Finally, if $\nabla_g f(y) = 0$, then the function $t \mapsto \rho(x, y, t)$ is non-increasing. Indeed, taking $t_1 \le t_2$, from a path $\gamma_1: [0, t_1] \to \mathcal{M}$ such that $\gamma_1(0) = x$ and $\gamma_1(t_1) = y$, we can construct the path $\gamma_2: [0, t_2] \to \mathcal{M}$ by $\gamma_2(s) = \gamma_1(s)$ for $s \in [0, t_1]$ and $\gamma_2(s) = y$ for $s \in [t_1, t_2]$. This yields a path in $W^{1,\infty}([0, t_2]; \mathcal{M})$ if $\gamma_1 \in W^{1,\infty}([0, t_1]; \mathcal{M})$, and thus the set of admissible paths on $[0, t_2]$ is larger than the set of admissible paths on $[0, t_1]$. Since the contribution $\int_{t_1}^{t_2} |\dot{\gamma}_2(s)|^2 + |\nabla_g \gamma_2(s)|_g^2 ds = 0$ we deduce that $\rho(x, y, t_2) \le \rho(x, y, t_1)$. This proves that the inf is actually a $\lim_{t \to +\infty}$ in this case. $\square$

Note that related properties are proved in the appendix of [HS85]. For instance, [HS85, Lem. A2.2] with our notations can be loosely stated as follows: If $W(x, y) = 0$, then every minimizing geodesic of $d_A$ is a generalized integral curve of $\nabla_g f$.

Finally, we state a last result that explains that $d_{\nabla_g f}(x, y, t)$ measures how far $x$ is the final state of a path of the vector field at time $t$ and starting at $y$. Part of this result is contained in the last statement of Lemma A.2; we here give a different proof, which, we believe, is interesting in itself.

**PROPOSITION A.4.** — With $d_{\nabla_g f}(x, y, t)$ defined in (5.5), we have for all $(x, y, t) \in \mathcal{M}^2 \times (0, +\infty)$,

(1) $d_{\nabla_g f}(x, y, t) \ge 0$;

(2) $d_{\nabla_g f}(x, y, t) = 0$ if and only if there exists a trajectory of $\dot{\gamma}(s) = \nabla_g f(\gamma(s))$ with $\gamma(0) = x, \gamma(t) = y$, that is if and only if $y = \phi_t(x)$.

In particular, $(\mathcal{M}, \nabla_g f, \omega, T)$ satisfies (GCC) if and only if for any $x \in \mathcal{M}$, there exist $y \in \omega$ and $t \in (0, T)$ so that $d_{\nabla_g f}(x, y, t) = 0$.

Recall that the flow $(\phi_t)_{t \in \mathbb{R}}$ is defined in (2.14) and the Geometric Control Condition (GCC) is defined in Definition 2.10.

*Proof.* — Statement 1 follows from the definition of $d_{\nabla_g f}$ in (A.4). Let us now consider Statement (2). Assume first that there exists a trajectory of $\dot{\gamma}(s) = \nabla_g f(\gamma(s))$ with $\gamma(0) = x$ and $\gamma(t) = y$. Then, by definition of the infimum in (A.4), this yields $d_{\nabla_g f}(x, y, t) \le 0$ and hence $d_{\nabla_g f}(x, y, t) = 0$. Conversely, assume $d_{\nabla_g f}(x, y, t) = 0$. Take a minimizing sequence in (A.4), that is to say that $\gamma_n \in W^{1,\infty}([0, t]; \mathcal{M})$ such that $\gamma_n(0) = x$, $\gamma_n(t) = y$ and $\dot{\gamma}_n - \nabla_g f(\gamma_n) = R_n$ (bounded continuous with values in the tangent bundle of $\mathcal{M}$) with

