We consider Kolmogorov-type equations on a rectangle domain , that combine diffusion in variable and transport in variable at speed , , with Dirichlet boundary conditions in . We study the null controllability of this equation with a distributed control as source term, localized on a subset of . When the control acts on a horizontal strip with , then the system is null controllable in any time when , and only in large time when (see [K. Beauchard, Math. Control Signals Syst. 26 (2014) 145–176]). In this article, we prove that, when , the system is not null controllable (whatever is) in this configuration. This is due to the diffusion weakening produced by the first order term. When the control acts on a vertical strip with ω̅1⊂��, we investigate the null controllability on a toy model, where is replaced by , and is an open subset of . As the original system, this toy model satisfies the controllability properties listed above. We prove that, for and for appropriate domains , then null controllability does not hold (whatever is), when the control acts on a vertical strip with ω̅1⊂��. Thus, a geometric control condition is required for the null controllability of this toy model. This indicates that a geometric control condition may be necessary for the original model too.
Keywords: Null controllability, degenerate parabolic equation, hypoelliptic operator, geometric control condition
@article{COCV_2015__21_2_487_0, author = {Beauchard, Karine and Helffer, Bernard and Henry, Raphael and Robbiano, Luc}, title = {Degenerate parabolic operators of {Kolmogorov} type with a geometric control condition}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {487--512}, publisher = {EDP-Sciences}, volume = {21}, number = {2}, year = {2015}, doi = {10.1051/cocv/2014035}, zbl = {1311.93042}, mrnumber = {3348409}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014035/} }
TY - JOUR AU - Beauchard, Karine AU - Helffer, Bernard AU - Henry, Raphael AU - Robbiano, Luc TI - Degenerate parabolic operators of Kolmogorov type with a geometric control condition JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 487 EP - 512 VL - 21 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014035/ DO - 10.1051/cocv/2014035 LA - en ID - COCV_2015__21_2_487_0 ER -
%0 Journal Article %A Beauchard, Karine %A Helffer, Bernard %A Henry, Raphael %A Robbiano, Luc %T Degenerate parabolic operators of Kolmogorov type with a geometric control condition %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 487-512 %V 21 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014035/ %R 10.1051/cocv/2014035 %G en %F COCV_2015__21_2_487_0
Beauchard, Karine; Helffer, Bernard; Henry, Raphael; Robbiano, Luc. Degenerate parabolic operators of Kolmogorov type with a geometric control condition. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 2, pp. 487-512. doi : 10.1051/cocv/2014035. http://archive.numdam.org/articles/10.1051/cocv/2014035/
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas graphs and mathematical tables. Edited by Milton. New York, Dover (1972). | MR | Zbl
Carleman estimates for degenerate parabolic operators with applications to null controllability. J. Evol. Equ. 6 (2006) 161–204. | DOI | MR | Zbl
, , and ,Uniqueness and nonuniqueness of the Cauchy problem for hyperbolic operators with double characteristics. Commun. Partial Differ. Equ. 6 (1981) 799–828. | DOI | Zbl
and ,The stability of the normal state of superconductors in the presence of electric currents. Siam J. Math. Anal. 40 (2008) 824–850. | DOI | Zbl
,Global stability of the normal state of superconductors in the presence of a strong electric current. Commun. Math. Phys. 330 (2014) 1021–1094. | DOI | Zbl
and ,Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field II: The large conductivity limit. SIAM J. Math. Anal. 44 (2012) 3671–3733. | DOI | Zbl
, and ,Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field. Trans. Amer. Math. Soc. 365 (2013) 1183–1217. | DOI | Zbl
, and ,Superconductivity near the normal state under the action of electric currents and induced magnetic fields in . Commun. Math. Phys. 300 (2010) 147–184. | DOI | Zbl
, and ,Null controllability of Kolmogorov-type equations. Math. Control Signals Syst. 26 (2014) 145–176. | DOI | Zbl
,Some controllability results for the 2D Grushin equations. J. Eur. Math. Soc. 16 (2014) 67–101.
