introduction for Scientists and Engineers\n                      , Princeton University Press, Princeton, NJ, 2021."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1109\/TCST.2019.2943458"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.4310\/MAA.2019.v26.n2.a1"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2014.12.025"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2017.12.007"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2019.2915003"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/S036301290342471X"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1142\/S0219199706002209"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1109\/9.566660"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2007.09.022"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1109\/TCST.2014.2299407"},{"key":"atypb14","unstructured":"I. Gohberg and M. G. Krein,\n                      Introduction to the Theory of Linear Nonselfadjoint Operators\n                      , vol. 18, American Mathematical Society, Providence, RI, 1978."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012999354880"},{"key":"atypb16","unstructured":"A. Hayat,\n                      PI Controller for the General Saint-Venant Equations\n                      , preprint, hal-01827988, 2019."},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"P.O. Lamare and N. Bekiaris-Liberis,\n                      Control of 2 x 2 linear hyperbolic systems: Backstepping-based trajectory generation and PI-based tracking\n                      , Systems Control Lett., 86 (2015), pp. 24-33.","DOI":"10.1016\/j.sysconle.2015.09.009"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"H. Lhachemi and C. Prieur,\n                      Finite-dimensional observer-based PI regulation control of a reaction-diffusion equation\n                      , IEEE Trans. Automat. Control, (2021),https:\/\/doi.org\/10.1109\/TAC.2021.3130874.","DOI":"10.1109\/TAC.2020.2996598"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2020.2975003"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2021.109955"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2020.2996598"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2020.104651"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1137\/0330033"},{"key":"atypb24","unstructured":"A. Pazy,\n                      Semigroups of linear operators and applications to partial differential equations\n                      , vol. 44, Springer, New York, 2012."},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1982.1102887"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(85)90239-2"},{"key":"atypb27","first-page":"3895","author":"Rodrigues M.","year":"2013","journal-title":"IEEE"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/1020095"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2006.5.201"},{"key":"atypb30","first-page":"4631","author":"Terrand-Jeanne A.","year":"2018","journal-title":"Miami Beach, FL"},{"key":"atypb31","unstructured":"A. Terrand-Jeanne, V. Andrieu, V. D. S. Martins, and C.Z. Xu,\n                      Adding Integral Action for Open-Loop Exponentially Stable Semigroups and Application to Boundary Control of PDE Systems\n                      , preprint, arXiv:1901.02208, 2019,https:\/\/arxiv.org\/abs\/1901.02208."},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2019.2907792"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2017.2666040"},{"key":"atypb34","doi-asserted-by":"crossref","unstructured":"M. Tucsnak and G. Weiss,\n                      Observation and Control for Operator Semigroups\n                      , Springer Science & Business Media, New York, 2009.","DOI":"10.1007\/978-3-7643-8994-9"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1080\/00207179508921891"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.3934\/mcrf.2014.4.501"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1346857","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:41:53Z","timestamp":1787316113000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1346857"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,4]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["10.1137\/20M1346857"],"URL":"https:\/\/doi.org\/10.1137\/20m1346857","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,4]]}}}