On uniform observability of gradient flows in the vanishing viscosity limit
[Sur l’observabilité uniforme des flots de gradient dans la limite de viscosité évanescente]
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 439-506.

Nous considérons l’équation de transport par un champ de gradient avec une petite perturbation visqueuse -εΔ g . Nous étudions la propriété d’observabilité (resp. de contrôlabilité) uniforme dans la limite (singulière) de viscosité évanescente ε0 + , c’est-à-dire la possibilité d’avoir une constante d’observabilité (resp. un coût du contrôle) uniforme. Nous prouvons avec une série d’exemples que le temps minimal pour l’observabilité uniforme peut être bien plus grand que le temps minimal pour l’équation limite ε=0. Nous montrons aussi que les deux temps minimaux coïncident pour les solutions positives. Les preuves reposent sur une reformulation semiclassique du problème ainsi que (a) des estimées d’Agmon de décroissance des fonctions propres dans la zone classiquement interdite [HS84] (b) des estimées fines du noyaux de la chaleur semiclassique [LY84].

We consider a transport equation by a gradient vector field with a small viscous perturbation -εΔ g . We study uniform observability (resp. controllability) properties in the (singular) vanishing viscosity limit ε0 + , that is, the possibility of having a uniformly bounded observation constant (resp. control cost). We prove with a series of examples that in general, the minimal time for uniform observability may be much larger than the minimal time needed for the observability of the limit equation ε=0. We also prove that the two minimal times coincide for positive solutions. The proofs rely on a semiclassical reformulation of the problem together with (a) Agmon estimates concerning the decay of eigenfunctions in the classically forbidden region [HS84] (b) fine estimates of the kernel of the semiclassical heat equation [LY84].

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.151
Classification : 93B07,  93B05,  35B25,  35F05,  35K05,  93C73
Mots clés : Équation de transport, flot gradient, limite de viscosité évanescente, équation parabolique, temps minimal de contrôle, opérateur de Schrödinger semiclassique
@article{JEP_2021__8__439_0,
     author = {Camille Laurent and Matthieu L\'eautaud},
     title = {On uniform observability of gradient flows in the vanishing viscosity limit},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {439--506},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.151},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.151/}
}
Camille Laurent; Matthieu Léautaud. On uniform observability of gradient flows in the vanishing viscosity limit. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 439-506. doi : 10.5802/jep.151. https://jep.centre-mersenne.org/articles/10.5802/jep.151/

[ABB20] A. Agrachev, D. Barilari & U. Boscain - A comprehensive introduction to sub-Riemannian geometry, Cambridge Studies in Advanced Math., vol. 181, Cambridge University Press, Cambridge, 2020 | MR 3971262 | Zbl 07073879

[All98] B. Allibert - “Contrôle analytique de l’équation des ondes et de l’équation de Schrödinger sur des surfaces de révolution”, Comm. Partial Differential Equations 23 (1998) no. 9-10, p. 1493-1556 | Article | MR 1641772 | Zbl 0954.35028

[AM19a] Y. Amirat & A. Münch - “Asymptotic analysis of an advection-diffusion equation and application to boundary controllability”, Asymptot. Anal. 112 (2019) no. 1-2, p. 59-106 | Article | MR 3932910 | Zbl 07087637

[AM19b] Y. Amirat & A. Münch - “On the controllability of an advection-diffusion equation with respect to the diffusion parameter: asymptotic analysis and numerical simulations”, Acta Math. Appl. Sinica (English Ser.) 35 (2019) no. 1, p. 54-110 | Article | MR 3918636 | Zbl 1414.35098

[Bes78] A. L. Besse - Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb. (3), vol. 93, Springer-Verlag, Berlin-New York, 1978 | MR 496885 | Zbl 0387.53010

[BP20a] J. A. Bárcena-Petisco - “Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit”, SIAM J. Control Optim. 58 (2020) no. 3, p. 1597-1625 | Article | MR 4111667 | Zbl 1444.35016