, and .Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier 19 (1969) 277–304. | DOI | Numdam | Zbl
,H. Brézis, Analyse Fonctionnelle, Théorie et Applications. Masson, Paris (1983). | Zbl
Feedback stabilization of a boundary layer equation, part2: Nonhomogeneous state equations and numerical simulations. Appl. Math. Res. Express 2009 (2010) 87–122. | Zbl
and ,Feedback stabilization of a boundary layer equation, part 1. ESAIM:COCV 17 (2011) 506–551. | Numdam | Zbl
and ,Controllability of 1-D coupled degenerate parabolic equations. Electron. J. Differ. Equ. 73 (2009) 21. | Zbl
and ,Null controllability of degenerate parabolic operators with drift. Netw. Heterog. Media 2 (2007) 695–715. | DOI | Zbl
, and ,Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form. J. Evol. Equ. 8 (2008) 583–616. | DOI | Zbl
, and ,Persistent regional null controllability for a class of degenerate parabolic equations. Commun. Pure Appl. Anal. 3 (2004) 607–635. | DOI | Zbl
, and ,Null controllability of degenerate heat equations. Adv. Differ. Equ. 10 (2005) 153–190. | Zbl
, and ,Carleman estimates for a class of degenerate parabolic operators. SIAM J. Control Optim. 47 (2008) 1–19. | DOI | Zbl
, and ,Carleman estimates and null controllability for boundary-degenerate parabolic operators. C. R. Math. Acad. Sci. Paris 347 (2009) 147–152. | DOI | Zbl
, and ,Wild spectral behaviour of anharmonic oscillators. Bull. London Math. Soc. 32 (2000) 432–438. | DOI | Zbl
,S. Didelot, Etude d’une perturbation singulière elliptique dégénérée. Thèse de doctorat, Reims (1999).
Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43 (1971) 272–292. | DOI | Zbl
and ,Carleman estimates for degenerate parabolic equations with first order terms and applications. C. R. Math. Acad. Sci. Paris 348 (2010) 391–396. | DOI | Zbl
and ,A.V. Fursikov and O.Y. Imanuvilov, Controllability of evolution equations. Vol. 34 of Lect. Notes Series. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1996). | Zbl
B. Helffer, Spectral Theory and its Applications. Cambridge University Press (2013). | Zbl
Propriétés asymptotiques du spectre d’opérateurs pseudo-différentiels sur . Commun. Partial Differ. Eq. 7 (1982) 795–882. | Zbl
and ,B. Helffer and J. Sjöstrand, From resolvent bounds to semigroup bounds, Appendix of a course by Sjöstrand. Proc. of the Evian Conference (2009). Preprint arXiv:1001.4171
R. Henry, On the semi-classical analysis of Schrödinger operators with purely imaginary electric potentials in a bounded domain. Preprint arXiv:1405.6183
Boundary controllability of parabolic equations. Uspekhi. Mat. Nauk 48 (1993) 211–212. | Zbl
,Controllability of parabolic equations. Mat. Sb. 186 (1995) 109–132. | Zbl
,T. Kato, Perturbation Theory for Linear operators. Springer-Verlag, Berlin New-York (1966). | Zbl
Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Eq. 20 (1995) 335–356. | DOI | Zbl
and ,On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM:COCV 18 (2012) 712–747. | Numdam | Zbl
and ,Carleman estimates for one-dimensional degenerate heat equations. J. Evol. Equ. 6 (2006) 325–362. | DOI | Zbl
and ,Regional null controllability of a linearized Crocco type equation. SIAM J. Control Optim. 42 (2003) 709–728. | DOI | Zbl
, and ,Localization of laplacian eigenfunctions in circular and elliptical domains. SIAM J. Appl. Math. 73 780–803. | DOI | Zbl
and ,O.A. Oleinik and V.N. Samokhin, Mathematical Models in Boundary Layer Theory. In vol. 15 of Appl. Math. Math. Comput. Chapman Hall CRC, Boca Raton, London, New York (1999). | Zbl
A. Pazy, Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. Springer Verlag, New-York (1983). | Zbl
A complete study of the pseudo-spectrum for the rotated harmonic oscillator. J. London Math. Soc. 73 (2006) 745–761. | DOI | Zbl
,Y. Sibuya, Global theory of a second order linear ordinary differential equation with a polynomial coefficient. Amsterdam, North-Holland (1975). | Zbl
An inequality involving Bessel functions of argument nearly equal to their orders. Proc. Amer. Math. Soc. 4 (1953) 858–859. | DOI | Zbl
,Counting nodal lines wich touch the boundary of an analytic domain. J. Differ. Geometry 81 (2009) 649–686. | DOI | Zbl
and ,Cited by Sources: