Insensitizing controls for the Navier–Stokes equations

Gueye, Mamadou

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 5, pp. 825-844.

In this paper, we deal with the existence of insensitizing controls for the Navier–Stokes equations in a bounded domain with Dirichlet boundary conditions. We prove that there exist controls insensitizing the $L^2$-norm of the observation of the solution in an open subset $𝒪$ of the domain, under suitable assumptions on the data. This problem is equivalent to an exact controllability result for a cascade system. First we prove a global Carleman inequality for the linearized Navier–Stokes system with right-hand side, which leads to the null controllability at any time $T>0$. Then, we deduce a local null controllability result for the cascade system.

[1] V.M. Alekseev, V.M. Tikhomirov, S.V. Fomin, Optimal Control, Contemporary Soviet Mathematics, Consultants Bureau, New York (1987) | MR | Zbl

[2] F. Ammar Khodja, A. Benabdallah, C. Dupaix, I. Kostin, Null-controllability of some systems of parabolic type by one control force, ESAIM Control Optim. Calc. Var. 11 (2005), 426-448 | EuDML | Numdam | MR | Zbl

[3] O. Bodart, C. Fabre, Controls insensitizing the norm of the solution of a semilinear heat equation, J. Math. Anal. Appl. 195 no. 3 (1995), 658-683 | MR | Zbl

[4] O. Bodart, M. González-Burgos, R. Pérez-García, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Comm. Partial Differential Equations 29 no. 7–8 (2004), 39-72 | MR | Zbl

[5] J.-M. Coron, S. Guerrero, Local null controllability of the two-dimensional Navier–Stokes system in the torus with a control force having a vanishing component, J. Math. Pures Appl. (9) 92 no. 5 (2009), 528-545 | MR | Zbl

[6] R. Dáger, Insensitizing controls for the 1-D wave equation, SIAM J. Control Optim. 45 no. 5 (2006), 1758-1768 | MR | Zbl

[7] L. De Teresa, O. Kavian, Unique continuation principle for systems of parabolic equations, ESAIM Control Optim. Calc. Var. 16 (2010), 247-274 | EuDML | Numdam | MR | Zbl

[8] L. De Teresa, Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations 25 no. 1–2 (2000), 39-72 | MR | Zbl

[9] L. De Teresa, E. Zuazua, Identification of the class of initial data for the insensitizing control of the heat equation, Commun. Pure Appl. Anal. 8 no. 1 (2009), 457-471 | MR | Zbl

[10] C. Fabre, G. Lebeau, Prolongement unique des solutions de lʼéquation de Stokes, Comm. Partial Differential Equations 21 (1996), 573-596 | MR | Zbl

[11] E. Fernández-Cara, C. Galina, A. Osses, Insensitizing controls for a large scale Ocean circulation model, C. R. Math. Acad. Sci. 337 no. 4 (2003), 265-270 | MR | Zbl

[12] E. Fernández-Cara, M. González-Burgos, S. Guerrero, J.-P. Puel, Null controllability of the heat equation with boundary Fourier conditions: the linear case, ESAIM Control Optim. Calc. Var. 12 no. 3 (2006), 442-465 | EuDML | Numdam | MR | Zbl

[13] E. Fernández-Cara, S. Guerrero, O.Yu. Imanuvilov, J.-P. Puel, Local exact controllability of the Navier–Stokes system, J. Math. Pures Appl. (9) 83 no. 12 (2004), 1501-1542 | MR | Zbl

[14] A.V. Fursikov, O.Y. Imanuvilov, Controllability of Evolution Equations, Lect. Notes vol. 34, Seoul National University, Korea (1996) | MR | Zbl

[15] M. González-Burgos, R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal. 46 (2006), 123-162 | MR | Zbl

[16] S. Guerrero, Null controllability of some systems of two parabolic equation with one control force, SIAM J. Control Optim. 46 no. 2 (2007), 379-394 | MR | Zbl

[17] S. Guerrero, Controllability of systems of Stokes equations with one control force: existence of insensitizing controls, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 no. 6 (2007), 1029-1054 | EuDML | Numdam | MR | Zbl

[18] O.Y. Imanuvilov, Remarks on exact controllability for the Navier–Stokes equation, ESAIM Control Optim. Calc. Var. 6 (2001), 39-72 | EuDML | Numdam | MR | Zbl

[19] O.A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York–London (1963) | MR

[20] J.-L. Lions, Quelques notions dans lʼanalyse et le contrôle de systèmes à données incomplètes, Proceedings of the XIth Congress on Differential Equations and Applications/First Congress on Applied Mathematics, Málaga, 1989, Univ. Málaga (1990), 43-54 | MR | Zbl

[21] J.-L. Lions, Sentinelles pour les systèmes distribué à données incomplètes, Rech. Math. Appl. vol. 21, Masson, Paris (1992) | MR | Zbl

[22] S. Micu, J.H. Ortega, L. De Teresa, An example of ϵ-insensitizing controls for the heat equation with no intersecting observation and control regions, Appl. Math. Lett. 17 no. 8 (2004), 927-932 | MR | Zbl

[23] L. Tebou, Some results on the controllability of coupled semilinear wave equations: the desensitizing control case, SIAM J. Control Optim. 49 no. 3 (2011), 1221-1238 | MR | Zbl

[24] R. Temam, Navier–Stokes Equation. Theory and Numerical Analysis, Stud. Math. Appl. vol. 2, North Holland Publishing Co., Amsterdam (1977) | MR | Zbl

[25] Y. Yan, F. Sun, Insensitizing controls for a forward stochastic heat equation, J. Math. Anal. Appl. (2011) | MR | Zbl