Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T16:53:48.937Z Has data issue: false hasContentIssue false
  • English
  • Français

On the cost of null-control of an artificialadvection-diffusion problem

Published online by Cambridge University Press:  27 August 2013

Pierre Cornilleau  and
Sergio Guerrero
Pierre Cornilleau
Affiliation:
Teacher at Lycée du parc des Loges, 1, boulevard des Champs-Élysées, 91012 Évry, France. pierre.cornilleau@ens-lyon.org
Sergio Guerrero
Affiliation:
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris Cédex 05, France; guerrero@ann.jussieu.fr
Get access

Abstract

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

Keywords

Vanishing viscositycontrollabilityheat equationCarleman
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 1209 - 1224
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Coron, J.-M. and Guerrero, S., Singular optimal control: a linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44 (2005) 237–257. Google Scholar
Cornilleau, P. and Guerrero, S., Controllability and observability of an artificial advection-diffusion problem. Math. Control Signals Syst. 24 (2012) 265–294 Google Scholar
Danchin, R., Poches de tourbillon visqueuses. J. Math. Pures Appl. 76 (1997) 609–647. Google Scholar
Dolecki, S. and Russell, D., A general theory of observation and control. SIAM J. Control Optim. 15 (1977) 185–220. Google Scholar
K.J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations. Graduate texts in mathematics. Springer-Verlag (2000).
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).
Glass, O., 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. Google Scholar
Glass, O. and Guerrero, S., Uniform controllability of a transport equation in zero diffusion-dispersion limit. M3AS 19 (2009) 1567–1601. Google Scholar
P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, London (1985).
Guerrero, S. and Lebeau, G., Singular optimal control for a transport-diffusion equation. Commun. Partial Differ. Equ. 32 (2007) 1813–1836. Google Scholar
Halpern, L., Artificial boundary for the linear advection diffusion equation. Math. Comput. 46 (1986) 425–438. Google Scholar
Miller, L., On the null-controllability of the heat equation in unbounded domains. Bull. Sci. Math. 129 (2005) 175–185. Google Scholar