[BP20b] J. A. Bárcena-Petisco - “Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term”, 2020 | HAL:hal-02455632

[CF96] J.-M. Coron & A. V. Fursikov - “Global exact controllability of the 2-D Navier-Stokes equations on a manifold without boundary”, Russian J. Math. Phys. 4 (1996), p. 429-448 | MR 1470445 | Zbl 0938.93030

[CFKS87] H. L. Cycon, R. G. Froese, W. Kirsch & B. Simon - Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987 | Zbl 0619.47005

[CG05] J.-M. Coron & S. Guerrero - “Singular optimal control: A linear 1-D parabolic-hyperbolic example”, Asymptot. Anal. 44 (2005), p. 237-257 | MR 2176274

[Cha43] S. Chandresekhar - “Stochastic problems in physics and astronomy”, Rev. Modern Phys. 15 (1943), p. 1-89 | Article | MR 8130

[Cha09] M. Chapouly - “On the global null controllability of a Navier-Stokes system with Navier slip boundary conditions”, J. Differential Equations 247 (2009), p. 2094-2123 | Article | MR 2560050 | Zbl 1178.35285

[CMS20] J.-M. Coron, F. Marbach & F. Sueur - “Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions”, J. Eur. Math. Soc. (JEMS) 22 (2020) no. 5, p. 1625-1673 | Article | MR 4081730 | Zbl 1447.93024

[Cor96] J.-M. Coron - “On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions”, ESAIM Control Optim. Calc. Var. 1 (1996), p. 35-75 | Article | Numdam | Zbl 0872.93040

[Cor07] J.-M. Coron - Control and nonlinearity, Math. Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007 | MR 2302744 | Zbl 1140.93002

[Daf00] C. M. Dafermos - Hyperbolic conservation laws in continuum physics, Springer-Verlag, Berlin, 2000 | Zbl 0940.35002

[DGLLPN20] G. Di Gesù, T. Lelièvre, D. Le Peutrec & B. Nectoux - “The exit from a metastable state: Concentration of the exit point distribution on the low energy saddle points, part 1”, J. Math. Pures Appl. (9) 138 (2020), p. 242-306 | Article | MR 4098769 | Zbl 1445.60055

[DL09] B. Dehman & G. Lebeau - “Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time”, SIAM J. Control Optim. 48 (2009) no. 2, p. 521-550 | Article | MR 2486082 | Zbl 1194.35268

[DR77] S. Dolecki & D. L. Russell - “A general theory of observation and control”, SIAM J. Control Optim. 15 (1977) no. 2, p. 185-220 | Article | MR 451141 | Zbl 0353.93012

[DR20] N. V. Dang & G. Rivière - “Pollicott-Ruelle spectrum and Witten Laplacians”, J. Eur. Math. Soc. (JEMS) (2020), to appear

[DS99] M. Dimassi & J. Sjöstrand - Spectral asymptotics in the semi-classical limit, London Math. Society Lect. Note Series, vol. 268, Cambridge University Press, Cambridge, 1999 | Article | MR 1735654 | Zbl 0926.35002

[ET74] I. Ekeland & R. Temam - Analyse convexe et problèmes variationnels, Dunod–Gauthier-Villars, Paris, 1974 | Zbl 0281.49001

[Eva98] L. C. Evans - Partial differential equations, Graduate Studies in Math., American Mathematical Society, Providence, RI, 1998 | Zbl 0902.35002

[FI96] A. V. Fursikov & O. Y. Imanuvilov - Controllability of evolution equations, Lect. Notes Series, vol. 34, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996 | MR 1406566 | Zbl 0862.49004

[GG07] O. Glass & S. Guerrero - “On the uniform controllability of the Burgers equation”, SIAM J. Control Optim. 46 (2007), p. 1211-1238 | Article | MR 2346380