$$ (A.10) \qquad \int_0^t |R_n(s)|_g^2 ds \to 0. $$

Since $\nabla_g f$ is bounded on $\mathcal{M}$, the sequence $\int_0^t |\dot{\gamma}_n(s)|_g^2 ds$ is then uniformly bounded in $\mathbb{R}$. As a consequence, the sequence of paths $(\gamma_n)_{n \in \mathbb{N}}$ is equicontinuous. From Ascoli's theorem, we may extract a subsequence (which we do not relabel) $(\gamma_n)_{n \in \mathbb{N}}$ which converges strongly for the topology $C^0([0, t]; \mathcal{M})$ to a limit $\gamma \in C^0([0, t]; \mathcal{M})$. The latter thus has the same and endpoints $\gamma(0) = x$ and $\gamma(t) = y$. It is solution of $\dot{\gamma} = \nabla_g f(\gamma)$ in
the distributional sense according to (A.10). Bootstrapping in the differential equation
implies $\gamma \in W^{1,\infty}([0, 1]; \mathcal{M})$ and $\gamma$ is a strong solution to $\dot{\gamma} = \nabla_g f(\gamma)$. This concludes
the proof of Statement 2.

According to Lemma 2.11 Item (4), that $(\mathcal{M}, X, \omega, T)$ satisfies (GCC) is equivalent
to the fact that for any $x \in \mathcal{M}$, there exist $t \in (0, T)$ such that $y := \phi_t(x) \in \omega$. As a
consequence of Item (2), this is equivalent to having $d_{\nabla_g f}(x, y, t) = 0$. $\square$

Note that an analogue statement for the function $W$ in (A.9) is proved in [HS85, Lem. A2.2], and could also be deduced from Proposition A.4.

## REFERENCES

[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.

[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.

[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.

[AM19b] ———, “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.

[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.

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

[Bes78] A. L. BESSE – *Manifolds all of whose geodesics are closed*, Ergeb. Math. Grenzgeb. (3), vol. 93, Springer-Verlag, Berlin-New York, 1978.

[Cha43] S. CHANDRESEKHAR – “Stochastic problems in physics and astronomy”, *Rev. Modern Phys.* **15** (1943), p. 1–89.

[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.

[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.

[Cor07] ———, *Control and nonlinearity*, Math. Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007.

[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.

[CG05] J.-M. CORON & S. GUERRERO – “Singular optimal control: A linear 1-D parabolic-hyperbolic example”, *Asymptot. Anal.* **44** (2005), p. 237–257.

[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.

[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.

[Daf00] C. M. DAFERMOS – *Hyperbolic conservation laws in continuum physics*, Springer-Verlag, Berlin, 2000.
[DR20] N. V. DANG & G. RIVIÈRE – “Pollicott-Ruelle spectrum and Witten Laplacians”, *J. Eur. Math. Soc. (JEMS)* (2020), to appear.

[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.

[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.

[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.

[DR77] S. DOLECKI & D. I. RUSSELL – “A general theory of observation and control”, *SIAM J. Control Optim.* **15** (1977), no. 2, p. 185–220.

[ET74] I. EKELAND & R. TEMAM – *Analyse convexe et problèmes variationnels*, Dunod–Gauthier-Villars, Paris, 1974.

[Eva98] L. C. EVANS – *Partial differential equations*, Graduate Studies in Math., American Mathematical Society, Providence, RI, 1998.

[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.

[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.

[GG07] O. GLASS & S. GUERRERO – “On the uniform controllability of the Burgers equation”, *SIAM J. Control Optim.* **46** (2007), p. 1211–1238.

[GG08] – – –, “Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit”, *Asymptot. Anal.* **60** (2008), p. 61–100.

[GG09] – – –, “Uniform controllability of a transport equation in zero diffusion-dispersion limit”, *Math. Models Methods Appl. Sci.* **19** (2009), p. 1567–1601.

[GL07] S. GUERRERO & G. LEBEAU – “Singular optimal control for a transport-diffusion equation”, *Comm. Partial Differential Equations* **32** (2007), p. 1813–1836.

[HKL15] D. HAN-KWAN & M. LÉAUTAUD – “Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium”, *Ann. PDE* **1** (2015), no. 1, article no. 3 (84 pages).

[Hel88] B. HELFFER – *Semi-classical analysis for the Schrödinger operator and applications*, Lect. Notes in Math., vol. 1336, Springer-Verlag, Berlin, 1988.

[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.

[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.

