×

zbMATH — the first resource for mathematics

Remarks on the internal exponential stabilization to a nonstationary solution for 1D Burgers equations. (English) Zbl 1312.93052

MSC:
93B52 Feedback control
93C20 Control/observation systems governed by partial differential equations
93D15 Stabilization of systems by feedback
35Q53 KdV equations (Korteweg-de Vries equations)
PDF BibTeX Cite
Full Text: DOI
References:
[1] J. A. Atwell and B. B. King, Stabilized finite element methods and feedback control for Burgers’ equation, Bull. Sci. Math., 4 (2000), pp. 2745–2749.
[2] M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier–Stokes system, SIAM J. Control Optim., 49 (2011), pp. 420–463. · Zbl 1217.93137
[3] H. T. Banks and K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. Control Optim., 22 (1984), pp. 684–698. · Zbl 0548.49017
[4] V. Barbu, Stabilization of Navier–Stokes Flows, Comm. Control Engrg. Ser., Springer-Verlag, London, 2011. · Zbl 1213.76001
[5] V. Barbu, Stabilization of Navier–Stokes equations by oblique boundary feedback controllers, SIAM J. Control Optim., 50 (2012), pp. 2288–2307. · Zbl 1292.93100
[6] V. Barbu, I. Lasiecka, and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal., 64 (2006), pp. 2704–2746. · Zbl 1098.35025
[7] V. Barbu, S. S. Rodrigues, and A. Shirikyan, Internal exponential stabilization to a nonstationary solution for \(3\)D Navier–Stokes equations, SIAM J. Control Optim., 49 (2011), pp. 1454–1478. · Zbl 1231.35141
[8] V. Barbu and R. Triggiani, Internal stabilization of Navier–Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), pp. 1443–1494. · Zbl 1073.76017
[9] P. Benner, MORLAB Package Software, http://www-user.tu-chemnitz.de/\string benner/software.php.
[10] P. Benner, A MATLAB repository for model reduction based on spectral projection, in Proceedings of the 2006 IEEE Conference on Computer Aided Control Systems Design, 2006, pp. 19–24.
[11] M. Braack, E. Burman, V. John, and G. Lube, Stabilized finite element methods for the generalized Oseen problem, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 853–866. · Zbl 1120.76322
[12] R. Curtain and A. J. Pritchard, The infinite-dimensional Riccati equation for systems defined by evolution operators, SIAM J. Control Optim., 14 (1976), pp. 951–983. · Zbl 0352.49003
[13] G. Da Prato and A. Ichikawa, Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Control Optim., 28 (1990), pp. 359–381.
[14] R. Datko, Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal., 3 (1972), pp. 428–445. · Zbl 0241.34071
[15] M. C. Delfour and S. K. Mitter, Controllability, observability and optimal feedback control of affine hereditary differential systems, SIAM J. Control, 10 (1972), pp. 298–328. · Zbl 0242.93011
[16] J. Donea and A. Huerta, Finite Element Methods for Flow Problems, John Wiley & Sons, New York, 2003.
[17] A. Doubova, E. Fernández-Cara, M. González-Burgos, and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim., 41 (2002), pp. 798–819. · Zbl 1038.93041
[18] T. Duyckaerts, X. Zhang, and E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), pp. 1–41. · Zbl 1248.93031
[19] C. Foias, O. Manley, R. Rosa, and R. Temam, Navier–Stokes Equations and Turbulence, Encyclopedia Math. Appl., Cambridge University Press, Cambridge, 2001.
[20] A. V. Fursikov and A. A. Kornev, Feedback stabilization for the Navier–Stokes equations: Theory and calculations, in Mathematical Aspects of Fluid Mechanics, London Math. Soc. Lecture Note Ser. 402, Cambridge University Press, Cambridge, 2012, pp. 130–172. · Zbl 1296.76049
[21] M. Gunzburger, Perspectives in Flow Control and Optimization, Advances in Design and Control, SIAM, Philadelphia, 2003. · Zbl 1088.93001
[22] M. Hinze and S. Volkwein, Analysis of instantaneous control for the Burgers equation, Nonlinear Anal., 50 (2002), pp. 1–26. · Zbl 1022.49001
