Degenerate parabolic operators of Kolmogorov type with a geometric control condition

Beauchard, Karine; Helffer, Bernard; Henry, Raphael; Robbiano, Luc

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 2, pp. 487-512.

We consider Kolmogorov-type equations on a rectangle domain $(x,v)\in \Omega =T\times (-1,1)$, that combine diffusion in variable $v$ and transport in variable $x$ at speed $v^{\gamma }$, $\gamma \in N^{*}$, with Dirichlet boundary conditions in $v$. We study the null controllability of this equation with a distributed control as source term, localized on a subset $\omega$ of $\Omega$. When the control acts on a horizontal strip $\omega =T\times (a,b)$ with $0<a<b<1$, then the system is null controllable in any time $T>0$ when $\gamma =1$, and only in large time $T>T_{min}>0$ when $\gamma =2$ (see [K. Beauchard, Math. Control Signals Syst. 26 (2014) 145–176]). In this article, we prove that, when $\gamma >3$, the system is not null controllable (whatever $T$ 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 $\omega =\omega {}_1\times (-1,1)$ with ω̅₁⊂��, we investigate the null controllability on a toy model, where $(\partial {}_x,x\in T)$ is replaced by $(i{(-\Delta )}^{1/2},x\in \Omega {}_1)$, and $\Omega {}_1$ is an open subset of $R^N$. As the original system, this toy model satisfies the controllability properties listed above. We prove that, for $\gamma =1,2$ and for appropriate domains $(\Omega {}_1,\omega {}_1)$, then null controllability does not hold (whatever $T>0$ is), when the control acts on a vertical strip $\omega =\omega {}_1\times (-1,1)$ 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.

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

F. Alabau-Boussouira, P. Cannarsa, and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability. J. Evol. Equ. 6 (2006) 161–204. | DOI | MR | Zbl

S. Alinhac and C. Zuily, Uniqueness and nonuniqueness of the Cauchy problem for hyperbolic operators with double characteristics. Commun. Partial Differ. Equ. 6 (1981) 799–828. | DOI | Zbl

Y. Almog, The stability of the normal state of superconductors in the presence of electric currents. Siam J. Math. Anal. 40 (2008) 824–850. | DOI | Zbl

Y. Almog and B. Helffer, 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

Y. Almog, B. Helffer and X. Pan, 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

Y. Almog, B. Helffer and X. Pan, 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

Y. Almog, B. Helffer and X.-B. Pan, Superconductivity near the normal state under the action of electric currents and induced magnetic fields in $R^2$. Commun. Math. Phys. 300 (2010) 147–184. | DOI | Zbl

K. Beauchard, Null controllability of Kolmogorov-type equations. Math. Control Signals Syst. 26 (2014) 145–176. | DOI | Zbl

K. Beauchard, P. Cannarsa and R. Guglielmi. Some controllability results for the 2D Grushin equations. J. Eur. Math. Soc. 16 (2014) 67–101.

J.-M. Bony, 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

J.-M. Buchot and J.-P. Raymond, Feedback stabilization of a boundary layer equation, part2: Nonhomogeneous state equations and numerical simulations. Appl. Math. Res. Express 2009 (2010) 87–122. | Zbl

J.-M. Buchot and J.-P. Raymond, Feedback stabilization of a boundary layer equation, part 1. ESAIM:COCV 17 (2011) 506–551. | Numdam | Zbl

P. Cannarsa and L. De Teresa, Controllability of 1-D coupled degenerate parabolic equations. Electron. J. Differ. Equ. 73 (2009) 21. | Zbl

P. Cannarsa, G. Fragnelli and D. Rocchetti, Null controllability of degenerate parabolic operators with drift. Netw. Heterog. Media 2 (2007) 695–715. | DOI | Zbl

P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form. J. Evol. Equ. 8 (2008) 583–616. | DOI | Zbl

P. Cannarsa, P. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations. Commun. Pure Appl. Anal. 3 (2004) 607–635. | DOI | Zbl

P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations. Adv. Differ. Equ. 10 (2005) 153–190. | Zbl

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators. SIAM J. Control Optim. 47 (2008) 1–19. | DOI | Zbl

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates and null controllability for boundary-degenerate parabolic operators. C. R. Math. Acad. Sci. Paris 347 (2009) 147–152. | DOI | Zbl

E.B. Davies, 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).

H.O. Fattorini and D. Russel, Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Ration. Mech. Anal. 43 (1971) 272–292. | DOI | Zbl

C. Flores and L. De Teresa, Carleman estimates for degenerate parabolic equations with first order terms and applications. C. R. Math. Acad. Sci. Paris 348 (2010) 391–396. | DOI | Zbl

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

B. Helffer and D. Robert, Propriétés asymptotiques du spectre d’opérateurs pseudo-différentiels sur $R^n$. Commun. Partial Differ. Eq. 7 (1982) 795–882. | Zbl

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

O.Y. Imanuvilov, Boundary controllability of parabolic equations. Uspekhi. Mat. Nauk 48 (1993) 211–212. | Zbl

O.Y. Imanuvilov, 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

G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Eq. 20 (1995) 335–356. | DOI | Zbl

G. Lebeau and J. Le Rousseau, 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

P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations. J. Evol. Equ. 6 (2006) 325–362. | DOI | Zbl

P. Martinez, J. Vancostonoble and J.-P. Raymond, Regional null controllability of a linearized Crocco type equation. SIAM J. Control Optim. 42 (2003) 709–728. | DOI | Zbl

B.-T. Nguyen and D.S. Grebekov, Localization of laplacian eigenfunctions in circular and elliptical domains. SIAM J. Appl. Math. 73 780–803. | DOI | Zbl

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

K. Pravda-Starov, 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

K.M. Siegel, An inequality involving Bessel functions of argument nearly equal to their orders. Proc. Amer. Math. Soc. 4 (1953) 858–859. | DOI | Zbl

J. Toth and S. Zelditch, Counting nodal lines wich touch the boundary of an analytic domain. J. Differ. Geometry 81 (2009) 649–686. | DOI | Zbl