Comptes Rendus
Control theory
On uniform controllability of 1D transport equations in the vanishing viscosity limit
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 265-312.

We consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit.

We assume that the vector field varies on the whole interval except at one point. The upper/lower estimates we obtain depend on geometric quantities such as an Agmon distance and the spectral gap of an associated semiclassical Schrödinger operator. They improve, in this particular situation, the results obtained in the companion paper [38].

The proofs rely on a reformulation of the problem as a uniform observability question for the semiclassical heat equation together with a fine analysis of localization of eigenfunctions both in the semiclassically allowed and forbidden regions [40], together with estimates on the spectral gap [33, 1]. Along the proofs, we provide with a construction of biorthogonal families with fine explicit bounds, which we believe is of independent interest.

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Accepted:
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DOI: 10.5802/crmath.405
Classification: 93B07, 93B05, 35B25, 35F05, 35K05, 93C73

Camille Laurent 1; Matthieu Léautaud 2

1 CNRS UMR 7598 and Sorbonne Universités UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
2 Laboratoire de Mathématiques d’Orsay, UMR 8628, Université Paris-Saclay, CNRS, Bâtiment 307, 91405 Orsay Cedex France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Camille Laurent; Matthieu Léautaud. On uniform controllability of 1D transport equations in the vanishing viscosity limit. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 265-312. doi : 10.5802/crmath.405. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.405/

[1] Brice Allibert Contrôle analytique de l’équation des ondes et de l’équation de Schrödinger sur des surfaces de révolution, Commun. Partial Differ. Equations, Volume 23 (1998) no. 9-10, pp. 1493-1556 | DOI | MR | Zbl

[2] Youcef Amirat; Arnaud Münch Asymptotic analysis of an advection-diffusion equation and application to boundary controllability, Asymptotic Anal., Volume 112 (2019) no. 1-2, pp. 59-106 | DOI | MR | Zbl

[3] Youcef Amirat; Arnaud Münch On the controllability of an advection-diffusion equation with respect to the diffusion parameter: asymptotic analysis and numerical simulations, Acta Math. Appl. Sin., Engl. Ser., Volume 35 (2019) no. 1, pp. 54-110 | DOI | MR | Zbl

[4] Farid Ammar-Khodja; Assia Benabdallah; Manuel González-Burgos; Luz de Teresa The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., Volume 96 (2011) no. 6, pp. 555-590 | DOI | MR

[5] Karine Beauchard; Jérémi Dardé; Sylvain Ervedoza Minimal time issues for the observability of Grushin-type equations, Ann. Inst. Fourier, Volume 70 (2020) no. 1, pp. 247-312 | DOI | MR | Zbl

[6] Piermarco Cannarsa; Patrick Martinez; Judith Vancostenoble Precise estimates for biorthogonal families under asymptotic gap conditions, Discrete Contin. Dyn. Syst., Ser. S, Volume 13 (2020) no. 5, pp. 1441-1472 | DOI | MR | Zbl

[7] Subrahmanyan Chandresekhar Stochastic problems in physics and astronomy, Rev. Mod. Phys., Volume 15 (1943), pp. 1-89 | DOI | MR

[8] Marianne Chapouly On the global null controllability of a Navier–Stokes system with Navier slip boundary conditions, J. Differ. Equations, Volume 247 (2009), pp. 2094-2123 | DOI | MR | Zbl

[9] Jean-Michel Coron On the controllability of the 2-D incompressible Navier–Stokes equations with the Navier slip boundary conditions, ESAIM, Control Optim. Calc. Var., Volume 1 (1996), pp. 35-75 | DOI | Zbl

[10] Jean-Michel Coron Control and nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007

[11] Jean-Michel Coron; Andrei V. Fursikov Global exact controllability of the 2D Navier–Stokes equations on a manifold without boundary, Russ. J. Math. Phys., Volume 4 (1996), pp. 429-448 | Zbl

[12] Jean-Michel Coron; Sergio Guerrero Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymptotic Anal., Volume 44 (2005), pp. 237-257 | MR

[13] Jean-Michel Coron; Frédéric Marbach; Franck Sueur Small-time global exact controllability of the Navier–Stokes equation with Navier slip-with-friction boundary conditions, J. Eur. Math. Soc., Volume 22 (2020) no. 5, pp. 1625-1673 | DOI | MR | Zbl

[14] Constantine M. Dafermos Hyperbolic conservation laws in continuum physics, Springer, 2000 | DOI

[15] Jérémi Dardé; Sylvain Ervedoza On the cost of observability in small times for the one-dimensional heat equation, Anal. PDE, Volume 12 (2019) no. 6, pp. 1455-1488 | DOI | MR | Zbl

[16] Mouez Dimassi; Johannes Sjöstrand Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, 1999, xii+227 pages | DOI | MR

[17] Szymon Dolecki; David L. Russell A general theory of observation and control, SIAM J. Control Optim., Volume 15 (1977) no. 2, pp. 185-220 | DOI | MR | Zbl

[18] Sylvain Ervedoza; Enrique Zuazua Observability of heat processes by transmutation without geometric restrictions, Math. Control Relat. Fields, Volume 1 (2011) no. 2, pp. 177-187 | DOI | MR | Zbl

[19] Sylvain Ervedoza; Enrique Zuazua Sharp observability estimates for heat equations, Arch. Ration. Mech. Anal., Volume 202 (2011) no. 3, pp. 975-1017 | DOI | MR | Zbl

