Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports

Karine Beauchard; Michela Egidi; Karel Pravda-Starov

doi:10.5802/crmath.79

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Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports

Karine Beauchard ¹ ; Michela Egidi ² ; Karel Pravda-Starov ¹

¹ Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
² Ruhr Universität Bochum, Fakultät für Mathematik, Gebaude IB 3/55, Universitätsstr. 150, 44780 Bochum, Germany

Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 651-700.

Résumé

We study the null-controllability of some hypoelliptic quadratic parabolic equations posed on the whole Euclidean space with moving control supports, and provide necessary or sufficient geometric conditions on the moving control supports to ensure null-controllability. The first class of equations is the one associated to non-autonomous Ornstein–Uhlenbeck operators satisfying a generalized Kalman rank condition. In particular, when the moving control supports comply with the flow associated to the transport part of the Ornstein–Uhlenbeck operators, a necessary and sufficient condition for null-controllability on the moving control supports is established. The second class of equations is the class of accretive non-selfadjoint quadratic operators with zero singular spaces for which some sufficient geometric conditions on the moving control supports are also given to ensure null-controllability.

Reçu le : 2019-08-27
Révisé le : 2020-05-25
Accepté le : 2020-05-26
Publié le : 2020-10-02

DOI : 10.5802/crmath.79

Classification : 93B05, 35H10

Affiliations des auteurs :

Karine Beauchard ¹ ; Michela Egidi ² ; Karel Pravda-Starov ¹

¹ Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
² Ruhr Universität Bochum, Fakultät für Mathematik, Gebaude IB 3/55, Universitätsstr. 150, 44780 Bochum, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Karine Beauchard and Michela Egidi and Karel Pravda-Starov},
     title = {Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {651--700},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {6},
     year = {2020},
     doi = {10.5802/crmath.79},
     language = {en},
}

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Karine Beauchard; Michela Egidi; Karel Pravda-Starov. Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 651-700. doi : 10.5802/crmath.79. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.79/

Bibliographie
Cité par

[1] Paul Alphonse Quadratic differential equations: partial Gelfand–Shilov smoothing effect and null-controllability, J. Inst. Math. Jussieu (2020), pp. 1-53 | DOI

[2] Jone Apraiz; Luis Escauriaza Zubiria; Gengsheng Wang; Can Zhang Observability inequalities and measurable sets, J. Eur. Math. Soc., Volume 16 (2014) no. 11, pp. 2433-2475 | DOI | MR | Zbl

[3] Karine Beauchard; Philippe Jaming; Karel Pravda-Starov Spectral inequality for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations (2018) (https://arxiv.org/abs/1804.04895)

[4] Karine Beauchard; Karel Pravda-Starov Null-controllability of non-autonomous Ornstein–Uhlenbeck equations, J. Math. Anal. Appl., Volume 456 (2017) no. 1, pp. 496-524 | DOI | MR | Zbl

[5] Karine Beauchard; Karel Pravda-Starov Null-controllability of hypoelliptic quadratic differential equations, J. Éc. Polytech., Math., Volume 5 (2018), pp. 1-43 | DOI | MR | Zbl

[6] Henri Cartan Calcul différentiel, Hermann, 1967 | Zbl

[7] A. Chang An algebraic characterization of controllability, IEEE Trans. Autom. Control (1965), pp. 112-113 | DOI

[8] Felipe W. Chaves-Silva; Lionel Rosier; Enrique Zuazua Null-controllability of a system of viscoelasticity with moving control, J. Math. Pures Appl., Volume 101 (2014) no. 2, pp. 198-222 | DOI | MR | Zbl

[9] Jean-Michel Coron Control and nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007 | MR | Zbl

[10] Michela Egidi; Ivan Veselić Sharp geometric condition for null-controllability of the heat equation on $ℝ^{d}$ and consistent estimates on the control cost, Arch. Math., Volume 111 (2018) no. 1, pp. 85-99 | DOI | MR | Zbl

[11] Israilʼ M. Gelʼfand; Georgiĭ E. Shilov Generalized Functions. Vol. 2: Spaces of fundamental and generalized functions, Academic Press Inc., 1968 | Zbl

