Skip to main content Accessibility help
×
Home
Hostname: page-component-5c569c448b-6g96d Total loading time: 0.31 Render date: 2022-07-02T15:54:00.077Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

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
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Evolution Equations
Long Time Behavior and Control
, pp. 118 - 136
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×