Abstract
This paper deals with the boundary controllability of inviscid incompressible fluids for which thermal effects are important. They will be modeled through the so-called Boussinesq approximation. In the zero heat diffusion case, by adapting and extending some ideas from J.-M. Coron and O. Glass, we establish the simultaneous global exact controllability of the velocity field and the temperature for 2D and 3D flows. When the heat diffusion coefficient is positive, we present some additional results concerning exact controllability for the velocity field and local null controllability of the temperature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bardos C, Frisch U (1976) Finite-time regularity for bounded and unbounded ideal incompressible fluids using Hölder estimates. In: Turbulence and Navier–Stokes equations. Proc. conf., Univ. Paris-Sud, Orsay. Lecture notes in math., vol 565. Springer, Berlin, pp 1–13
Coron J-M (1992) Global asymptotic stabilization for controllable systems without drift. Math Control Signals Syst 5:295–312
Coron J-M (1993) Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles bidimensionnels. C R Acad Sci Paris Sér I Math 317:271–276
Coron J-M (1996) On the controllability of \(2\)-D incompressible perfect fluids. J Math Pures Appl (9) 75:155–188
Coron J-M (2007) Control and nonlinearity. In: Mathematical surveys and monographs, vol 136. American Mathematical Society, Providence
Fernández-Cara E, Guerrero S (2006) Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J Control Optim 45:1395–1446
Glass O (1997) Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles en dimension 3. C R Acad Sci Paris Sér I Math 325:987–992
Glass O (1998) Contrôlabilité de l’équation d’Euler tridimensionnelle pour les fluides parfaits incompressibles. In: Séminaire sur les Équations aux Dérivées Partielles, 1997–1998, École Polytech., Palaiseau, pp Exp. No. XV, 11
Glass O (2000) Exact boundary controllability of 3-D Euler equation. ESAIM Control Optim Calc Var 5:1–44 (electronic)
Glowinski R, Lions JL, He J (2008) Exact and approximate controllability for distributed parameter systems. In: Encyclopedia of mathematics and its applications. A numerical approach, vol 117. Cambridge University Press, Cambridge
Hamilton RS (1982) The inverse function theorem of Nash and Moser. Bull Am Math Soc (NS) 7:65–222
Kato T (1967) On classical solutions of the two-dimensional nonstationary Euler equation. Arch Rat Mech Anal 25:188–200
Kazhikhov AV (1980) The correctness of the nonstationary problem about the flow of the ideal fluid through the given domain. Dinamika Sploshn Sredy 47:37–56 (Russian)
Kazhikhov AV (1991) Initial-boundary value problems for the Euler equations of an ideal incompressible fluid (Russian). Vestnik Moskov Univ Ser I Mat Mekh 5:13–19, 96 (translation in Moscow Univ. Math. Bull. 46(5):10–14)
Lasiecka I, Triggiani R (2000) Control theory for partial differential equations: continuous and approximation theories. I and II. In: Encyclopedia of mathematics and its applications, vol 74. Cambridge University Press, Cambridge
Littman W (1978) Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients. Ann Scuola Norm Sup Pisa Cl Sci (4) 5(3):567–580
Majda AJ, Bertozzi AL (2002) Vorticity and incompressible flow. In: Cambridge texts in applied mathematics, vol 27. Cambridge University Press, Cambridge
Pedlosky J (1987) Geophysical fluid dynamics. Springer, New York
Russell DL (1974) Exact boundary value controllability theorems for wave and heat processes in star-complemented regions. In: Differential games and control theory (Proc. NSF—CBMS Regional Res. Conf., Univ. Rhode Island, Kingston, R.I., 1973). Lecture notes in pure appl. math., vol 10. Dekker, New York , pp 291–319
Russell DL (1978) Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev 20(4):639–739
Simon J (1987) Compact sets in the space \(L^p\)(0, T;B). Ann Mat Pura Appl (4) 146:65–96
Tucsnak M, Weiss G (2009) Observation and control for operator semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher (Birkhäuser advanced texts: basel textbooks). Birkhäuser Verlag, Basel
Acknowledgments
The authors are grateful to the anonymous referees for their valuable comments and suggestions that have allowed to improve a previous version of the paper.
Author information
Authors and Affiliations
Corresponding authors
Additional information
E. Fernández-Cara was partially supported by Grant MTM2010-15592 (DGI-MICINN, Spain). M. C. Santos was partially supported by CAPES. D. A. Souza was partially supported by CAPES (Brazil) and Grant MTM2010-15592 (DGI-MICINN, Spain).
Rights and permissions
About this article
Cite this article
Fernández-Cara, E., Santos, M.C. & Souza, D.A. Boundary controllability of incompressible Euler fluids with Boussinesq heat effects. Math. Control Signals Syst. 28, 7 (2016). https://doi.org/10.1007/s00498-015-0158-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00498-015-0158-x