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5 - About Least-Squares Type Approach to Address Direct and Controllability Problems

Published online by Cambridge University Press:  25 October 2017

Arnaud Münch
Affiliation:
Blaise Pascal University
Pablo Pedregal
Affiliation:
none
Kaïs Ammari
Affiliation:
Université de Monastir, Tunisia
Stéphane Gerbi
Affiliation:
Université Savoie Mont Blanc, France
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Chapter
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Evolution Equations
Long Time Behavior and Control
, pp. 118 - 136
Publisher: Cambridge University Press
Print publication year: 2017

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References

[1] P.B., Bochev and M.D., Gunzburger, Analysis of least squares finite element methods for the Stokes equations, Math. Comput., 63(208) (1994), 479–506.Google Scholar
[2] B., Bochev and M., Gunzburger, Least-Squares Finite Element Methods, Applied Mathematical Sciences, 166. Springer, New York, 2009, xxii+660 pp.
[3] J.M., Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, AMS, Vol. 136, (2007).
[4] C., Fabre. Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems. ESAIM:COCV, 1 1995/6), 267–302.Google Scholar
[5] C., Fabre and G., Lebeau, Prolongement unique des solutions de l’équation de Stokes (French), Comm. PDE, 21 (1996), 573–96.Google Scholar
[6] E. Fernández-Cara, S., Guerrero, O.Yu., Imanuvilov, and J.-P., Puel, Local exact controllability of the Navier–Stokes system, J. Math. Pures Appl., 83(12)2004), 1501–42.Google Scholar
[7] E. Fernández-Cara and A., Münch, Numerical null controllability of semilinear 1D heat equations: Fixed points, least squares and Newton methods, Math. Control Relat. Fields, 2(3) (2012), 217–46.Google Scholar
[8] E. Fernández-Cara and A., Münch, Numerical null controllability of the 1D heat equation: Primal algorithms, Séma J., 61(1) (2013), 49–78.Google Scholar
[9] E. Fernández-Cara and A., Münch. Numerical null controllability of the 1D heat equation: Duality and Carleman weights. J. Optim. Theory Appl., 163(01) (2014), 253–85.Google Scholar
[10] A.V., Fursikov and O. Yu., Imanuvilov, On approximate controllability of the Stokes system, Annales de la Faculté des Sciences de Toulouse, II(2) (1993), 205–32.Google Scholar
[11] A.V., Fursikov and O. Yu., Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Korea (1996), 1–163.
[12] R., Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer Series in Computational Physics, (1983).
[13] R., Glowinski, J.L., Lions, and J., He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach, Encyclopedia of Mathematics and Its Applications, 117. Cambridge University Press, Cambridge, (2008).
[14] O.Yu., Imanuvilov, Remarks on exact controllability for the Navier–Stokes equations, ESAIM Control Optim. Cal. Var., 6 (2001), 39–72.Google Scholar
[15] O.Yu., Imanuvilov, J.-P., Puel, and M., Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. 30B(4), 2009, 333–78.Google Scholar
[16] I., Lasiecka and R., Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems. Encyclopedia of Mathematics and Its Applications, 74. Cambridge University Press, Cambridge, (2000).
[17] J.-L., Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Recherches en Mathématiques Appliquées, Tomes 1 et 2, Masson, Paris, (1988).
[18] A., Münch, A variational approach to approximate controls for systems with essential spectrum: Application to membranal arch, Evolut. Eq. Control Theory, 2(1) (2013), 119–51.Google Scholar
[19] A., Münch. A least-squares formulation for the approximation of controls for the Stokes system. Math. Controls Signals Syst., 27 (2015), 49–75.Google Scholar
[20] A., Münch and E., Zuazua, Numerical approximation of null controls for the heat equation: Ill-posedness and remedies, Inverse Problems, 26(8) (2010), 085018, 39 pp.
[21] A., Münch and P., Pedregal, Numerical null controllability of the heat equation through a least squares and variational approach, Eur. J. Appl. Math., 25(03) (2014), 277–306.Google Scholar
[22] A., Münch and P., Pedregal, A Least-Squares Formulation for the Approximation of Null Controls for the Stokes system, C.R. Acad. Sci. Série, 1, 351 (2013), 545–50.Google Scholar
[23] A., Münch and P., Pedregal, A least-squares formulation for the approximation of null controls for the Navier–Stokes system. In preparation.
[24] P., Pedregal, A variational perspective on controllability, Inverse Problems, 26(1) (2010), 015004, 17 pp.Google Scholar
[25] P., Pedregal, A variational approach for the Navier–Stokes system, J. Math. Fluid Mech., 14(1) (2012), 159–76.Google Scholar
[26] P., Pedregal, On error functionals, S⃗eMA J., 65 (2014), 13–22.Google Scholar
[27] R., Temam, Navier–Stokes Equations. Theory and Numerical Analysis. Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001, xiv+408 pp.
[28] D.L., Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev., 20 (1978), 639–739.Google Scholar

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