In this paper, we prove controllability results for some linear and semilinear systems where we find two parabolic PDEs and one elliptic PDE and we act through one locally supported in space scalar control. The arguments rely on a careful analysis of the linear case and an application of an inverse function theorem. The facts that we act through a single scalar control and one of the PDEs has no time derivative are the main novelties and introduce several nontrivial difficulties.
Accepté le :
DOI : 10.1051/cocv/2016031
Mots clés : Null controllability, parabolic-elliptic linear and semilinear systems, Carleman estimates
@article{COCV_2016__22_4_1017_0,
author = {Fern\'andez-Cara, E. and Limaco, J. and de Menezes, S. B.},
title = {Controlling linear and semilinear systems formed by one elliptic and two parabolic {PDEs} with one scalar control},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1017--1039},
publisher = {EDP-Sciences},
volume = {22},
number = {4},
year = {2016},
doi = {10.1051/cocv/2016031},
zbl = {1355.35021},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2016031/}
}
TY - JOUR AU - Fernández-Cara, E. AU - Limaco, J. AU - de Menezes, S. B. TI - Controlling linear and semilinear systems formed by one elliptic and two parabolic PDEs with one scalar control JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 1017 EP - 1039 VL - 22 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016031/ DO - 10.1051/cocv/2016031 LA - en ID - COCV_2016__22_4_1017_0 ER -
%0 Journal Article %A Fernández-Cara, E. %A Limaco, J. %A de Menezes, S. B. %T Controlling linear and semilinear systems formed by one elliptic and two parabolic PDEs with one scalar control %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 1017-1039 %V 22 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016031/ %R 10.1051/cocv/2016031 %G en %F COCV_2016__22_4_1017_0
Fernández-Cara, E.; Limaco, J.; de Menezes, S. B. Controlling linear and semilinear systems formed by one elliptic and two parabolic PDEs with one scalar control. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1017-1039. doi : 10.1051/cocv/2016031. http://archive.numdam.org/articles/10.1051/cocv/2016031/
V.M. Alekseev, V.M. Tikhomorov and S.V. Formin, Optimal control. Consultants Bureau, New York (1987). | Zbl
, , and , A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems. J. Evol. Equ. 9 (2009) 267–291. | DOI | Zbl
, , and , Recent results on the controllability of linear coupled parabolic problems: a survey. Math. Control Relat. Fields 1 (2011) 267–306. | DOI | Zbl
, , and , The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials. J. Math. Pures Appl. 96 (2011) 555–590. | DOI | Zbl
, , and , Minimal time of controllability of two parabolic equations with disjoint control and coupling domains. C. R. Math. Acad. Sci. Paris 352 (2014), 391–396. | DOI | Zbl
, , , Manuel and , Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the -dimensional boundary null controllability in cylindrical domains. SIAM J. Control Optim. 52 (2014) 2970–3001. | DOI | Zbl
and , Null controllability of a degenerate reaction-diffusion system in cardiac electro-phisyology. C. R. Math. Acad. Sci. Paris 350 (11–12) (2012) 587–590. | DOI | Zbl
M. Bendahmane and F. W. Chaves-Silva, Uniform null controllability for a degenerating reaction-diffusion system approximating a simplified cardiac model, to appear.
, and , Controllability of fast diffusion coupled parabolic systems. Math. Control Relat. Fields 4 (2014) 465–479. | DOI | Zbl
and , Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components. Invent. Math. 198 (2014) 833–880. | DOI | MR | Zbl
, , and , On the controllability of parabolic systems with a nonlinear term involving the state and the gradient. SIAM J. Control Optim. 41 (2002) 798–819. | DOI | MR | Zbl
and , Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 17 (2000) 583–616. | DOI | Numdam | MR | Zbl
and , Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45 (2006) 1395–1446. | DOI | MR | Zbl
, , and , Local exact controllability of the Navier–Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. | DOI | MR | Zbl
, and , Null controllability for a parabolic-elliptic coupled system. Bull Braz Math Soc, New Series 44 (2013) 1–24. | DOI | MR | Zbl
, and , Theoretical and numerical local null controllability of a Ladyzhenskaya-Smagorinsky model of turbulence. J. Math. Fluid Mech. 17 (2015) 669–698. | DOI | MR | Zbl
A. Fursikov and O.Yu. Imanuvilov, Controllability of evolution equations. Vol. 34 of Lect. Notes Ser. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR | Zbl
B.Z. Guo and L. Zhang, Local exact controllability of a parabolic system of chemotaxis, Preprint, School of Computational and Applied Mathematics. University of the Witwatersrand, South Africa (2012).
B.-Z. Guo and L. Zhang, Local exact controllability of a parabolic system of chemotaxis. Preprint (2013). | arXiv
, and , Single input controllability of a simplified fluid-structure interaction model. ESAIM: COCV 19 (2013) 20–42. | Numdam | MR | Zbl
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