Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-20T00:16:39.578Z Has data issue: false hasContentIssue false
  • English
  • Français

QUADRATIC DIFFERENTIAL EQUATIONS: PARTIAL GELFAND–SHILOV SMOOTHING EFFECT AND NULL-CONTROLLABILITY

Part of: Partial differential equations Controllability, observability, and system structure

Published online by Cambridge University Press:  31 January 2020

Paul Alphonse Open the ORCID record for Paul Alphonse [Opens in a new window]
Paul Alphonse*
Affiliation:
Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000Rennes, France (paul.alphonse@univ-rennes1.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the partial Gelfand–Shilov regularizing effect and the exponential decay for the solutions to evolution equations associated with a class of accretive non-selfadjoint quadratic operators, which fail to be globally hypoelliptic on the whole phase space. By taking advantage of the associated Gevrey regularizing effects, we study the null-controllability of parabolic equations posed on the whole Euclidean space associated with this class of possibly non-globally hypoelliptic quadratic operators. We prove that these parabolic equations are null-controllable in any positive time from thick control subsets. This thickness property is known to be a necessary and sufficient condition for the null-controllability of the heat equation posed on the whole Euclidean space. Our result shows that this geometric condition turns out to be a sufficient one for the null-controllability of a large class of quadratic differential operators.

Keywords

quadratic operatorsGelfand–Shilov regularitygevrey regularitynull-controllability

