{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T06:26:37Z","timestamp":1773037597645,"version":"3.50.1"},"reference-count":34,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T00:00:00Z","timestamp":1698278400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,12,24]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This study presents a novel approach for the state and output feedback exponential control of cascaded unstable ODE-wave equations with nonlinear boundary condition. The new approach can be also used to investigate the state and output feedback stabilization of other cascaded ODE-partial differential equation (PDE) systems with nonlinear boundary condition. The interconnection between the ODE and wave PDE with the nonlinear boundary is bi-directional at two points. First, an exponential controller is planned and the well-posed and exponential stability of the closed-loop system is derived. Next, an exponential observer is designed, whereby the exponential stability of the overall error system is shown. Then, an observer-based output feedback control that causes the overall real system to be exponentially stable is built. The existence and exponential stability of the solution to the overall closed-loop system are further deduced. Finally, simulation results are given to verify the effectiveness, feasibility and validity of the proposed technique.<\/jats:p>","DOI":"10.1093\/imamci\/dnad027","type":"journal-article","created":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T11:55:23Z","timestamp":1698407723000},"page":"714-745","source":"Crossref","is-referenced-by-count":2,"title":["On a new method for the state and output feedback exponential stabilization of a cascaded wave PDE-ODE system with nonlinear boundary condition"],"prefix":"10.1093","volume":"40","author":[{"given":"Najmeh","family":"Ghaderi","sequence":"first","affiliation":[{"name":"Applied Mathematics department , University of Guilan, Rasht 4199613776 , Iran"}]}],"member":"286","published-online":{"date-parts":[[2023,10,26]]},"reference":[{"key":"2023122615292497200_ref1","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.automatica.2018.09.008","article-title":"Boundary observability of wave equations with van der Pol type boundary conditions","volume":"98","author":"Cai","year":"2018","journal-title":"Automatica J. IFAC"},{"key":"2023122615292497200_ref2","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.automatica.2016.01.043","article-title":"Nonlinear stabilization through wave PDE dynamics with a moving uncontrolled boundary","volume":"68","author":"Cai","year":"2016","journal-title":"Automatica J. IFAC"},{"key":"2023122615292497200_ref3","doi-asserted-by":"crossref","first-page":"4401","DOI":"10.1109\/TAC.2020.3046576","article-title":"Boundary control of nonlinear ODE\/wave PDE with spatially-varying propagation speed","volume":"66","author":"Cai","year":"2021","journal-title":"IEEE Trans. Automat. Contr."},{"key":"2023122615292497200_ref4","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1142\/S0218127402004504","article-title":"Nonisotropic spatiotemporal chaotic vibration of the wave equation duo to mixing energy transport and a van der Pol boundary condition","volume":"12","author":"Chen","year":"2002","journal-title":"Int. J. Bifurcation Chaos"},{"key":"2023122615292497200_ref5","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4899-2846-7","volume-title":"Nonlinear Partial Differential Equations for Scientists and Engineers","author":"Debnath","year":"1997"},{"key":"2023122615292497200_ref6","doi-asserted-by":"crossref","first-page":"109338","DOI":"10.1016\/j.automatica.2020.109338","article-title":"A backstepping approach to output regulation for coupled linear wave-ODE systems","volume":"123","author":"Deutscher","year":"2021","journal-title":"Automatica J. IFAC"},{"key":"2023122615292497200_ref7","doi-asserted-by":"crossref","first-page":"2436","DOI":"10.1137\/15M1045296","article-title":"Stabilization of one-dimensional wave equation with van der Pol type boundary condition","volume":"54","author":"Feng","year":"2016","journal-title":"SIAM J. Control Optim."},{"key":"2023122615292497200_ref8","doi-asserted-by":"crossref","first-page":"105273","DOI":"10.1016\/j.sysconle.2022.105273","article-title":"A novel observer-based approach for the exponential stabilization of the string PDE with cubic nonlinearities","volume":"165","author":"Ghaderi","year":"2022","journal-title":"Syst. Control Lett."