{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T04:34:23Z","timestamp":1768451663779,"version":"3.49.0"},"reference-count":39,"publisher":"EDP Sciences","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["ESAIM: COCV"],"published-print":{"date-parts":[[2015,4]]},"DOI":"10.1051\/cocv\/2014037","type":"journal-article","created":{"date-parts":[[2015,1,7]],"date-time":"2015-01-07T16:07:30Z","timestamp":1420646850000},"page":"535-560","source":"Crossref","is-referenced-by-count":13,"title":["Coupling estimation and control for a two dimensional Burgers type equation"],"prefix":"10.1051","volume":"21","author":[{"given":"Jean-Marie","family":"Buchot","sequence":"first","affiliation":[]},{"given":"Jean-Pierre","family":"Raymond","sequence":"additional","affiliation":[]},{"given":"Jorge","family":"Tiago","sequence":"additional","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2015,3,9]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"Badra M., Lyapunov function and local feedback boundary stabilization of the Navier\u2013Stokes equations.SIAM J. Control Optim.48(2009) 1797\u20131830.","DOI":"10.1137\/070682630"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"Badra M., Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier\u2013Stokes and Boussinesq equations with Neumann or Dirichlet control.Discrete Contin. Dyn. Syst.32(2012) 1169\u20131208.","DOI":"10.3934\/dcds.2012.32.1169"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"Badra M. and Takahashi T., Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: application to the Navier\u2013Stokes system.SIAM J. Control Optim.49(2011) 420\u2013463.","DOI":"10.1137\/090778146"},{"key":"R4","unstructured":"H.T. Banks, R.C. Smith and Y. Wang, Smart Material Structures, modeling, estimation and control. Masson\/John Wiley, Paris\/Chichester (1996)."},{"key":"R5","doi-asserted-by":"crossref","unstructured":"A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems. Vol. 1. Birkh\u00e4user, Boston (1992).","DOI":"10.1007\/978-1-4612-2750-2"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems. Vol. 2. Birkh\u00e4user, Boston (1992).","DOI":"10.1007\/978-1-4612-2750-2"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"J. Burns and H. Marrekchi, Optimal fixed-finite-dimensional compensator for Burgers\u2019 equation with unbounded input\/output operators, Computation and Control III.Progress System Control Theory. Vol. 15. Birkhauser, Boston, MA (1993) 83\u2013104.","DOI":"10.1007\/978-1-4612-0321-6_6"},{"key":"R8","unstructured":"Burns J., Balogh A., Gilliam D.S. and Shubov V.I., Numerical Stationary Solutions for a Viscous Burgers\u2019 Equation.J. Math. Syst. Estim. Control8(1998) 1\u201316."},{"key":"R9","doi-asserted-by":"crossref","unstructured":"Chen C.M. and Thom\u00e9e V., The lumped mass finite element method for a parabolic problem.J. Austral. Math. Soc. Ser. B26(1985) 329\u2013354.","DOI":"10.1017\/S0334270000004549"},{"key":"R10","unstructured":"J.-M. Coron, Control and nonlinearity. American Mathematical Society. Providence, RI (2007)."},{"key":"R11","doi-asserted-by":"crossref","unstructured":"Curtain R., Finite dimensional compensators for parabolic distributed systems with unbounded control and observation.SIAM J. Control Optim.22(1984) 255\u2013276.","DOI":"10.1137\/0322018"},{"key":"R12","unstructured":"Curtain R., A comparison of finite-dimensional controller designs for distributed parameter systems,Control Theory Adv. Technol.9(1993) 609\u2013628."},{"key":"R13","doi-asserted-by":"crossref","unstructured":"Curtain R. and Salamon D., Finite dimensional compensators for infinite dimensional systems with unbounded input operators.SIAM J. Control Optim.24(1986) 797\u2013816.","DOI":"10.1137\/0324050"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"R. Curtain and H.J. Zwart, An introduction to infinite dimensional linear systems theory. Springer-Verlag, New York (1995).","DOI":"10.1007\/978-1-4612-4224-6"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"H.C. Elman, D.J. Silvester and A.J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford University Press, New York (2005).","DOI":"10.1093\/oso\/9780198528678.001.0001"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"A.V. Fursikov, Optimal control of distributed systems. Theory and Applications. AMS Providence, Rhode Island (2000).","DOI":"10.1090\/mmono\/187"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"Grisvard P., Caract\u00e9risation de quekques espaces d\u2019interpolation.Arch. Rat. Mech. An.25(1967) 40\u201363.","DOI":"10.1007\/BF00281421"},{"key":"R18","unstructured":"P. Grisvard, Elliptic problems in nonsmooth domains.Pitman Monogr. Studies in Mathematics. Vol. 24. Advanced Publishing Program, Boston, MA (1985)."},{"key":"R19","unstructured":"P. Grisvard, Singularities in boundary value problems.Recherches en Math\u00e9matiques Appliqu\u00e9es [Research in Applied Mathematics]. Vol. 22. Masson, Paris; Springer-Verlag, Berlin (1992)."