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

Beauchard, Karine; Egidi, Michela; Pravda-Starov, Karel

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

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.

Classification : 93B05, 35H10

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

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

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

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

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

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

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

[8] Chaves-Silva, Felipe W.; Rosier, Lionel; Zuazua, Enrique 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] Coron, Jean-Michel Control and nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007 | MR | Zbl

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

[12] Gramchev, Todor; Pilipović, Stevan; Rodino, Luigi 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] Havin, Victor; Jöricke, Burglind The uncertainty principle in harmonic analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete, 28, Springer, 1994 | MR | Zbl

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

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

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

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

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

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

[23] Le Rousseau, Jérôme; Lebeau, Gilles; Terpolilli, Peppino; Trélat, Emmanuel 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] Lerner, Nicolas; Morimoto, Yoshinori; Pravda-Starov, Karel; Xu, Chao-Jiang 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] Lions, Jacques-Louis Optimal control of systems governed by partial differential equations, Grundlehren der Mathematischen Wissenschaften, 170, Springer, 1971 | MR | Zbl

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

[27] Martin, Jérémy; Pravda-Starov, Karel 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] Martin, Philippe; Rosier, Lionel; Rouchon, Pierre 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] Miller, Luc 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] Miller, Luc 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] Muscalu, Camil; Schlag, Wilhelm Classical and Multilinear Harmonic Analysis, Cambridge Studies in Advanced Mathematics, 137, Cambridge University Press, 2013 | MR | Zbl

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

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

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

[37] Phung, Kim Dang; Wang, Gengsheng 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] Pravda-Starov, Karel Subelliptic estimates for quadratic differential operators, Am. J. Math., Volume 133 (2011) no. 1, pp. 39-89 | DOI | MR | Zbl

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

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

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

[42] Sjöstrand, Johannes 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] Toft, Joachim; Khrennikov, Andrei; Nilsson, Börje; Nordebo, Sven 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] Viola, Joe 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] Viola, Joe 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] Viola, Joe 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] Wang, Gengsheng $L^{\infty }$-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] Wang, Gengsheng; Wang, Ming; Zhang, Can; Zhang, Yubiao 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] Zhang, Yubiao 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