{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T17:15:20Z","timestamp":1776446120918,"version":"3.51.2"},"reference-count":40,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2020,3,16]],"date-time":"2020-03-16T00:00:00Z","timestamp":1584316800000},"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":[[2021,3,16]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we investigate the stabilization problem of a cascade of a fractional ordinary differential equation (FODE) and a fractional reaction\u2013diffusion (FRD) equation where the interconnections are of Neumann type. We exploit the partial differential equation backstepping method for designing a controller, which guarantees the Mittag\u2013Leffler stability of the FODE-FRD cascade. Moreover, we propose an observer that is Mittag\u2013Leffler convergent. Also, we propose an output feedback boundary controller, and we prove that the closed-loop FODE-FRD system is Mittag\u2013Leffler stable in the sense of the corresponding norm. Finally, numerical simulations are presented to verify the results.<\/jats:p>","DOI":"10.1093\/imamci\/dnaa002","type":"journal-article","created":{"date-parts":[[2020,1,28]],"date-time":"2020-01-28T04:21:19Z","timestamp":1580185279000},"page":"90-124","source":"Crossref","is-referenced-by-count":10,"title":["Observer-based output feedback control design for a coupled system of fractional ordinary and reaction\u2013diffusion equations"],"prefix":"10.1093","volume":"38","author":[{"given":"Shadi","family":"Amiri","sequence":"first","affiliation":[{"name":"Faculty of Mathematical Sciences, University of Guilan, Rasht 4193833697, Iran"}]},{"given":"Mohammad","family":"Keyanpour","sequence":"additional","affiliation":[{"name":"Faculty of Mathematical Sciences, and Center of Excellence for Mathematical Modelling, Optimization and Combinational Computing (MMOCC), University of Guilan, Rasht 4193833697, Iran"}]},{"given":"Asadollah","family":"Asaraii","sequence":"additional","affiliation":[{"name":"Faculty of Mathematical Sciences, University of Guilan, Rasht 4193833697, Iran"}]}],"member":"286","published-online":{"date-parts":[[2020,3,16]]},"reference":[{"key":"2021021910075779500_ref1","first-page":"2409","article-title":"Stabilization of a Ginzburg\u2013Landau model of vortex shedding by output feedback boundary control","volume":"3","author":"Aamo","year":"2004","journal-title":"43rd IEEE Conference on Decision and Control. IEEE"},{"key":"2021021910075779500_ref2","doi-asserted-by":"crossref","first-page":"1953","DOI":"10.1137\/S036301290342601X","article-title":"Boundary control of the linearized Ginzburg\u2013Landau model of vortex shedding","volume":"43","author":"Aamo","year":"2005","journal-title":"SIAM J. Control Optim."},{"key":"2021021910075779500_ref3","doi-asserted-by":"crossref","first-page":"2951","DOI":"10.1016\/j.cnsns.2014.01.022","article-title":"Lyapunov functions for fractional order systems","volume":"19","author":"Aguila-Camacho","year":"2014","journal-title":"Comm. Nonlinear Sci. Numer. Simulat."},{"key":"2021021910075779500_ref4","first-page":"27","article-title":"Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality","volume":"187","author":"Ahn","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"2021021910075779500_ref5","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.automatica.2015.01.032","article-title":"Boundary control of coupled reaction\u2013diffusion processes with constant parameters","volume":"54","author":"Baccoli","year":"2015","journal-title":"Automatica"},{"key":"2021021910075779500_ref6","doi-asserted-by":"crossref","first-page":"165","DOI":"10.3166\/ejc.8.165-175","article-title":"Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability","volume":"8","author":"Balogh","year":"2002","journal-title":"Eur. J. Control"},{"key":"2021021910075779500_ref7","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/S0167-6911(03)00222-6","article-title":"Stability of partial difference equations governing control gains in infinite-dimensional backstepping","volume":"51","author":"Balogh","year":"2004","journal-title":"Syst. Control Lett."},{"key":"2021021910075779500_ref8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.neunet.2013.11.016","article-title":"Global Mittag\u2013Leffler stability and synchronization of memristor-based fractional-order neural networks","volume":"51","author":"Chen","year":"2014","journal-title":"Neural Networks"},{"key":"2021021910075779500_ref9","doi-asserted-by":"crossref","first-page":"2964","DOI":"10.1049\/iet-cta.2017.0227","article-title":"Backstepping-based boundary feedback control for a fractional reaction diffusion system with mixed or robin boundary conditions","volume":"11","author":"Chen","year":"2017","journal-title":"IET Control Theory Appl."