MSC classification

Primary: 35B65: Smoothness and regularity of solutions
Secondary: 93B05: Controllability
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 6 , November 2021 , pp. 1749 - 1801
Check for updates
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alphonse, P. and Bernier, J., Smoothing properties of fractional Ornstein-Uhlenbeck semigroups and null-controllability, preprint, 2018, arXiv:1810.02629.Google Scholar
Beauchard, K., Egidi, M. and Pravda-Starov, K., Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports, preprint, 2019, arXiv:1908.10603.10.5802/crmath.79CrossRefGoogle Scholar
Beauchard, K., Jaming, P. and Pravda-Starov, K., Spectral inequality for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations, preprint, 2018, arXiv:1804.04895.10.5802/jep.62CrossRefGoogle Scholar
Beauchard, K. and Pravda-Starov, K., Null-controllability of hypoelliptic quadratic differential equations, J. Éc. polytech. Math. 5 (2018), 1–43.10.5802/jep.62CrossRefGoogle Scholar
Boulkhemair, A., L 2 estimates for Weyl quantization, J. Funct. Anal. 165(1) (1999), 173–204.10.1006/jfan.1999.3423CrossRefGoogle Scholar
Bramanti, M., Cupini, G., Lanconelli, E. and Priola, E., Global L p estimates for degenerate Ornstein-Uhlenbeck operators, Math. Z. 266(4) (2010), 789–816.10.1007/s00209-009-0599-3CrossRefGoogle Scholar
Carypis, E. and Wahlberg, P., Propagation of exponential phase space singularities for Schrödinger equations with quadratic Hamiltonians, J. Fourier Anal. Appl. 23(3) (2017), 530–571.10.1007/s00041-016-9478-6CrossRefGoogle Scholar
Coron, J.-M., Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, (American Mathematical Society, Providence, RI, 2007).Google Scholar
Egidi, M. and Veselić, I., Sharp geometric condition for null-controllability of the heat equation on ℝd and consistent estimates on the control cost, Arch. Math. (Basel) 111(1) (2018), 85–99.10.1007/s00013-018-1185-xCrossRefGoogle Scholar
Farkas, B. and Lorenzi, L., On a class of hypoelliptic operators with unbounded coefficients in ℝN, Commun. Pure Appl. Anal. 8(4) (2009), 1159–1201.10.3934/cpaa.2009.8.1159CrossRefGoogle Scholar
Farkas, B. and Lunardi, A., Maximal regularity for Kolmogorov operators in L 2 spaces with respect to invariant measures, J. Math. Pures Appl. (9) 86(4) (2006), 310–321.10.1016/j.matpur.2006.06.002CrossRefGoogle Scholar
Hitrik, M. and Pravda-Starov, K., Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann. 344(4) (2009), 801–846.10.1007/s00208-008-0328-yCrossRefGoogle Scholar
Hitrik, M. and Pravda-Starov, K., Semiclassical hypoelliptic estimates for non-selfadjoint operators with double characteristics, Comm. Partial Differential Equations 35(6) (2010), 988–1028.10.1080/03605301003717092CrossRefGoogle Scholar
Hitrik, M. and Pravda-Starov, K., Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics, Ann. Inst. Fourier (Grenoble) 63(3) (2013), 985–1032.10.5802/aif.2782CrossRefGoogle Scholar
Hitrik, M., Pravda-Starov, K. and Viola, J., Short-time asymptotics of the regularizing effect for semigroups generated by quadratic operators, Bull. Sci. Math. 141(7) (2017), 615–675.10.1016/j.bulsci.2017.07.003CrossRefGoogle Scholar
Hitrik, M., Pravda-Starov, K. and Viola, J., From semigroups to subelliptic estimates for quadratic operators, Trans. Amer. Math. Soc. 370(10) (2018), 7391–7415.10.1090/tran/7251CrossRefGoogle Scholar
Hörmander, L., Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z. 219(3) (1995), 413–449.10.1007/BF02572374CrossRefGoogle Scholar
Hörmander, L., The analysis of linear partial differential operators, in Classics in Mathematics, vol. I, Reprint of the 1990 edition (Springer Verlag, Berlin, 2003).Google Scholar
Hörmander, L., The analysis of linear partial differential operators, in Classics in Mathematics, vol. III, Reprint of the 1994 edition (Springer Verlag, Berlin, 2007).Google Scholar
Kolmogoroff, A., Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2) 35(1) (1934), 116–117.10.2307/1968123CrossRefGoogle Scholar
Kovrijkine, O., Some results related to the Logvinenko-Sereda theorem, Proc. Amer. Math. Soc. 129(10) (2001), 3037–3047.10.1090/S0002-9939-01-05926-3CrossRefGoogle Scholar
Lanconelli, E. and Polidoro, S., On a class of hypoelliptic evolution operators, Partial differential equations, II (Turin, 1993), Rend. Sem. Mat. Univ. Politec. Torino 52(1) (1994), 29–63.Google Scholar
Lerner, N., Metrics on the phase space and non-selfadjoint pseudo-differential operators, in Pseudo-Differential Operators, Theory and Applications, vol. 3 (Birkhäuser Verlag, Basel, 2010).Google Scholar
Lorenzi, L. and Bertoldi, M., Analytical Methods for Markov Semigroups, Pure and Applied Mathematics (Boca Raton), vol. 283, (Chapman & Hall/CRC, Boca Raton, FL, 2007).Google Scholar
Lunardi, A., Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in R n , Ann. Sc. Norm. Super Pisa Cl. Sci. (4) 24(1) (1997), 133–164.Google Scholar
Metafune, G., Pallara, D. and Priola, E., Spectrum of Ornstein-Uhlenbeck operators in L p spaces with respect to invariant measures, J. Funct. Anal. 196(1) (2002), 40–60.10.1006/jfan.2002.3978CrossRefGoogle Scholar
Nicola, F. and Rodino, L., Global pseudo-differential calculus on Euclidean spaces, in Pseudo-Differential Operators, Theory and Applications, vol. 4 (Birkhäuser Verlag, Basel, 2010).Google Scholar
Ottobre, M., Pavliotis, G. A. and Pravda-Starov, K., Some remarks on degenerate hypoelliptic Ornstein-Uhlenbeck operators, J. Math. Anal. Appl. 429(2) (2015), 676–712.10.1016/j.jmaa.2015.04.019CrossRefGoogle Scholar
Pravda-Starov, K., Subelliptic estimates for quadratic differential operators, Amer. J. Math. 133(1) (2011), 39–89.10.1353/ajm.2011.0003CrossRefGoogle Scholar
Pravda-Starov, K., Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities, Math. Ann. 372(3–4) (2018), 1335–1382.10.1007/s00208-018-1667-yCrossRefGoogle Scholar
Pravda-Starov, K., Rodino, L. and Wahlberg, P., Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians, Math. Nachr. 291(1) (2018), 128–159.10.1002/mana.201600410CrossRefGoogle Scholar
Sjöstrand, J., Resolvent estimates for non-selfadjoint operators via semigroups, in Around the Research of Vladimir Maz’ya. III, International Mathematical Series (New York), vol. 13, pp. 359–384 (Springer, New York, 2010).10.1007/978-1-4419-1345-6_13CrossRefGoogle Scholar
Viola, J., Non-elliptic quadratic forms and semiclassical estimates for non-selfadjoint operators, Int. Math. Res. Not. IMRN 20 (2013), 4615–4671.10.1093/imrn/rns188CrossRefGoogle Scholar
Viola, J., Spectral projections and resolvent bounds for partially elliptic quadratic differential operators, J. Pseudo-Differ. Oper. Appl. 4(2) (2013), 145–221.10.1007/s11868-013-0066-0CrossRefGoogle Scholar
Wahlberg, P., Propagation of polynomial phase space singularities for Schrödinger equations with quadratic Hamiltonians, Math. Scand. 122(1) (2018), 107–140.10.7146/math.scand.a-97187CrossRefGoogle Scholar
Wang, G., Wang, M., Zhang, C. and Zhang, Y., Observable set, observability, interpolation inequality and spectral inequality for the heat equation in ℝ n , J. Math. Pures Appl. (9) 126 (2019), 144–194.10.1016/j.matpur.2019.04.009CrossRefGoogle Scholar