[HS84] B. HELFFER & J. SJÖSTRAND – “Multiple wells in the semiclassical limit. I”, *Comm. Partial Differential Equations* **9** (1984), no. 4, p. 337–408.

[HS85] – – –, “Puits multiples en mécanique semi-classique IV. Étude du complexe de Witten”, *Comm. Partial Differential Equations* **10** (1985), no. 3, p. 245–340.

[Kru70] S. N. KRUŽKOV – “First order quasilinear equations with several independent variables. (russian)”, *Mat. Sb. (N.S.)* **81** (1970), p. 228–255.

[LSU68] O. A. LADYŽENSKAJA, V. A. SOLONIKOV & N. N. URAL'CEVA – *Linear and quasilinear equations of parabolic type*, Translations of Math. Monographs, vol. 23, American Mathematical Society, Providence, R.I., 1968.

[LL16] C. LAURENT & M. LÉAUTAUD – “Uniform observability estimates for linear waves”, *ESAIM Control Optim. Calc. Var.* **22** (2016), no. 4, p. 1097–1136.

[LL21a] – – –, “Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller”, *Anal. PDE* (2021), to appear.
[LL21b] —, "On uniform controllability of 1D transport equations in the vanishing viscosity limit", work in progress, 2021.

[LL21c] —, "The cost function for the approximate control of waves", work in progress, 2021.

[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.

[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.

[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.

[Léa12] —, "Uniform controllability of scalar conservation laws in the vanishing viscosity limit", *SIAM J. Control Optim.* **50** (2012), no. 3, p. 1661–1699.

[Léa18] —, "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.

[LR95] G. LEBEAU & L. ROBBIANO — "Contrôle exact de l'équation de la chaleur", *Comm. Partial Differential Equations* **20** (1995), p. 335–356.

[Lee13] J. M. LEE — *Introduction to smooth manifolds*, 2nd ed., Graduate Texts in Math., vol. 218, Springer, New York, 2013.

[LY86] P. LI & S.-T. YAU — "On the parabolic kernel of the Schrödinger operator", *Acta Math.* **156** (1986), no. 3-4, p. 153–201.

[Lio88] J.-L. LIONS — *Contrôlabilité exacte, perturbations et stabilisation de systèmes dis-tribués 2. Perturbations*, Recherches en Math. Appliquées, vol. 9, Masson, Paris, 1988.

[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.

[Lis14] —, "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.

[Lis15] —, "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.

[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.

[Mic19] L. MICHEL — "About small eigenvalues of the Witten Laplacian", *Pure Appl. Anal. 1* (2019), no. 2, p. 149–206.

[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.

[Mün18] A. MÜNCH — "Numerical estimations of the cost of boundary controls for the equation $y_t - \varepsilon y_{xx} + My_x = 0$ with respect to $\varepsilon$", 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*, 2nd ed., Academic Press, Inc., New York, 1980.

[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.

[Sim82] B. SIMON — "Schrödinger semigroups", *Bull. Amer. Math. Soc. (N.S.)* **7** (1982), no. 3, p. 447–526.

[Sim83] —, "Instantons, double wells and large deviations", *Bull. Amer. Math. Soc. (N.S.)* **8** (1983), no. 2, p. 323–326.
[Var67] S. R. S. VARADHAN – “Diffusion processes in a small time interval”, *Comm. Pure Appl. Math.* **20** (1967), p. 659–685.

[Wit82] E. WITTEN – “Supersymmetry and Morse theory”, *J. Differential Geometry* **17** (1982), no. 4, p. 661–692 (1983).

Manuscript received 3rd March 2020
accepted 1st February 2021

CAMILLE LAURENT, CNRS UMR 7598 and Sorbonne Universités UPMC Univ Paris 06, Laboratoire
Jacques-Louis Lions
F-75005, Paris, France
E-mail: laurent@ann.jussieu.fr
Url: https://www.ljll.math.upmc.fr/~laurent/

MATTHIEU LÉAUTAUD, Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, CNRS, Université
Paris-Saclay
Bâtiment 307, 91405 Orsay Cedex France
E-mail: matthieu.leautaud@math.u-psud.fr
Url: http://leautaud.perso.math.cnrs.fr/