On the cost of null-control of an artificial advection-diffusion problem

Pierre Cornilleau; Sergio Guerrero

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 4, page 1209-1224
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Cornilleau, Pierre, and Guerrero, Sergio. "On the cost of null-control of an artificial advection-diffusion problem." ESAIM: Control, Optimisation and Calculus of Variations 19.4 (2013): 1209-1224. <http://eudml.org/doc/272842>.

@article{Cornilleau2013,
abstract = {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.},
author = {Cornilleau, Pierre, Guerrero, Sergio},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {vanishing viscosity; controllability; heat equation; Carleman; Carleman estimate},
language = {eng},
number = {4},
pages = {1209-1224},
publisher = {EDP-Sciences},
title = {On the cost of null-control of an artificial advection-diffusion problem},
url = {http://eudml.org/doc/272842},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Cornilleau, Pierre
AU - Guerrero, Sergio
TI - On the cost of null-control of an artificial advection-diffusion problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 4
SP - 1209
EP - 1224
AB - 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.
LA - eng
KW - vanishing viscosity; controllability; heat equation; Carleman; Carleman estimate
UR - http://eudml.org/doc/272842
ER -

References

top
  1. [1] J.-M. Coron and S. Guerrero, Singular optimal control: a linear 1-D parabolic-hyperbolic example. Asymptot. Anal.44 (2005) 237–257. Zbl1078.93009MR2176274
  2. [2] P. Cornilleau and S. Guerrero, Controllability and observability of an artificial advection-diffusion problem. Math. Control Signals Syst.24 (2012) 265–294 Zbl1244.93023MR2935463
  3. [3] R. Danchin, Poches de tourbillon visqueuses. J. Math. Pures Appl.76 (1997) 609–647. MR1472116
  4. [4] S. Dolecki and D. Russell, A general theory of observation and control. SIAM J. Control Optim.15 (1977) 185–220. Zbl0353.93012MR451141
  5. [5] K.J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations. Graduate texts in mathematics. Springer-Verlag (2000). Zbl0952.47036MR1721989
  6. [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). Zbl0862.49004MR1406566
  7. [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. Zbl1180.93015MR2558179
  8. [8] O. Glass and S. Guerrero, Uniform controllability of a transport equation in zero diffusion-dispersion limit. M3AS 19 (2009) 1567–1601. Zbl1194.93025MR2571687
  9. [9] P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, London (1985). Zbl0695.35060MR775683
  10. [10] S. Guerrero and G. Lebeau, Singular optimal control for a transport-diffusion equation. Commun. Partial Differ. Equ.32 (2007) 1813–1836. Zbl1135.35017MR2372489
  11. [11] L. Halpern, Artificial boundary for the linear advection diffusion equation. Math. Comput.46 (1986) 425–438. Zbl0649.35041MR829617
  12. [12] L. Miller, On the null-controllability of the heat equation in unbounded domains. Bull. Sci. Math.129 (2005) 175–185. Zbl1079.35018MR2123266

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.