[12] Todor Gramchev; Stevan Pilipović; Luigi Rodino Classes of degenerate elliptic operators in Gelfand–Shilov spaces, New developments in pseudo-differential operators (Operator Theory: Advances and Applications), Volume 189, Birkhäuser, 2009, pp. 15-31 | MR | Zbl

[13] Victor Havin; Burglind Jöricke The uncertainty principle in harmonic analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete, 28, Springer, 1994 | MR | Zbl

[14] Michael Hitrik; Karel Pravda-Starov Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann., Volume 344 (2009) no. 4, pp. 801-846 | DOI | MR | Zbl

[15] Michael Hitrik; Karel Pravda-Starov Semiclassical hypoelliptic estimates for non-selfadjoint operators with double characteristics, Commun. Partial Differ. Equations, Volume 35 (2010) no. 6, pp. 988-1028 | DOI | MR | Zbl

[16] Michael Hitrik; Karel Pravda-Starov Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics, Ann. Inst. Fourier, Volume 63 (2013) no. 3, pp. 985-1032 | DOI | Numdam | MR | Zbl

[17] Michael Hitrik; Karel Pravda-Starov; Joe Viola Short-time asymptotics of the regularizing effect for semigroups generated by quadratic operators, Bull. Sci. Math., Volume 141 (2017) no. 7, pp. 615-675 | DOI | MR | Zbl

[18] Michael Hitrik; Karel Pravda-Starov; Joe Viola From semigroups to subelliptic estimates for quadratic operators, Trans. Am. Math. Soc., Volume 370 (2018) no. 10, pp. 7391-7415 | DOI | MR | Zbl

[19] Lars Hörmander Symplectic classification of quadratic forms and general Mehler formulas, Math. Z., Volume 219 (1995) no. 3, pp. 413-449 | DOI | MR | Zbl

[20] V. E. Kacnelʼson Equivalent norms in spaces of entire functions, Mat. Sb., N. Ser., Volume 92 (1973) no. 134, pp. 34-54 | MR | Zbl

[21] Oleg Kovrijkine Some results related to the Logvinenko–Sereda Theorem, Proc. Am. Math. Soc., Volume 129 (2001) no. 10, pp. 3037-3047 | DOI | MR | Zbl

[22] Michael Langenbruch Hermite functions and weighted spaces of generalized functions, Manuscr. Math., Volume 119 (2006) no. 3, pp. 269-285 | DOI | MR | Zbl

[23] Jérôme Le Rousseau; Gilles Lebeau; Peppino Terpolilli; Emmanuel Trélat Geometric control condition for the wave equation with a time-dependent observation domain, Anal. PDE, Volume 10 (2017) no. 4, pp. 983-1015 | DOI | MR | Zbl

[24] Nicolas Lerner; Yoshinori Morimoto; Karel Pravda-Starov; Chao-Jiang Xu Gelfand–Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation, J. Funct. Anal., Volume 269 (2015) no. 2, pp. 459-535 | DOI | MR | Zbl

[25] Jacques-Louis Lions Optimal control of systems governed by partial differential equations, Grundlehren der Mathematischen Wissenschaften, 170, Springer, 1971 | MR | Zbl

[26] Vladimir N. Logvinenko; Yu. F. Sereda Equivalent norms in spaces of entire functions of exponential type, Teor. Funkts. Funkts. Anal. Prilozh. (1974), pp. 102-111

[27] Jérémy Martin; Karel Pravda-Starov Spectral inequality for finite combinations of Hermite functions and null-controllability from thick control subsets with respect to unbounded densities (2019) (work in preparation)

[28] Philippe Martin; Lionel Rosier; Pierre Rouchon Null-controllability of the structurally damped wave equation with moving control, SIAM J. Control Optimization, Volume 51 (2013) no. 1, pp. 660-684 | DOI | MR | Zbl

[29] Luc Miller A direct Lebeau–Robbiano strategy for the observability of heat-like semigroups, Discrete Contin. Dyn. Syst., Volume 14 (2010) no. 4, pp. 1465-1485 | MR | Zbl

[30] Luc Miller Spectral inequalities for the control of linear PDEs, PDE’s, dispersion, scattering theory and control theory (Séminaires et Congrès), Volume 30, Société Mathématique de France, 2017, pp. 81-98 | MR | Zbl

