Insensitizing controls with two vanishing components for the three-dimensional Boussinesq system

Carreño, N.; Guerrero, S.; Gueye, M.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 73-100.

In this paper we prove the existence of insensitizing controls for a viscous newtonian fluid wherein thermic effects are taken into acount, the so called Boussinesq system. The proof relies on a standard approach introduced by Fursikov and Imanuvilov for the Navier−Stokes system which consists in solving a constrained extremal problem, and then on an inverse mapping theorem to deal with the nonlinear system. Furthermore, we use the coupling with the heat equation to get rid of two components of the control in the fluid equations.

Classification : 34B15, 35Q30, 93C10, 93B05
Mots clés : Navier−Stokes system, Boussinesq system, null controllability, Carleman inequalities, insensitizing controls

V.M. Alekseev, V.M. Tikhomirov and S.V. Fomin., Optimal Control, Translated from the Russian by V.M. Volosov, Contemporary Soviet Mathematics. Consultants Bureau, New York (1987). | MR | Zbl

O. Bodart and C. Fabre, Controls insensitizing the norm of the solution of a semilinear heat equation. J. Math. Anal. Appl. 195 (1995) 658–683. | DOI | MR | Zbl

N. Carreño, Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain. Math. Control Relat. Fields 2 (2012) 361–382. | DOI | MR | Zbl

N. Carreño and S. Guerrero, Local null controllability of the $N$-dimensional Navier−Stokes system with $N-1$ scalar controls in an arbitrary control domain. J. Math. Fluid Mech. 15 (2013) 139–153. | DOI | MR | Zbl

N. Carreño and M. Gueye, Insensitizing controls with one vanishing component for the Navier−Stokes system. J. Math. Pures Appl. 101 (2014) 27–53. | DOI | MR | Zbl

J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls. J. Differ. Equ. 246 (2009) 2908–2921. | DOI | MR | Zbl

J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier−Stokes system with a distributed control having two vanishing components. To appear in Invent. Math. | MR | Zbl

J.I. Díaz and A.V. Fursikov, Approximate controllability of the Stokes system on cylinders by external unidirectional forces. J. Math. Pures Appl. 9 (1997) 353–375. | DOI | MR | Zbl

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Some controllability results for the N-dimensional Navier−Stokes and Boussinesq systems with $N-1$ scalar controls. SIAM J. Control Optim. 45 (2006) 146–173. | DOI | MR | Zbl

A.V. Fursikov O. Yu. Imanuvilov. Controllability of Evolution Equations. Lecture Notes #34. Seoul National University, Korea (1996). | MR | Zbl

S. Guerrero, Controllability of systems of Stokes equations with one control force: existence of insensitizing controls. Ann. Inst. Henri Poincaré Anal. Non Linéaire 24 (2007) 1029–1054. | DOI | Numdam | MR | Zbl

M. Gueye, Insensitizing controls for the Navier−Stokes equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 30 (2013) 825–844. | DOI | Numdam | MR | Zbl

O. Yu. Imanuvilov, Remarks on exact controllability for the Navier−Stokes equation. ESAIM: COCV 6 (2001) 39–72. | Numdam | MR | Zbl

O. Yu. Imanuvilov, J.-P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions. Chin. Ann. Math. Ser. B 30 (2009) 333–378. | DOI | MR | Zbl

O. Kavian and L. De Teresa, Unique continuation principle for systems of parabolic equations. ESAIM: COCV 16 (2010) 247–274. | Numdam | MR | Zbl

O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, revised English edition, translated from the Russian by Richard A. Silverman. Gordon and Breach Science Publishers, New York, London (1963). | MR | Zbl

O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Vol. 23 of Translations of Mathematical Monographs. American Mathematical Society, Providence, R.I. (1968). | MR | Zbl

J.-L. Lions, Quelques notions dans l’analyse et le contrôle de systèmes à données incomplètes (Somes notions in the analysis and control of systems with incomplete data). Proc. of the XIth congress on Differential Equations and Applications/First Congress on Applied Mathematics, Málaga (1989). Univ. Málaga (1990) 43–54. | MR | Zbl

J.-L. Lions, Sentinelles pour les systèmes distribués à données incomplètes (Sentinelles for Distributed Systems with Incomplete Data). Vol. 21 of Recherches en Mathématiques Appliquées (Research in Applied Mathematics). Masson, Paris (1992). | MR | Zbl

J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 2, Travaux et Recherches Mathématiques, No. 18. Dunod, Paris (1968). | MR | Zbl

J.-L. Lions and E. Zuazua, A generic uniqueness result for the Stokes system and its control theoretical consequences, Partial differential equations and applications. Vol. 177 of Lect. Notes Pure Appl. Math. Dekker, New York (1996) 221–235. | MR | Zbl

S. Micu, J.H. Ortega and L. De Teresa, An example of $\epsilon$-insensitizing controls for the heat equation with no intersecting observation and control regions. Appl. Math. Lett. 17 (2004) 927–932. | DOI | MR | Zbl

R. Pérez-García, Nuevos resultados de control para algunos problemas parabólicos acoplados no lineales: controlabilidad y controles insensibilizantes. Ph.D. thesis, University of Seville, Spain, 2004.

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Differ. Equ. 66 (1987) 118–139. | DOI | MR | Zbl

R. Temam, Navier−Stokes Equations, Theory and Numerical Analysis. Vol. 2 of Study Math. Appl. North-Holland, Amsterdam, New York, Oxford (1977). | MR | Zbl

L. De Teresa, Insensitizing controls for a semilinear heat equation. Commun. Partial Differ. Equ. 25 (2000) 39–72. | DOI | MR | Zbl