[GG08] O. Glass & S. Guerrero - “Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit”, Asymptot. Anal. 60 (2008), p. 61-100 | Article | MR 2463799 | Zbl 1160.35063

[GG09] O. Glass & S. Guerrero - “Uniform controllability of a transport equation in zero diffusion-dispersion limit”, Math. Models Methods Appl. Sci. 19 (2009), p. 1567-1601 | Article | MR 2571687 | Zbl 1194.93025

[GL07] S. Guerrero & G. Lebeau - “Singular optimal control for a transport-diffusion equation”, Comm. Partial Differential Equations 32 (2007), p. 1813-1836 | Article | MR 2372489 | Zbl 1135.35017

[Gla10] O. Glass - “A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit”, J. Funct. Anal. 258 (2010), p. 852-868 | Article | MR 2558179 | Zbl 1180.93015

[Hel88] B. Helffer - Semi-classical analysis for the Schrödinger operator and applications, Lect. Notes in Math., vol. 1336, Springer-Verlag, Berlin, 1988 | Article | Zbl 0647.35002

[HKL15] D. Han-Kwan & M. Léautaud - “Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium”, Ann. PDE 1 (2015) no. 1, article ID 3, 84 pages | Article | MR 3479064 | Zbl 1398.35142

[HKN04] B. Helffer, M. Klein & F. Nier - “Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach”, Mat. Contemp. 26 (2004), p. 41-85 | MR 2111815 | Zbl 1079.58025

[HN06] B. Helffer & F. Nier - Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary, Mém. Soc. Math. France (N.S.), vol. 105, Société Mathématique de France, Paris, 2006 | Article | Numdam | Zbl 1108.58018

[HS84] B. Helffer & J. Sjöstrand - “Multiple wells in the semiclassical limit. I”, Comm. Partial Differential Equations 9 (1984) no. 4, p. 337-408 | Article | MR 740094

[HS85] B. Helffer & J. Sjöstrand - “Puits multiples en mécanique semi-classique IV. Étude du complexe de Witten”, Comm. Partial Differential Equations 10 (1985) no. 3, p. 245-340 | Article | Zbl 0597.35024

[Kru70] S. N. Kružkov - “First order quasilinear equations with several independent variables. (Russian)”, Mat. Sb. (N.S.) 81 (1970), p. 228-255

[LB20] K. Le Balc’h - “Global null-controllability and nonnegative-controllability of slightly superlinear heat equations”, J. Math. Pures Appl. (9) 135 (2020), p. 103-139 | Article | MR 4066102 | Zbl 1436.93065

[Lee13] J. M. Lee - Introduction to smooth manifolds, Graduate Texts in Math., vol. 218, Springer, New York, 2013 | MR 2954043 | Zbl 1258.53002

[Lio88] J.-L. Lions - Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués 2. Perturbations, Recherches en Math. Appliquées, vol. 9, Masson, Paris, 1988 | Zbl 0653.93003

[Lis12] P. 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”, Comptes Rendus Mathématique 350 (2012) no. 11-12, p. 591-595 | Article | MR 2956149 | Zbl 1246.93019

[Lis14] P. 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”, Systems Control Lett. 69 (2014), p. 98-102 | Article | MR 3212827

[Lis15] P. 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. Differential Equations 259 (2015) no. 10, p. 5331-5352 | Article | MR 3377528 | Zbl 1331.35352

[LL16] C. Laurent & M. Léautaud - “Uniform observability estimates for linear waves”, ESAIM Control Optim. Calc. Var. 22 (2016) no. 4, p. 1097-1136 | Article | MR 3570496 | Zbl 1368.35163

[LL21a] C. Laurent & M. Léautaud - “Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller”, Anal. PDE (2021), to appear | Article

[LL21b] C. Laurent & M. Léautaud - “On uniform controllability of 1D transport equations in the vanishing viscosity limit” (2021), work in progress