},{"key":"2021021910075779500_ref10","doi-asserted-by":"crossref","first-page":"3097","DOI":"10.1109\/TAC.2013.2274723","article-title":"Stabilization of a system of $n+1$ coupled first-order hyperbolic linear PDEs with a single boundary input","volume":"58","author":"Di Meglio","year":"2013","journal-title":"IEEE Trans. Automat. Control"},{"key":"2021021910075779500_ref11","doi-asserted-by":"crossref","first-page":"6920","DOI":"10.1109\/CDC.2014.7040476","article-title":"Adaptive Mittag\u2013Leffler stabilization of commensurate fractional-order nonlinear systems","author":"Ding","year":"2014","journal-title":"53rd IEEE Conference on Decision and Control. IEEE"},{"key":"2021021910075779500_ref12","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1049\/iet-cta.2014.0642","article-title":"Non-linear Mittag\u2013Leffler stabilisation of commensurate fractional-order non-linear systems","volume":"9","author":"Ding","year":"2015","journal-title":"IET Control Theory Appl."},{"key":"2021021910075779500_ref13","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/978-1-4614-0457-6_3","article-title":"Application of backstepping control technique to fractional order dynamic systems","volume-title":"Fraction. Dynam. Control","author":"Efe","year":"2012"},{"key":"2021021910075779500_ref14","doi-asserted-by":"crossref","first-page":"1250","DOI":"10.1049\/iet-cta.2015.0882","article-title":"Boundary feedback stabilisation for the time fractional-order anomalous diffusion system","volume":"10","author":"Ge","year":"2016","journal-title":"IET Control Theory Appl."},{"key":"2021021910075779500_ref15","article-title":"Stability of nonlinear fractional-order time varying systems","volume":"11","author":"Huang","year":"2016","journal-title":"J. Comput. Nonlinear Dyn."},{"key":"2021021910075779500_ref16","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1007\/s11071-016-3288-x","article-title":"Stability and stabilization of a class of fractional-order nonlinear systems for $0$","volume":"88","author":"Huang","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"2021021910075779500_ref17","first-page":"252","article-title":"Boundary control of delayed ODE-heat cascade under actuator saturation","volume-title":"Automatica","author":"Kang","year":"2017"},{"key":"2021021910075779500_ref18","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. Control"},{"key":"2021021910075779500_ref19","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":"2021021910075779500_ref20","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":"2021021910075779500_ref21","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-69689-3","volume-title":"Chemical Oscillations, Waves, and Turbulence","author":"Kuramoto","year":"1984"},{"key":"2021021910075779500_ref22","doi-asserted-by":"crossref","first-page":"1965","DOI":"10.1016\/j.automatica.2009.04.003","article-title":"Mittag\u2013Leffler stability of fractional order nonlinear dynamic systems","volume":"45","author":"Li","year":"2009","journal-title":"Automatica"},{"key":"2021021910075779500_ref23","doi-asserted-by":"crossref","first-page":"1810","DOI":"10.1016\/j.camwa.2009.08.019","article-title":"Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag\u2013Leffler stability","volume":"59","author":"Li","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"2021021910075779500_ref24","doi-asserted-by":"crossref","first-page":"1062","DOI":"10.1109\/TAC.2012.2218064","article-title":"Stability and stabilization of fractional-order linear systems subject to input saturation","volume":"58","author":"Lim","year":"2013","journal-title":"IEEE Trans. Automat. Control"},{"key":"2021021910075779500_ref25","doi-asserted-by":"crossref","first-page":"1033","DOI":"10.1137\/S0363012902402414","article-title":"Boundary feedback stabilization of an unstable heat equation","volume":"42","author":"Liu","year":"2003","journal-title":"SIAM J. Control Optimiz."},{"key":"2021021910075779500_ref26","doi-asserted-by":"crossref","DOI":"10.1119\/1.13295","volume-title":"The Fractal Geometry of Nature","author":"Mandelbrot","year":"1983"},{"key":"2021021910075779500_ref27","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1109\/9.341815","article-title":"Boundary fractional derivative control of the wave equation","volume":"40","author":"Mbodje","year":"1995","journal-title":"IEEE Trans. Automat. Control"},{"key":"2021021910075779500_ref28","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/S0378-4371(00)00085-6","article-title":"Reaction\u2013diffusion modelling of bacterial colony patterns","volume":"282","author":"Mimura","year":"2000","journal-title":"Physica A Stat. Mech. Appl."