In this article, we give a completely constructive proof of the observability/controllability of the wave equation on a compact manifold under optimal geometric conditions. This contrasts with the original proof of Bardos–Lebeau–Rauch [C. Bardos, G. Lebeau and J. Rauch, SIAM J. Control Optim. 30 (1992) 1024–1065], which contains two non-constructive arguments. Our method is based on the Dehman-Lebeau [B. Dehman and G. Lebeau, SIAM J. Control Optim. 48 (2009) 521–550] Egorov approach to treat the high-frequencies, and the optimal unique continuation stability result of the authors [C. Laurent and M. Léautaud. Preprint arXiv:1506.04254 (2015)] for the low-frequencies. As an application, we first give estimates of the blowup of the observability constant when the time tends to the limit geometric control time (for wave equations with possibly lower order terms). Second, we provide (on manifolds with or without boundary) with an explicit dependence of the observability constant with respect to the addition of a bounded potential to the equation.
Accepted:
DOI: 10.1051/cocv/2016046
Mots-clés : Wave equation, observability, controllability, geometric control conditions, uniform estimates
@article{COCV_2016__22_4_1097_0, author = {Laurent, Camille and L\'eautaud, Matthieu}, title = {Uniform observability estimates for linear waves}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1097--1136}, publisher = {EDP-Sciences}, volume = {22}, number = {4}, year = {2016}, doi = {10.1051/cocv/2016046}, mrnumber = {3570496}, zbl = {1368.35163}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016046/} }
TY - JOUR AU - Laurent, Camille AU - Léautaud, Matthieu TI - Uniform observability estimates for linear waves JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 1097 EP - 1136 VL - 22 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016046/ DO - 10.1051/cocv/2016046 LA - en ID - COCV_2016__22_4_1097_0 ER -
%0 Journal Article %A Laurent, Camille %A Léautaud, Matthieu %T Uniform observability estimates for linear waves %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 1097-1136 %V 22 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016046/ %R 10.1051/cocv/2016046 %G en %F COCV_2016__22_4_1097_0
Laurent, Camille; Léautaud, Matthieu. Uniform observability estimates for linear waves. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 4, pp. 1097-1136. doi : 10.1051/cocv/2016046. http://archive.numdam.org/articles/10.1051/cocv/2016046/
C. Bardos, G. Lebeau and J. Rauch, Un exemple d’utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques. Nonlinear hyperbolic equations in applied sciences. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue) (1989) 11–31. | MR | Zbl
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Contr. Opt. 30 (1992) 1024–1065. | DOI | MR | Zbl
, and ,Global Carleman estimates for waves and applications. Comm. Partial Differ. Eq. 38 (2013) 823–859. | DOI | MR | Zbl
, and ,R. Bosi, Y. Kurylev and M. Lassas, Stability of the unique continuation for the wave operator via tataru inequality and applications. Preprint (2015). | arXiv | MR
Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 749–752. | DOI | MR | Zbl
and ,F. Chaves-Silva and G. Lebeau, Announcement (2015).
Singular optimal control: A linear -D parabolic-hyperbolic example. Asympt. Anal. 44 (2005) 237–257. | MR | Zbl
and ,J.-M. Coron, Control and nonlinearity. Vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007). | MR | Zbl
Dependence of high-frequency waves with respect to potentials. SIAM J. Control Optim. 52 (2014) 3722–3750. | DOI | MR | Zbl
and ,Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time. SIAM J. Control Optim. 48 (2009) 521–550. | DOI | MR | Zbl
and ,Controllability of two coupled wave equations on a compact manifold. Arch. Ration. Mech. Anal. 211 (2014) 113–187. | DOI | MR | Zbl
, and ,A general theory of observation and control. SIAM J. Control Optim. 15 (1977) 185–220. | DOI | MR | Zbl
and ,On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25 (2008) 1–41. | DOI | Numdam | MR | Zbl
, and ,A systematic method for building smooth controls for smooth data. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1375–1401. | MR | Zbl
and ,S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry. Universitext, 3rd edn. Springer-Verlag, Berlin (2004). | MR
Microlocal defect measures. Comm. Partial Differ. Eq. 16 (1991) 1761–1794. | DOI | MR | Zbl
,A. Haraux, T. Liard and Y. Privat, How to estimate observability constants of one-dimensional wave equations? propagation versus spectral methods. J. Evol. Equ. (2016) 1–32. | MR
L. Hörmander, The Analysis of Linear Partial Differential Operators, Volume III. Springer-Verlag (1985). Second printing (1994). | MR | Zbl
