Null-controllability of hypoelliptic quadratic differential equations

Karine Beauchard; Karel Pravda-Starov

doi:10.5802/jep.62

Null-controllability of hypoelliptic quadratic differential equations
[Contrôlabilité à zéro d’équations aux dérivées partielles quadratiques hypoelliptiques]

Karine Beauchard ¹ ; Karel Pravda-Starov ²

¹ IRMAR, École Normale Supérieure de Rennes, UBL, CNRS, Campus de Ker Lann Avenue Robert Schumann, 35170 Bruz, France
² IRMAR, CNRS UMR 6625, Université de Rennes 1 Campus de Beaulieu, 263 avenue du Général Leclerc, CS 74205, 35042 Rennes cedex, France

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 1-43.

Résumé
Abstract

Nous étudions la contrôlabilité à zéro d’équations paraboliques associées à une classe générale d’opérateurs différentiels quadratiques hypoelliptiques. Les opérateurs différentiels quadratiques sont les opérateurs définis, en quantification de Weyl, par un symbole quadratique à valeurs complexes. Dans ce travail, nous considérons la classe des opérateurs quadratiques accrétifs avec espace singulier réduit au singleton zéro. Ces opérateurs différentiels, possiblement dégénérés et non auto-adjoints, sont hypoelliptiques et génèrent des semi-groupes de contractions, régularisant dans des espaces de Gelfand-Shilov particuliers, en tout temps strictement positif. Grâce à cet effet régularisant, nous démontrons, en adaptant la méthode de Lebeau-Robbiano, que les équations paraboliques associées sont contrôlables à zéro en tout temps strictement positif, lorsque les contrôles sont localisés sur un sous domaine, assurant classiquement la contrôlabilité à zéro de l’équation de la chaleur. Nous déduisons de ce résultat la contrôlabilité à zéro d’équations paraboliques associées à des opérateurs hypoelliptiques de Ornstein-Uhlenbeck agissant sur des espaces $L^{2}$ à poids, dont le poids est la mesure invariante. La même stratégie fournit la contrôlabilité à zéro, en tout temps strictement positif, avec le même support de contrôle, pour les équations paraboliques associées aux opérateurs de Ornstein-Uhlenbeck hypoelliptiques agissant sur l’espace $L^{2}$ plat, étendant ainsi le résultat connu pour l’équation de la chaleur et l’équation de Kolmogorov posées sur tout l’espace.

We study the null-controllability of parabolic equations associated with a general class of hypoelliptic quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. We consider in this work the class of accretive quadratic operators with zero singular spaces. These possibly degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in specific Gelfand-Shilov spaces for any positive time. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that parabolic equations associated with these operators are null-controllable in any positive time from control regions, for which null-controllability is classically known to hold in the case of the heat equation on the whole space. Some applications of this result are then given to the study of parabolic equations associated with hypoelliptic Ornstein-Uhlenbeck operators acting on weighted $L^{2}$ spaces with respect to invariant measures. By using the same strategy, we also establish the null-controllability in any positive time from the same control regions for parabolic equations associated with any hypoelliptic Ornstein-Uhlenbeck operator acting on the flat $L^{2}$ space extending in particular the known results for the heat equation or the Kolmogorov equation on the whole space.

Reçu le : 2016-03-15
Accepté le : 2017-10-26
Publié le : 2017-11-28

MR Zbl

DOI : 10.5802/jep.62

Classification : 93B05, 35H10
Keywords: Null-controllability, observability, quadratic differential operators, Ornstein-Uhlenbeck operators, Fokker-Planck operators, hypoellipticity
Mot clés : Contrôlabilité à zéro, observabilité, opérateurs différentiels quadratiques, opérateurs de Ornstein-Uhlenbeck, opérateurs de Fokker-Planck, hypoellipticité

Affiliations des auteurs :

Karine Beauchard ¹ ; Karel Pravda-Starov ²

¹ IRMAR, École Normale Supérieure de Rennes, UBL, CNRS, Campus de Ker Lann Avenue Robert Schumann, 35170 Bruz, France
² IRMAR, CNRS UMR 6625, Université de Rennes 1 Campus de Beaulieu, 263 avenue du Général Leclerc, CS 74205, 35042 Rennes cedex, France

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Null-controllability of hypoelliptic quadratic differential equations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Karine Beauchard; Karel Pravda-Starov. Null-controllability of hypoelliptic quadratic differential equations. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 1-43. doi : 10.5802/jep.62. https://jep.centre-mersenne.org/articles/10.5802/jep.62/