[20] Hector O. Fattorini; David L. Russell Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal., Volume 43 (1971), pp. 272-292 | DOI | MR | Zbl

[21] Hector O. Fattorini; David L. Russell Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Q. Appl. Math., Volume 32 (1974/75), pp. 45-69 | DOI | Zbl

[22] Andrei V. Fursikov; Oleg Yu. Imanuvilov Controllability of evolution equations, Lecture Notes Series, Seoul, 34, Seoul National University, 1996

[23] Olivier Glass A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit, J. Funct. Anal., Volume 258 (2010), pp. 852-868 | DOI | MR | Zbl

[24] Olivier Glass; Sergio Guerrero On the uniform controllability of the Burgers equation, SIAM J. Control Optim., Volume 46 (2007), pp. 1211-1238 | DOI | MR

[25] Olivier Glass; Sergio Guerrero Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptotic Anal., Volume 60 (2008), pp. 61-100 | DOI | MR | Zbl

[26] Olivier Glass; Sergio Guerrero Uniform controllability of a transport equation in zero diffusion-dispersion limit, Math. Models Methods Appl. Sci., Volume 19 (2009), pp. 1567-1601 | DOI | MR | Zbl

[27] Israel C. Gohberg; Mark G. Krein Introduction to the theory of linear non-selfadjoint operators, Translations of Mathematical Monographs, 18, American Mathematical Society, 1969

[28] Sergio Guerrero; Gilles Lebeau Singular optimal control for a transport-diffusion equation, Commun. Partial Differ. Equations, Volume 32 (2007), pp. 1813-1836 | DOI | MR | Zbl

[29] Scott W. Hansen Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, J. Math. Anal. Appl., Volume 158 (1991) no. 2, pp. 487-508 | DOI | MR | Zbl

[30] Alain Haraux Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire, J. Math. Pures Appl., Volume 68 (1989) no. 4, pp. 457-465 | MR | Zbl

[31] Bernard Helffer Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics, 1336, Springer, 1988, vi+107 pages | DOI | MR

[32] Bernard Helffer; Didier Robert Puits de potentiel généralisés et asymptotique semi-classique, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 41 (1984) no. 3, pp. 291-331 | MR | Zbl

[33] Bernard Helffer; Johannes Sjöstrand Multiple wells in the semiclassical limit. I, Commun. Partial Differ. Equations, Volume 9 (1984) no. 4, pp. 337-408 | DOI | MR | Zbl

[34] Bernard Helffer; Johannes Sjöstrand Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten, Commun. Partial Differ. Equations, Volume 10 (1985) no. 3, pp. 245-340 | DOI | MR | Zbl

[35] Vilmos Komornik; Paola Loreti Fourier series in control theory, Springer Monographs in Mathematics, Springer, 2005, x+226 pages | DOI | MR

[36] Paul Koosis The logarithmic integral. I, Cambridge Studies in Advanced Mathematics, 12, Cambridge University Press, 1988, xvi+606 pages | DOI | MR

[37] Stanislav N. Kružkov First order quasilinear equations with several independent variables. (Russian), Mat. Sb., N. Ser., Volume 81 (1970), pp. 228-255

[38] Camille Laurent; Matthieu Léautaud On uniform observability of gradient flows in the vanishing viscosity limit, J. Éc. Polytech., Math., Volume 8 (2021), pp. 439-506 | DOI | MR | Zbl

[39] Camille Laurent; Matthieu Léautaud Tunneling estimates and approximate controllability for hypoelliptic equations, Mem. Am. Math. Soc., Volume 276 (2022) no. 1357, p. vi+95 | DOI | MR | Zbl

[40] Camille Laurent; Matthieu Léautaud Uniform observation of semiclassical Schrödinger eigenfunctions on an interval (2022) (https://arxiv.org/abs/2203.03271, to appear in Tunis. J. Math.)

[41] Matthieu Léautaud Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems, J. Funct. Anal., Volume 258 (2010), pp. 2739-2778 | DOI | MR | Zbl

[42] Matthieu Léautaud Uniform controllability of scalar conservation laws in the vanishing viscosity limit, SIAM J. Control Optim., Volume 50 (2012) no. 3, pp. 1661-1699 | DOI | MR | Zbl

[43] Gilles Lebeau; Luc Robbiano Contrôle exact de l’équation de la chaleur, Commun. Partial Differ. Equations, Volume 20 (1995), pp. 335-356 | DOI | Zbl

[44] Pierre Lissy A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 11-12, pp. 591-595 | DOI | MR | Zbl

[45] Pierre Lissy An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation in the vanishing viscosity limit, Syst. Control Lett., Volume 69 (2014), pp. 98-102 | DOI | MR

[46] Pierre Lissy Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation, J. Differ. Equations, Volume 259 (2015) no. 10, pp. 5331-5352 | DOI | MR | Zbl

[47] Luc Miller The control transmutation method and the cost of fast controls, SIAM J. Control Optim., Volume 45 (2006) no. 2, pp. 762-772 | DOI | MR | Zbl

[48] Zeev Schuss; Bernard J. Matkowsky The exit problem: a new approach to diffusion across potential barriers, SIAM J. Appl. Math., Volume 36 (1979) no. 3, pp. 604-623 | DOI | MR | Zbl

[49] Edward Witten Supersymmetry and Morse theory, J. Differ. Geom., Volume 17 (1982) no. 4, pp. 661-692 | MR | Zbl

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