},{"key":"R20","doi-asserted-by":"crossref","unstructured":"Ji G. and Lasiecka I., Partially observed analytic systems with fully unbounded actuators and sensors-FEM algorithms.Comput. Optim. Appl.11(1998) 111\u2013136.","DOI":"10.1023\/A:1018681526852"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"Jund S. and Salmon S., Arbitrary high-order finite element schemes and high-order mass lumping.Int. J. Appl. Math. Comput. Sci.17(2007) 375\u2013393.","DOI":"10.2478\/v10006-007-0031-2"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"T. Kato, Perturbation theory for linear operators, Reprint of the 1980 Edition. Springer-Verlag (1995).","DOI":"10.1007\/978-3-642-66282-9"},{"key":"R23","unstructured":"Kesavan S. and Raymond J.P., On a degenerate Riccati equation.Control Cybern.38(2009) 1393\u20131410."},{"key":"R24","doi-asserted-by":"crossref","unstructured":"Lasiecka I., Galerkin approximations of infinite dimensional compensators for flexible structures with unbounded control action.Acta Appl. Math.28(1992) 101\u2013133.","DOI":"10.1007\/BF00047552"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"Lasiecka I., Finite element approximations of compensator design for analytic generators with fully unbound controls\/observations.SIAM J. Control Optim.33(1995) 67\u201388.","DOI":"10.1137\/S0363012992232208"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximations Theories. Vol. I. Cambridge University Press, Cambridge (2000).","DOI":"10.1017\/CBO9780511574801"},{"key":"R27","doi-asserted-by":"crossref","unstructured":"Laub A.J., A Schur method for solving algebraic Riccati equations.IEEE Trans. Automat. Control24(1979) 913\u2013925.","DOI":"10.1109\/TAC.1979.1102178"},{"key":"R28","unstructured":"J.-L. Lions, Contr\u00f4le optimal des \u00e9quations aux d\u00e9riv\u00e9es partielles. Dunod, Paris (1968)."},{"key":"R29","doi-asserted-by":"crossref","unstructured":"J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. 2. Springer, Berlin-Heidelberg-New York (1972).","DOI":"10.1007\/978-3-642-65161-8"},{"key":"R30","doi-asserted-by":"crossref","unstructured":"A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New-York (1983).","DOI":"10.1007\/978-1-4612-5561-1"},{"key":"R31","doi-asserted-by":"crossref","unstructured":"Raymond J.P., Boundary feedback stabilization of the two dimensional Navier\u2013Stokes equations.SIAM J. Control Optim.45(2006) 790\u2013828.","DOI":"10.1137\/050628726"},{"key":"R32","doi-asserted-by":"crossref","unstructured":"Raymond J.P., Feedback boundary stabilization of the three dimensional incompressible Navier\u2013Stokes equations.J. Math. Pures Appl.87(2007) 627\u2013669.","DOI":"10.1016\/j.matpur.2007.04.002"},{"key":"R33","unstructured":"J.P. Raymond, Stabilizability of infinite dimensional systems by finite dimensional controls, submitted."},{"key":"R34","doi-asserted-by":"crossref","unstructured":"Raymond J.-P. and Thevenet L., Boundary feedback stabilization of the two dimensional Navier\u2013Stokes equations with finite dimensional controllers.Discrete Contin. Dyn. Syst.27(2010) 1159\u20131187.","DOI":"10.3934\/dcds.2010.27.1159"},{"key":"R35","doi-asserted-by":"crossref","unstructured":"L.F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with MATLAB. Cambridge University Press, Cambridge (2003).","DOI":"10.1017\/CBO9780511615542"},{"key":"R36","doi-asserted-by":"crossref","unstructured":"Schumacher J.M., A direct approach to compensator design for distributed parameter systems.SIAM J. Control and Optim.21(1983) 823\u2013836.","DOI":"10.1137\/0321050"},{"key":"R37","unstructured":"L. Thevenet,Lois de feedback pour le contr\u00f4le d\u2019\u00e9coulements. Ph.D. Thesis, Universit\u00e9 de Toulouse (2009)."},{"key":"R38","doi-asserted-by":"crossref","unstructured":"Thevenet L., Buchot J.-M. and Raymond J.-P., Nonlinear feedback stabilization of a two-dimensional Burgers equation.ESAIM: COCV16(2010) 929\u2013955.","DOI":"10.1051\/cocv\/2009028"},{"key":"R39","doi-asserted-by":"crossref","unstructured":"M. Tucsnak and G. Weiss, Observation and control for operator semigroups. Birkhuser Verlag, Basel (2009).","DOI":"10.1007\/978-3-7643-8994-9"}],"container-title":["ESAIM: Control, Optimisation and Calculus of Variations"],"original-title":[],"link":[{"URL":"http:\/\/www.esaim-cocv.org\/10.1051\/cocv\/2014037\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T12:06:20Z","timestamp":1717675580000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.esaim-cocv.org\/10.1051\/cocv\/2014037"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,9]]},"references-count":39,"journal-issue":{"issue":"2"},"alternative-id":["cocv140037"],"URL":"https:\/\/doi.org\/10.1051\/cocv\/2014037","relation":{},"ISSN":["1292-8119","1262-3377"],"issn-type":[{"value":"1292-8119","type":"print"},{"value":"1262-3377","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,3,9]]}}}