Bibliographie
Cité par

[1] F. Alabau-Boussouira, P. Cannarsa & G. Fragnelli - “Carleman estimates for degenerate parabolic operators with applications to null controllability”, J. Evol. Equ. 6 (2006) no. 2, p. 161-204 | DOI | MR | Zbl

[2] J. Apraiz, L. Escauriaza, G. Wang & C. Zhang - “Observability inequalities and measurable sets”, J. Eur. Math. Soc. (JEMS) 16 (2014) no. 11, p. 2433-2475 | DOI | MR | Zbl

[3] K. Beauchard - “Null controllability of Kolmogorov-type equations”, Math. Control Signals Systems 26 (2014) no. 1, p. 145-176 | MR | Zbl

[4] K. Beauchard & P. Cannarsa - “Heat equation on the Heisenberg group: observability and applications”, J. Differential Equations 262 (2017) no. 8, p. 4475-4521 | DOI | MR | Zbl

[5] K. Beauchard, P. Cannarsa & R. Guglielmi - “Null controllability of Grushin-type operators in dimension two”, J. Eur. Math. Soc. (JEMS) 16 (2014) no. 1, p. 67-101 | DOI | MR | Zbl

[6] K. Beauchard, P. Cannarsa & M. Yamamoto - “Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type”, Inverse Problems 30 (2014) no. 2, 025006, 26 p. | DOI | MR | Zbl

[7] K. Beauchard, B. Helffer, R. Henry & L. Robbiano - “Degenerate parabolic operators of Kolmogorov type with a geometric control condition”, ESAIM Contrôle Optim. Calc. Var. 21 (2015) no. 2, p. 487-512 | DOI | MR | Zbl

[8] M. Bramanti, G. Cupini, E. Lanconelli & E. Priola - “Global $L^{p}$ estimates for degenerate Ornstein-Uhlenbeck operators”, Math. Z. 266 (2010) no. 4, p. 789-816 | DOI | MR | Zbl

[9] P. Cannarsa, G. Fragnelli & D. Rocchetti - “Null controllability of degenerate parabolic operators with drift”, Netw. Heterog. Media 2 (2007) no. 4, p. 695-715 | DOI | MR | Zbl

[10] P. Cannarsa, G. Fragnelli & D. Rocchetti - “Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form”, J. Evol. Equ. 8 (2008) no. 4, p. 583-616 | DOI | MR | Zbl

[11] P. Cannarsa, P. Martinez & J. Vancostenoble - “Null controllability of degenerate heat equations”, Adv. Differential Equations 10 (2005) no. 2, p. 153-190 | MR | Zbl

[12] P. Cannarsa, P. Martinez & J. Vancostenoble - “Carleman estimates for a class of degenerate parabolic operators”, SIAM J. Control Optim. 47 (2008) no. 1, p. 1-19 | MR | Zbl

[13] P. Cannarsa, P. Martinez & J. Vancostenoble - “Carleman estimates and null controllability for boundary-degenerate parabolic operators”, Comptes Rendus Mathématique 347 (2009) no. 3-4, p. 147-152 | DOI | MR | Zbl

[14] E. Carypis & P. Wahlberg - “Propagation of exponential phase space singularities for Schrödinger equations with quadratic Hamiltonians”, J. Fourier Anal. Appl. 23 (2017) no. 3, p. 530-571 | DOI | Zbl

[15] J.-M. Coron - Control and nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007 | MR

[16] G. Da Prato & J. Zabczyk - Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992 | MR | Zbl

[17] T. Duyckaerts & L. Miller - “Resolvent conditions for the control of parabolic equations”, J. Funct. Anal. 263 (2012) no. 11, p. 3641-3673 | DOI | MR | Zbl

[18] J.-P. Eckmann & M. Hairer - “Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators”, Comm. Math. Phys. 212 (2000) no. 1, p. 105-164 | MR | Zbl

[19] J.-P. Eckmann & M. Hairer - “Spectral properties of hypoelliptic operators”, Comm. Math. Phys. 235 (2003) no. 2, p. 233-253 | MR | Zbl

[20] J.-P. Eckmann, C.-A. Pillet & L. Rey-Bellet - “Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures”, Comm. Math. Phys. 201 (1999) no. 3, p. 657-697 | MR | Zbl

[21] B. Farkas & L. Lorenzi - “On a class of hypoelliptic operators with unbounded coefficients in $ℝ^{N}$ ”, Comm. Pure Appl. Math. 8 (2009) no. 4, p. 1159-1201 | MR