[31] Camil Muscalu; Wilhelm Schlag Classical and Multilinear Harmonic Analysis, Cambridge Studies in Advanced Mathematics, 137, Cambridge University Press, 2013 | MR | Zbl

[32] Fabio Nicola; Luigi Rodino Global pseudo-differential calculus on Euclidean spaces, Pseudo-Differential Operators. Theory and Applications, 4, Birkhäuser, 2010 | MR | Zbl

[33] Michela Ottobre; Grigorios A. Pavliotis; Karel Pravda-Starov Exponential return to equilibrium for hypoelliptic quadratic systems, J. Funct. Anal., Volume 262 (2012) no. 9, pp. 4000-4039 | DOI | MR | Zbl

[34] Boris P. Paneah On certain theorems of Paley–Wiener type, Dokl. Akad. Nauk SSSR, Volume 138 (1961), pp. 47-50 translated in Sov. Math., Dokl. 2 (1961), p. 533-536

[35] Boris P. Paneah On certain problems of harmonic analysis, Dokl. Akad. Nauk SSSR, Volume 142 (1962), pp. 1026-1029 translated in Sov. Math., Dokl. 2 (1961), p. 239-2442

[36] Amnon Pazy Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer, 1983 | MR | Zbl

[37] Kim Dang Phung; Gengsheng Wang An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., Volume 15 (2013) no. 2, pp. 681-703 | DOI | MR | Zbl

[38] Karel Pravda-Starov Subelliptic estimates for quadratic differential operators, Am. J. Math., Volume 133 (2011) no. 1, pp. 39-89 | DOI | MR | Zbl

[39] Karel Pravda-Starov; Luigi Rodino; Patrik Wahlberg Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians, Math. Nachr., Volume 291 (2018) no. 1, pp. 128-159 | DOI | Zbl

[40] Lee M. Silverman; Hollister E. Meadows Controllability and time-variable unilateral networks, IEEE Trans. Circuit Theory, Volume 12 (1965), pp. 308-314 | DOI | MR

[41] Johannes Sjöstrand Parametrices for pseudodifferential operators with multiple characteristics, Ark. Mat., Volume 12 (1974), pp. 85-130 | DOI | MR | Zbl

[42] Johannes Sjöstrand Resolvent estimates for non-selfadjoint operators via semigroups, Around the research of Vladimir Maz’ya. III. Analysis and applications (International Mathematical Series (New York)), Volume 13, Springer, 2010, pp. 359-384 | MR | Zbl

[43] Joachim Toft; Andrei Khrennikov; Börje Nilsson; Sven Nordebo Decompositions of Gelfand–Shilov kernels into kernels of similar class, J. Math. Anal. Appl., Volume 396 (2012) no. 1, pp. 315-322 | DOI | MR | Zbl

[44] Joe Viola Resolvent estimates for non-selfadjoint operators with double characteristics, J. Lond. Math. Soc., Volume 85 (2012) no. 1, pp. 41-78 | DOI | MR | Zbl

[45] Joe Viola Non-elliptic quadratic forms and semiclassical estimates for non-selfadjoint operators, Int. Math. Res. Not., Volume 2013 (2013) no. 20, pp. 4615-4671 | DOI | MR | Zbl

[46] Joe Viola Spectral projections and resolvent bounds for partially elliptic quadratic differential operators, J. Pseudo-Differ. Oper. Appl., Volume 4 (2013) no. 2, pp. 145-221 | DOI | MR | Zbl

[47] Gengsheng Wang $L^{\infty}$ -null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optimization, Volume 47 (2008) no. 4, pp. 1701-1720 | DOI | MR | Zbl

[48] Gengsheng Wang; Ming Wang; Can Zhang; Yubiao Zhang Observable set, observability, interpolation inequality and spectral inequality for the heat equation in $ℝ^{n}$ , J. Math. Pures Appl., Volume 126 (2019), pp. 144-194 | DOI | MR | Zbl

[49] Yubiao Zhang Unique continuation estimates for the Kolmogorov equation in the whole space, C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 4, pp. 389-393 | DOI | MR | Zbl

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