},{"key":"2021021910075779500_ref29","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1016\/j.camwa.2008.02.015","article-title":"Implicit finite difference approximation for time fractional diffusion equations","volume":"56","author":"Murio","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"2021021910075779500_ref30","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.1109\/TAC.2002.800737","article-title":"Discontinuous feedback stabilization of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor","volume":"47","author":"Orlov","year":"2002","journal-title":"IEEE Trans. Automat. Control"},{"key":"2021021910075779500_ref31","volume-title":"Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications","author":"Podlubny","year":"1999"},{"key":"2021021910075779500_ref32","doi-asserted-by":"crossref","first-page":"2185","DOI":"10.1109\/TAC.2004.838495","article-title":"Closed-form boundary state feedbacks for a class of 1-d partial integro-differential equations","volume":"49","author":"Smyshlyaev","year":"2004","journal-title":"IEEE Trans. Automat. Control"},{"key":"2021021910075779500_ref33","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/j.jfranklin.2009.09.005","article-title":"Control of PDE\u2013ODE cascades with neumann interconnections","volume":"347","author":"Susto","year":"2010","journal-title":"J. Franklin I."},{"key":"2021021910075779500_ref34","doi-asserted-by":"crossref","first-page":"2142","DOI":"10.1016\/j.jfranklin.2011.06.008","article-title":"Stabilization for a coupled PDE\u2013ODE control system","volume":"348","author":"Tang","year":"2011","journal-title":"J. Franklin I."},{"key":"2021021910075779500_ref35","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":"Tang","year":"2011","journal-title":"Syst. Control Lett."},{"key":"2021021910075779500_ref36","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1115\/1.3167615","article-title":"On the appearance of the fractional derivative in the behavior of real materials","volume":"51","author":"Torvik","year":"1984","journal-title":"J. Appl. Mech."},{"key":"2021021910075779500_ref37","doi-asserted-by":"crossref","first-page":"4937","DOI":"10.1109\/CDC.2011.6160338","article-title":"Backstepping boundary stabilization and state estimation of a 2$\\times $ 2 linear hyperbolic system. 2011 50th","author":"Vazquez","year":"2011","journal-title":"IEEE Conference on Decision and Control and European Control Conference. IEEE"},{"key":"2021021910075779500_ref38","doi-asserted-by":"crossref","first-page":"2026","DOI":"10.1109\/TAC.2016.2590506","article-title":"Boundary control of coupled reaction\u2013advection\u2013diffusion systems with spatially-varying coefficients","volume":"62","author":"Vazquez","year":"2017","journal-title":"IEEE Trans. Automat. Control"},{"key":"2021021910075779500_ref39","doi-asserted-by":"crossref","DOI":"10.1201\/9781420033588","volume-title":"Reaction\u2013Diffusion Problems in the Physics of Hot Plasmas","author":"Wilhelmsson","year":"2000"},{"key":"2021021910075779500_ref40","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1137\/15M1048999","article-title":"Boundary feedback stabilization for an unstable time fractional reaction diffusion equation","volume":"56","author":"Zhou","year":"2018","journal-title":"SIAM J. Control Optimiz."}],"container-title":["IMA Journal of Mathematical Control and Information"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/imamci\/article-pdf\/38\/1\/90\/36320013\/dnaa002.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/imamci\/article-pdf\/38\/1\/90\/36320013\/dnaa002.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,21]],"date-time":"2021-02-21T04:04:36Z","timestamp":1613880276000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/imamci\/article\/38\/1\/90\/5807632"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,16]]},"references-count":40,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,3,16]]},"published-print":{"date-parts":[[2021,3,16]]}},"URL":"https:\/\/doi.org\/10.1093\/imamci\/dnaa002","relation":{},"ISSN":["1471-6887"],"issn-type":[{"value":"1471-6887","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2021,3]]},"published":{"date-parts":[[2020,3,16]]}}}