[22] B. Farkas & A. Lunardi - “Maximal regularity for Kolmogorov operators in $L^{2}$ spaces with respect to invariant measures”, J. Math. Pures Appl. (9) 86 (2006) no. 4, p. 310-321 | DOI | MR | Zbl

[23] H. O. Fattorini & D. L. Russell - “Exact controllability theorems for linear parabolic equations in one space dimension”, Arch. Rational Mech. Anal. 43 (1971), p. 272-292 | DOI | MR | Zbl

[24] A. V. Fursikov & O. Y. Imanuvilov - Controllability of evolution equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996 | MR | Zbl

[25] I. M. Gelʼfand & G. E. Shilov - Generalized functions. Vol. 2. Spaces of fundamental and generalized functions, Academic Press, New York-London, 1968 | Zbl

[26] T. Gramchev, S. Pilipović & L. Rodino - “Classes of degenerate elliptic operators in Gelfand-Shilov spaces”, in New developments in pseudo-differential operators, Oper. Theory Adv. Appl., vol. 189, Birkhäuser, Basel, 2009, p. 15-31 | MR | Zbl

[27] F. Hérau, M. Hitrik & J. Sjöstrand - “Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications”, Internat. Math. Res. Notices (2008) no. 15, rnn057, 48 p. | MR | Zbl

[28] F. Hérau, M. Hitrik & J. Sjöstrand - “Supersymmetric structures for second order differential operators”, Algebra i Analiz 25 (2013) no. 2, p. 125-154 | MR | Zbl

[29] M. Hitrik & K. Pravda-Starov - “Spectra and semigroup smoothing for non-elliptic quadratic operators”, Math. Ann. 344 (2009) no. 4, p. 801-846 | DOI | MR | Zbl

[30] M. Hitrik & K. Pravda-Starov - “Semiclassical hypoelliptic estimates for non-selfadjoint operators with double characteristics”, Comm. Partial Differential Equations 35 (2010) no. 6, p. 988-1028 | DOI | MR | Zbl

[31] M. Hitrik & K. Pravda-Starov - “Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics”, Ann. Inst. Fourier (Grenoble) 63 (2013) no. 3, p. 985-1032 | DOI | Numdam | MR | Zbl

[32] M. Hitrik, K. Pravda-Starov & J. Viola - “Short-time asymptotics of the regularizing effect for semigroups generated by quadratic operators”, Bull. Sci. Math. 141 (2017) no. 7, p. 615-675 | DOI | MR | Zbl

[33] M. Hitrik, K. Pravda-Starov & J. Viola - “From semigroups to subelliptic estimates for quadratic operators”, Trans. Amer. Math. Soc. (to appear), arXiv:1510.02072 | Zbl

[34] L. Hörmander - “Quadratic hyperbolic operators”, in Microlocal analysis and applications (Montecatini Terme, 1989), Lect. Notes in Math., vol. 1495, Springer, Berlin, 1991, p. 118-160 | DOI | MR | Zbl

[35] L. Hörmander - “Symplectic classification of quadratic forms, and general Mehler formulas”, Math. Z. 219 (1995) no. 3, p. 413-449 | MR | Zbl

[36] A. Kolmogorov - “Zufällige Bewegungen. (Zur Theorie der Brownschen Bewegung.)”, Ann. of Math. 35 (1934) no. 1, p. 116-117 | DOI | Zbl

[37] E. Lanconelli & S. Polidoro - “On a class of hypoelliptic evolution operators”, Rend. Sem. Mat. Univ. e Politec. Torino 52 (1994) no. 1, p. 29-63 | MR | Zbl

[38] J. Le Rousseau & I. Moyano - “Null-controllability of the Kolmogorov equation in the whole phase space”, J. Differential Equations 260 (2016) no. 4, p. 3193-3233 | MR | Zbl

[39] G. Lebeau & L. Robbiano - “Contrôle exact de l’équation de la chaleur”, Comm. Partial Differential Equations 20 (1995) no. 1-2, p. 335-356 | DOI | Zbl

[40] J.-L. Lions - Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2: Perturbations., Recherches en mathématiques appliquées, vol. 9, Masson, Paris, 1988 | Zbl

[41] L. Lorenzi & M. Bertoldi - Analytical methods for Markov semigroups, Pure and Applied Mathematics (Boca Raton), vol. 283, Chapman & Hall/CRC, Boca Raton, FL, 2007 | MR | Zbl