},{"key":"2023122615292497200_ref9","doi-asserted-by":"crossref","first-page":"2017","DOI":"10.1002\/asjc.2104","article-title":"Backstepping design for a class of coupled parabolic PDEs with spatially varying coefficient","volume":"22","author":"Ghaderi","year":"2020","journal-title":"Asian J. Control"},{"key":"2023122615292497200_ref10","doi-asserted-by":"crossref","first-page":"100085","DOI":"10.1016\/j.ifacsc.2020.100085","article-title":"Output feedback stabilization for 1-D unstable wave equations with boundary control matched disturbance and van der pol nonlinear boundary","volume":"12","author":"Ghaderi","year":"2020","journal-title":"IFAC J. Syst. Contr."},{"key":"2023122615292497200_ref11","doi-asserted-by":"crossref","first-page":"3264","DOI":"10.1080\/00207179.2020.1757761","article-title":"Anti-collocated observer-based output feedback control of wave equation with cubic velocity nonlinear boundary and Dirichlet control","volume":"94","author":"Ghaderi","year":"2021","journal-title":"Internat. J. Control"},{"key":"2023122615292497200_ref12","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1093\/imamci\/dnab045","article-title":"Observer-based global output feedback controller design for an unstable wave PDE with nonlinear boundary condition","volume":"39","author":"Ghaderi","year":"2022","journal-title":"IMA J. Math. Control"},{"key":"2023122615292497200_ref13","doi-asserted-by":"crossref","first-page":"9154","DOI":"10.1016\/j.jfranklin.2020.06.028","article-title":"Observer-based finite-time output feedback control of heat equation with Neumann boundary condition","volume":"357","author":"Ghaderi","year":"2020","journal-title":"J. Franklin Inst. B"},{"key":"2023122615292497200_ref14","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1177\/01423312211032545","article-title":"Finite-time boundary stabilization of the reaction-diffusion system with switching time-delay input","volume":"44","author":"Ghaderi","year":"2022","journal-title":"Trans. Inst. Meas. Control"},{"key":"2023122615292497200_ref15","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/S0022-247X(03)00562-6","article-title":"A new characterization of nonisotropic chaotic vibrations of the one-dimensional linear wave equation with a van der Pol boundary condition","volume":"288","author":"Huang","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"2023122615292497200_ref16","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898718607","volume-title":"Boundary Control of PDEs: A Course on Backstepping Designs","author":"Krstic","year":"2008"},{"key":"2023122615292497200_ref17","doi-asserted-by":"crossref","first-page":"1362","DOI":"10.1109\/TAC.2009.2015557","article-title":"Compensating a string PDE in the actuation or sensing path of an unstable ODE","volume":"54","author":"Krstic","year":"2009","journal-title":"IEEE Trans. Automat. Contr."},{"key":"2023122615292497200_ref18","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1016\/j.sysconle.2009.01.006","article-title":"Compensating actuator and sensor dynamics governed by diffusion PDEs","volume":"58","author":"Krstic","year":"2009","journal-title":"Syst. Control Lett."},{"key":"2023122615292497200_ref19","first-page":"853","article-title":"Adaptive control of an anti-stable wave PDE. Dynamics of continuous, Discrete & Impulsive System","volume":"17","author":"Krstic","year":"2010","journal-title":"Dyn. Contin. Discrete Impuls. Syst. Ser. A: Math. Anal."},{"key":"2023122615292497200_ref20","doi-asserted-by":"crossref","first-page":"750","DOI":"10.1016\/j.sysconle.2008.02.005","article-title":"Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays","volume":"57","author":"Krstic","year":"2008","journal-title":"Syst. Control Lett."},{"key":"2023122615292497200_ref21","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.automatica.2007.05.012","article-title":"Output feedback stabilization of an unstable wave equation","volume":"44","author":"Krstic","year":"2008","journal-title":"Automatica J. IFAC"},{"key":"2023122615292497200_ref22","doi-asserted-by":"crossref","first-page":"1200","DOI":"10.1137\/11083037X","article-title":"Observer design for wave equations with van der Pol type boundary conditions","volume":"50","author":"Li","year":"2012","journal-title":"SIAM J. Control Optim."