[23] O. Yu. Imanuvilov, Controllability of parabolic equations, Sb. Math., 186 (1995), pp. 879–900. · Zbl 0845.35040
[24] C. Johnson, U. Nävert, and J. Pitkäranta, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg., 45 (1984), pp. 285–312.
[25] D. A. Jones and E. S. Titi, On the number of determining nodes for the \(2\)D Navier–Stokes equations, J. Math. Anal. Appl., 168 (1992), pp. 72–88. · Zbl 0773.35050
[26] D. A. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier–Stokes equations, Indiana Univ. Math. J., 42 (1993), pp. 875–887. · Zbl 0796.35128
[27] S. Kesavan and J.-P. Raymond, On a degenerate Riccati equation, Control Cybernet., 38 (2009), pp. 1393–1410. · Zbl 1236.49052
[28] A. A. Kornev, The method of asymptotic stabilization to a given trajectory based on a correction of the initial data, Comput. Math. Math. Phys., 46 (2006), pp. 34–48. · Zbl 1210.37020
[29] A. A. Kornev, A problem of asymptotic stabilization by the right-hand side, Russian J. Numer. Anal. Math. Modelling, 23 (2008), pp. 407–422. · Zbl 1152.65064
[30] M. Krstic, L. Magnis, and R. Vazquez, Nonlinear control of the viscous Burgers equation: Trajectory generation, tracking, and observer design, J. Dyn. Syst. Meas. Control, 131 (2009), 021012.
[31] K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), pp. 345–371. · Zbl 0949.93039
[32] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid mechanics, SIAM J. Numer. Anal., 40 (2002), pp. 492–515. · Zbl 1075.65118
[33] K. Kunisch, S. Volkwein, and L. Xie, HJB-POD based feedback design for the optimal control of evolution problems, SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 701–722. · Zbl 1058.35061
[34] K. Kunisch and L. Xie, POD-based feedback control of the Burgers equation by solving the evolutionary HJB equation, Comput. Math. Appl., 49 (2005), pp. 1113–1126. · Zbl 1080.93012
[35] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I, Abstract Parabolic Systems, Encyclopedia Math. Appl. 74, Cambridge University Press, Cambridge, 2000. · Zbl 0942.93001
[36] J.-L. Lions, Equations Differentielles Operationnelles et Problèmes aux Limites, Die Grundlehren Math. Wiss. Einzeldarstellungen 111, Springer-Verlag, London, 1961. · Zbl 0098.31101
[37] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod et Gauthier–Villars, Paris, 1969.
[38] S. S. Ravindran, Stabilization of Navier–Stokes equations by boundary feedback, Int. J. Numer. Anal. Model., 4 (2007), pp. 608–624. · Zbl 1131.76024
[39] J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two-dimensional Navier–Stokes equations with finite-dimensional controllers, Discrete Contin. Dyn. Syst., 27 (2010), pp. 1159–1187. · Zbl 1211.93103
[40] S. S. Rodrigues, Boundary observability inequalities for the 3D Oseen–Stokes system and applications, ESAIM Control Optim. Calc. Var., to appear; also available online from http://www.ricam.oeaw.ac.at/publications/reports/.
[41] S. S. Rodrigues, Local exact boundary controllability of 3D Navier–Stokes equations, Nonlinear Anal., 95 (2014), pp. 175–190. · Zbl 1426.76148
[42] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, 2nd ed., CBMS-NSF Regional Conf. Ser. Appl. Math. 66, SIAM, Philadelphia, 1995. · Zbl 0833.35110
[43] R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. · Zbl 0981.35001
[44] L. Thevenet, J.-M. Buchot, and J.-P. Raymond, Nonlinear feedback stabilization of a two-dimensional burgers equation, ESAIM Control Optim. Calc. Var., 16 (2010), pp. 929–955. · Zbl 1202.93129
[45] V. M. Ungureanu and V. Dragan, Nonlinear differential equations of Riccati type on ordered Banach spaces, Electron. J. Qual. Theory Differ. Equ., 17 (2012), pp. 1–22. · Zbl 1324.34118
[46] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013. · Zbl 1194.35512
[47] J. Zabczyk, Mathematical Control Theory: An Introduction, Systems Control Found. Appl., Birkhäuser, Boston, 1992. · Zbl 1071.93500
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.