[42] P. Martinez & J. Vancostenoble - “Carleman estimates for one-dimensional degenerate heat equations”, J. Evol. Equ. 6 (2006) no. 2, p. 325-362 | DOI | MR | Zbl

[43] G. Metafune, D. Pallara & E. Priola - “Spectrum of Ornstein-Uhlenbeck operators in $L^{p}$ spaces with respect to invariant measures”, J. Funct. Anal. 196 (2002) no. 1, p. 40-60 | DOI | MR | Zbl

[44] L. Miller - “On the null-controllability of the heat equation in unbounded domains”, Bull. Sci. Math. 129 (2005) no. 2, p. 175-185 | MR

[45] L. Miller - “Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds”, Math. Res. Lett. 12 (2005) no. 1, p. 37-47 | DOI | MR | Zbl

[46] L. Miller - “Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones” (2008), hal-00411840

[47] L. Miller - “A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups”, Discrete Contin. Dynam. Systems 14 (2010) no. 4, p. 1465-1485 | DOI | MR | Zbl

[48] L. Miller - “Spectral inequalities for the control of linear PDEs”, in PDE’s, dispersion, scattering theory and control theory (K. Ammari & G. Lebeau, eds.), Séminaires et Congrès, vol. 30, Société Mathématique de France, Paris, 2017, p. 81-98 | MR

[49] F. Nicola & L. Rodino - Global pseudo-differential calculus on Euclidean spaces, Pseudo-Differential Operators. Theory and Applications, vol. 4, Birkhäuser Verlag, Basel, 2010 | MR | Zbl

[50] M. Ottobre, G. A. Pavliotis & K. Pravda-Starov - “Exponential return to equilibrium for hypoelliptic quadratic systems”, J. Funct. Anal. 262 (2012) no. 9, p. 4000-4039 | DOI | MR | Zbl

[51] M. Ottobre, G. A. Pavliotis & K. Pravda-Starov - “Some remarks on degenerate hypoelliptic Ornstein-Uhlenbeck operators”, J. Math. Anal. Appl. 429 (2015) no. 2, p. 676-712 | DOI | MR | Zbl

[52] K. D. Phung & G. Wang - “An observability estimate for parabolic equations from a measurable set in time and its applications”, J. Eur. Math. Soc. (JEMS) 15 (2013) no. 2, p. 681-703 | DOI | MR | Zbl

[53] K. Pravda-Starov - “Subelliptic estimates for quadratic differential operators”, Amer. J. Math. 133 (2011) no. 1, p. 39-89 | DOI | MR | Zbl

[54] K. Pravda-Starov - “Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities” (2017), arXiv:1703.02797 | Zbl

[55] K. Pravda-Starov, L. Rodino & P. Wahlberg - “Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians”, Math. Nachr. (2017), published online, arXiv:1411.0251 | Zbl

[56] J. Sjöstrand - “Parametrices for pseudodifferential operators with multiple characteristics”, Ark. Mat. 12 (1974), p. 85-130 | DOI | MR | Zbl

[57] J. Sjöstrand - “Resolvent estimates for non-selfadjoint operators via semigroups”, in Around the research of Vladimir Maz’ya. III, Int. Math. Ser. (N.Y.), vol. 13, Springer, New York, 2010, p. 359-384 | DOI | MR | Zbl

[58] J. Toft, A. Khrennikov, B. Nilsson & S. Nordebo - “Decompositions of Gelfand-Shilov kernels into kernels of similar class”, J. Math. Anal. Appl. 396 (2012) no. 1, p. 315-322 | DOI | MR | Zbl

[59] J. Viola - “Resolvent estimates for non-selfadjoint operators with double characteristics”, J. London Math. Soc. (2) 85 (2012) no. 1, p. 41-78 | DOI | MR | Zbl

[60] J. Viola - “Non-elliptic quadratic forms and semiclassical estimates for non-selfadjoint operators”, Internat. Math. Res. Notices (2013) no. 20, p. 4615-4671 | DOI | MR | Zbl

[61] J. Viola - “Spectral projections and resolvent bounds for partially elliptic quadratic differential operators”, J. Pseudo-Differential. Oper. Appl. 4 (2013) no. 2, p. 145-221 | DOI | MR | Zbl

[62] P. Wahlberg - “Propagation of polynomial phase space singularities for Schrödinger equations with quadratic Hamiltonians” (2016), arXiv:1411.6518

[63] Y. Zhang - “Unique continuation estimates for the Kolmogorov equation in the whole space”, Comptes Rendus Mathématique 354 (2016) no. 4, p. 389-393 | DOI | MR

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