},{"key":"2023122615292497200_ref23","doi-asserted-by":"crossref","first-page":"1388","DOI":"10.1016\/j.jmaa.2017.01.095","article-title":"Chaotic oscillations of the 1D wave equation due to extreme imbalance of self-regulations","volume":"450","author":"Li","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"2023122615292497200_ref24","doi-asserted-by":"crossref","first-page":"3068","DOI":"10.1109\/TAC.2018.2874746","article-title":"Stabilization of one-dimensional wave equation with nonlinear boundary condition subject to boundary control matched disturbance","volume":"64","author":"Liu","year":"2019","journal-title":"IEEE Trans. Automat. Contr."},{"key":"2023122615292497200_ref25","volume-title":"Stability and Stabilisation of Infinite Dimensional Systems with Applications","author":"Luo","year":"1998"},{"key":"2023122615292497200_ref26","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1016\/j.sysconle.2013.03.003","article-title":"Stabilization of an ODE-Schr\u00f6dinger cascade","volume":"62","author":"Ren","year":"2013","journal-title":"Syst. Control Lett."},{"key":"2023122615292497200_ref27","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1016\/j.sysconle.2009.04.005","article-title":"Boundary control of an anti stable wave equation with anti-damping on the uncontrolled boundary","volume":"58","author":"Smyshlyaev","year":"2009","journal-title":"Syst. Control Lett."},{"key":"2023122615292497200_ref28","first-page":"540","article-title":"Control of PDE-ODE cascades with Neumann interconnections","volume":"347","author":"Susto","year":"2010","journal-title":"J. Frank. Inst."},{"key":"2023122615292497200_ref29","doi-asserted-by":"crossref","first-page":"109131","DOI":"10.1016\/j.automatica.2020.109131","article-title":"Delay-compensated control of sandwiched ODE-PDE-ODE hyperbolic systems for oil drilling and disaster relief","volume":"120","author":"Wang","year":"2020","journal-title":"Automatica J. IFAC"},{"key":"2023122615292497200_ref30","doi-asserted-by":"crossref","first-page":"110137","DOI":"10.1016\/j.automatica.2021.110137","article-title":"Delay-compensated event-triggered boundary control of hyperbolic PDEs for deep-sea construction","volume":"138","author":"Wang","year":"2022","journal-title":"Automatica J. IFAC"},{"key":"2023122615292497200_ref31","doi-asserted-by":"crossref","first-page":"1683","DOI":"10.1080\/00207179.2012.696704","article-title":"Boundary stabilization of a coupled wave-ODE system with internal anti-damping","volume":"85","author":"Zhou","year":"2016","journal-title":"Internat. J. Control"},{"key":"2023122615292497200_ref32","doi-asserted-by":"crossref","first-page":"2396","DOI":"10.1080\/00207179.2016.1158866","article-title":"Output feedback stabilization for a cascaded wave PDE-ODE system subject to boundary control matched disturbance","volume":"89","author":"Zhou","year":"2005","journal-title":"Internat. J. Control"},{"key":"2023122615292497200_ref33","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1016\/j.sysconle.2011.04.011","article-title":"State and output feedback boundary control for a coupled PDE\u2013ODE system","volume":"60","author":"Zhou","year":"2011","journal-title":"Syst. Control Lett."},{"key":"2023122615292497200_ref34","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1137\/S0036144503432862","article-title":"Propagation, observation, and control of waves approximated by finite difference methods","volume":"47","author":"Zuazua","year":"2005","journal-title":"SIAM Rev."}],"container-title":["IMA Journal of Mathematical Control and Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/imamci\/article-pdf\/40\/4\/714\/54804613\/dnad027.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/imamci\/article-pdf\/40\/4\/714\/54804613\/dnad027.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,26]],"date-time":"2023-12-26T15:29:51Z","timestamp":1703604591000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/imamci\/article\/40\/4\/714\/7330490"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,26]]},"references-count":34,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,10,26]]},"published-print":{"date-parts":[[2023,12,24]]}},"URL":"https:\/\/doi.org\/10.1093\/imamci\/dnad027","relation":{},"ISSN":["0265-0754","1471-6887"],"issn-type":[{"value":"0265-0754","type":"print"},{"value":"1471-6887","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,12]]},"published":{"date-parts":[[2023,10,26]]}}}