no. 4
On the cost of null-control of an artificial advection-diffusion problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1209-1224.

In this paper we study the null-controllability of an artificial advection-diffusion system in dimension n. Using a spectral method, we prove that the control cost goes to zero exponentially when the viscosity vanishes and the control time is large enough. On the other hand, we prove that the control cost tends to infinity exponentially when the viscosity vanishes and the control time is small enough.

DOI : 10.1051/cocv/2013048
Classification : 35K57, 93B05
Mots clés : vanishing viscosity, controllability, heat equation, Carleman 
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Cornilleau, Pierre; Guerrero, Sergio. On the cost of null-control of an artificial advection-diffusion problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1209-1224. doi : 10.1051/cocv/2013048. http://www.numdam.org/articles/10.1051/cocv/2013048/

[1] J.-M. Coron and S. Guerrero, Singular optimal control: a linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44 (2005) 237-257. | MR

[2] P. Cornilleau and S. Guerrero, Controllability and observability of an artificial advection-diffusion problem. Math. Control Signals Syst. 24 (2012) 265-294 | MR

[3] R. Danchin, Poches de tourbillon visqueuses. J. Math. Pures Appl. 76 (1997) 609-647. | MR

[4] S. Dolecki and D. Russell, A general theory of observation and control. SIAM J. Control Optim. 15 (1977) 185-220. | MR

[5] K.J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations. Graduate texts in mathematics. Springer-Verlag (2000). | MR

[6] A.V. Fursikov and O. Imanuvilov, Controllability of evolution equations. In Lect. Notes Ser. vol. 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR

[7] O. Glass, A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit. J. Funct. Anal. 258 (2010) 852-868. | MR

[8] O. Glass and S. Guerrero, Uniform controllability of a transport equation in zero diffusion-dispersion limit. M3AS 19 (2009) 1567-1601. | MR

[9] P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, London (1985). | MR

[10] S. Guerrero and G. Lebeau, Singular optimal control for a transport-diffusion equation. Commun. Partial Differ. Equ. 32 (2007) 1813-1836. | MR | Zbl

[11] L. Halpern, Artificial boundary for the linear advection diffusion equation. Math. Comput. 46 (1986) 425-438. | MR | Zbl

[12] L. Miller, On the null-controllability of the heat equation in unbounded domains. Bull. Sci. Math. 129 (2005) 175-185. | MR | Zbl

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