[LL21c] C. Laurent & M. Léautaud - “The cost function for the approximate control of waves” (2021), work in progress

[LP10] D. Le Peutrec - “Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian”, Ann. Fac. Sci. Toulouse Math. (6) 19 (2010) no. 3-4, p. 735-809 | Article | Numdam | MR 2790817 | Zbl 1213.58023

[LR95] G. Lebeau & L. Robbiano - “Contrôle exact de l’équation de la chaleur”, Comm. Partial Differential Equations 20 (1995), p. 335-356 | Article

[LSU68] O. A. Ladyženskaja, V. A. Solonnikov & N. N. Ural’ceva - Linear and quasilinear equations of parabolic type, Translations of Math. Monographs, vol. 23, American Mathematical Society, Providence, R.I., 1968 | MR 241822

[LY86] P. Li & S.-T. Yau - “On the parabolic kernel of the Schrödinger operator”, Acta Math. 156 (1986) no. 3-4, p. 153-201 | Zbl 0611.58045

[Léa10] M. Léautaud - “Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems”, J. Funct. Anal. 258 (2010), p. 2739-2778 | Article | MR 2593342 | Zbl 1185.35153

[Léa12] M. Léautaud - “Uniform controllability of scalar conservation laws in the vanishing viscosity limit”, SIAM J. Control Optim. 50 (2012) no. 3, p. 1661-1699 | Article | MR 2968071 | Zbl 1251.93033

[Léa18] M. Léautaud - Sur quelques questions de prolongement unique, de propagation et de contrôle, Habilitation à diriger des recherches, Université Paris Diderot, 2018, http://leautaud.perso.math.cnrs.fr/files/HdR.pdf

[Mar14] F. Marbach - “Small time global null controllability for a viscous Burgers’ equation despite the presence of a boundary layer”, J. Math. Pures Appl. (9) 102 (2014) no. 2, p. 364-384 | Article | MR 3227326 | Zbl 1291.93043

[Mic19] L. Michel - “About small eigenvalues of the Witten Laplacian”, Pure Appl. Anal. 1 (2019) no. 2, p. 149-206 | Article | MR 3949372 | Zbl 1447.35236

[Mil04] L. Miller - “Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time”, J. Differential Equations 204 (2004) no. 1, p. 202-226 | Article | MR 2076164 | Zbl 1053.93010

[Mün18] A. Münch - “Numerical estimations of the cost of boundary controls for the equation y t -εy xx +My x =0 with respect to ε, in Recent advances in PDEs: analysis, numerics and control, SEMA SIMAI Springer Ser., vol. 17, Springer, Cham, 2018, p. 159-191

[RS80] M. Reed & B. Simon - Methods of modern mathematical physics I. Functional analysis, Academic Press, Inc., New York, 1980 | Zbl 0459.46001

[Sim82] B. Simon - “Schrödinger semigroups”, Bull. Amer. Math. Soc. (N.S.) 7 (1982) no. 3, p. 447-526 | Article | Zbl 0524.35002

[Sim83] B. Simon - “Instantons, double wells and large deviations”, Bull. Amer. Math. Soc. (N.S.) 8 (1983) no. 2, p. 323-326 | Article | MR 684899 | Zbl 0529.35059

[SM79] Z. Schuss & B. J. Matkowsky - “The exit problem: a new approach to diffusion across potential barriers”, SIAM J. Appl. Math. 36 (1979) no. 3, p. 604-623 | Article | MR 531616 | Zbl 0406.60071

[Var67] S. R. S. Varadhan - “Diffusion processes in a small time interval”, Comm. Pure Appl. Math. 20 (1967), p. 659-685 | Article | MR 217881 | Zbl 0278.60051

[Wit82] E. Witten - “Supersymmetry and Morse theory”, J. Differential Geometry 17 (1982) no. 4, p. 661-692 (1983) | Article | MR 683171 | Zbl 0499.53056