A uniqueness theorem for second order hyperbolic differential equations. Commun. Partial Differ. Eq. 17 (1992) 699–714. | DOI | MR | Zbl
,L. Hörmander, On the uniqueness of the Cauchy problem under partial analyticity assumptions. In Geometrical optics and related topics (Cortona, 1996). Vol. 32 of Progr. Nonlin. Differ. Eq. Appl. Birkhäuser Boston, Boston, MA (1997) 179–219. | MR | Zbl
On linear partial differential equations with analytic coefficients. Unique continuation of data. Comm. Pure Appl. Math. 2 (1949) 209–253. | DOI | MR | Zbl
,V. Komornik, Exact controllability and stabilization. RAM: Res. Appl. Math. Masson, Paris (1994). The multiplier method. | MR | Zbl
C. Laurent and M. Léautaud, Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves. Preprint (2015). | arXiv | MR
Contrôle analytique. I. Estimations a priori. Duke Math. J. 68 (1992) 1–30. | DOI | MR | Zbl
,G. Lebeau, Équation des ondes amorties. In Algebraic and geometric methods in mathematical physics (Kaciveli, 1993). Vol. 19 of Math. Phys. Stud. Kluwer Acad. Publ., Dordrecht (1996) 73–109. | MR | Zbl
Uniqueness for an ill-posed problem. J. Differ. Eq. 71 (1988) 255–260. | DOI | MR | Zbl
,N. Lerner, Metrics on the phase space and non-selfadjoint pseudo-differential operators. Birkhäuser-Verlag, Basel (2010). | MR | Zbl
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Vol. 8 of Recherches en Mathématiques Appliquées. Masson, Paris (1988). | MR | Zbl
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 350 (2012) 591–595. | DOI | MR | Zbl
,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. Eq. 259 (2015) 5331–5352. | DOI | MR | Zbl
,Singularities of boundary value problems. I. Comm. Pure Appl. Math. 31 (1978) 593–617. | DOI | MR | Zbl
and ,Escape function conditions for the observation, control, and stabilization of the wave equation. SIAM J. Control Optim. 41 (2002) 1554–1566. | DOI | MR | Zbl
,Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time. J. Differ. Eq. 204 (2004) 202–226. | DOI | MR | Zbl
,How violent are fast controls for Schrödinger and plate vibrations? Arch. Ration. Mech. Anal. 172 (2004) 429–456. | DOI | MR | Zbl
,Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218 (2005) 425–444. | DOI | MR | Zbl
,K.D. Phung, Waves, damped wave and observation. In Some problems on nonlinear hyperbolic equations and applications. Vol. 15 of Ser. Contemp. Appl. Math. CAM. Higher Ed. Press (2010) 386–412. | MR | Zbl
Penetrations into shadow regions and unique continuation properties in hyperbolic mixed problems. Indiana Univ. Math. J. 22 (1972/73) 277–285. | DOI | MR | Zbl
and ,Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24 (1974) 79–86. | DOI | MR | Zbl
and ,Théorème d’unicité adapté au contrôle des solutions des problèmes hyperboliques. Commun. Partial Differ. Eq. 16 (1991) 789–800. | DOI | MR | Zbl
,Fonction de coût et contrôle des solutions des équations hyperboliques. Asymptotic Anal. 10 (1995) 95–115. | DOI | MR | Zbl
,J. Royer, Analyse haute fréquence de l’équation de Helmholtz dissipative. Thèse de Doctorat. Université de Nantes (2010). Available at http://www.math.sciences.univ-nantes.fr/˜jroyer/these.pdf.
Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients. Invent. Math. 131 (1998) 493–539. | DOI | MR | Zbl
and ,Limiting absorption principle for the dissipative Helmholtz equation. Commun. Partial Differ. Eq. 35 (2010) 1458–1489. | DOI | MR | Zbl
,R.T. Seeley, Complex powers of an elliptic operator. In Singular Integrals (Proc. of Sympos. Pure Math., Chicago, Ill., 1966). Amer. Math. Soc., Providence, R.I. (1967) 288–307. | MR | Zbl
T.I. Seidman, How violent are fast controls? Math. Control Signals Systems 1 (1988) 89–95. | MR | Zbl
M.A. Shubin, Pseudodifferential Operators and Spectral Theory, 2nd edn. Springer-Verlag, Berlin Heidelberg (2001). | MR | Zbl
-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990) 193–230. | DOI | MR | Zbl
,Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem. Commun. Partial Differ. Eq. 20 (1995) 855–884. | DOI | MR | Zbl
,D. Tataru, Carleman estimates, unique continuation and applications. Lecture notes. Available at https://math.berkeley.edu/˜tataru/papers/ucpnotes.ps (1999).
Unique continuation for operators with partially analytic coefficients. J. Math. Pures Appl. 78 (1999) 505–521. | DOI | Zbl
,Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28 (1975) 457–478. | DOI | Zbl
,M.E. Taylor, Partial differential equations II. Qualitative studies of linear equations, Vol. 116 of Applied Mathematical Sciences, 2nd edn. Springer, New York (2011). | Zbl
Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. Henri Poincaré, Anal. Non Lin. 10 (1993) 109–129. | DOI | Numdam | Zbl
,Cited by Sources: