On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials
Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 1, p. 1-41
@article{AIHPC_2008__25_1_1_0,
     author = {Duyckaerts, Thomas and Zhang, Xu and Zuazua, Enrique},
     title = {On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {25},
     number = {1},
     year = {2008},
     pages = {1-41},
     doi = {10.1016/j.anihpc.2006.07.005},
     zbl = {1248.93031},
     zbl = {pre05247877},
     mrnumber = {2383077},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2008__25_1_1_0}
}
Duyckaerts, Thomas; Zhang, Xu; Zuazua, Enrique. On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 1, pp. 1-41. doi : 10.1016/j.anihpc.2006.07.005. http://www.numdam.org/item/AIHPC_2008__25_1_1_0/

[1] Agmon S., Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schrödinger Operators, Princeton Univ. Press, Princeton, NJ, 1982. | MR 745286 | Zbl 0503.35001

[2] Alinhac S., Baouendi M.S., A nonuniqueness result for operators of principal type, Math. Z. 220 (1995) 561-568. | MR 1363855 | Zbl 0851.35003

[3] Alinhac S., Lerner N., Unicité forte à partir d'une variété de dimension quelconque pour des inégalités différentielles elliptiques, Duke Math. J. 48 (1981) 49-68. | MR 610175 | Zbl 0459.35095

[4] Alziary B., Takáč P., A pointwise lower bound for positive solutions of a Schrödinger equation in R N , J. Differential Equations 133 (1997) 280-295. | MR 1427854 | Zbl 0874.35030

[5] Bardos C., Lebeau G., Rauch J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992) 1024-1065. | MR 1178650 | Zbl 0786.93009

[6] Bardos C., Merigot M., Asymptotic decay of the solution of a second-order elliptic equation in an unbounded domain. Applications to the spectral properties of a Hamiltonian, Proc. Roy. Soc. Edinburgh Sect. A 76 (1977) 323-344. | MR 477432 | Zbl 0351.35009

[7] Cannarsa P., Komornik V., Loreti P., One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms, Discrete Contin. Dyn. Syst. B 8 (2002) 745-756. | MR 1897879 | Zbl 1005.35017

[8] Castro C., Zuazua E., Concentration and lack of observability of waves in highly heterogeneous media, Arch. Ration. Mech. Anal. 164 (2002) 39-72. | MR 1921162 | Zbl 1016.35003

[9] Doubova A., Fernández-Cara E., González-Burgos M., Zuazua E., On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim. 41 (2002) 798-819. | MR 1939871 | Zbl 1038.93041

[10] Fattorini H.O., Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation, in: New Trends in Systems Analysis, Proc. Internat. Sympos., Versailles, 1976, Lecture Notes in Control and Inform. Sci., vol. 2, Springer, Berlin, 1977, pp. 111-124. | MR 490213 | Zbl 0379.93030

[11] Fernández-Cara E., Zuazua E., The cost of approximate controllability for heat equations: the linear case, Adv. Differential Equations 5 (2000) 465-514. | MR 1750109 | Zbl 1007.93034

[12] Fernández-Cara E., Zuazua E., Null and approximate controllability for weakly blowing-up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 583-616. | Numdam | MR 1791879 | Zbl 0970.93023

[13] Fursikov A.V., Imanuvilov O.Yu., Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Research Institute of Mathematics, Seoul National University, Seoul, Korea, 1994. | MR 1406566 | Zbl 0862.49004

[14] X. Fu, J. Yong, X. Zhang, Exact controllability for multidimensional semilinear equations, Preprint, 2004.

[15] Hörmander L., Non-uniqueness for the Cauchy problem, in: Chazarain J. (Ed.), Fourier Integral Operators and Partial Differential Equations, Colloque International, Nice, Lecture Notes in Mathematics, vol. 459, Springer-Verlag, Berlin, 1974, pp. 36-72. | MR 419980 | Zbl 0315.35019

[16] Imanuvilov O.Yu., On Carleman estimates for hyperbolic equations, Asymptotic Anal. 32 (2002) 185-220. | MR 1993649 | Zbl 1050.35046

[17] Lions J.L., Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués, Tomes 1 & 2, RMA, vols. 8-9, Masson, Paris, 1988. | MR 963060 | Zbl 0653.93003

[18] Meshkov V.Z., On the possible rate of decrease at infinity of the solutions of second-order partial differential equations, Mat. Sb. 182 (1991) 364-383, (in Russian); Translation in, Math. USSR-Sb. 72 (1992) 343-361. | MR 1110071 | Zbl 0782.35010

[19] Miller L., Geometric bounds on the growth rate of null-controllability cost of the heat equation in small time, J. Differential Equations 204 (2004) 202-226. | MR 2076164 | Zbl 1053.93010

[20] Rosier L., Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM: Control Optim. Calc. Var. 2 (1997) 33-55. | Numdam | MR 1440078 | Zbl 0873.93008

[21] Russell D.L., A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Stud. Appl. Math. 52 (1973) 189-221. | MR 341256 | Zbl 0274.35041

[22] Seidman T., Avdonin S., Ivanov S.A., The “window problem” for series of complex exponentials, J. Fourier Anal. Appl. 6 (2000) 233-254. | MR 1755142 | Zbl 0960.42012

[23] Strauss W.A., Partial Differential Equations, Wiley, New York, 1992. | MR 1159712 | Zbl 0817.35001

[24] Uchiyama J., Lower bounds of decay order of eigenfunctions of second-order elliptic operators, Publ. Res. Inst. Math. Sci. 21 (1985) 1281-1297. | MR 842419 | Zbl 0616.35070

[25] Whittaker E.T., Watson G.N., A Course of Modern Analysis, Reprint of the fourth edition 1927, Cambridge Univ. Press, Cambridge, 1996. | JFM 53.0180.04 | MR 1424469 | Zbl 0951.30002

[26] Wolff T., A counterexample in a unique continuation problem, Comm. Anal. Geom. 2 (1994) 79-102. | MR 1312679 | Zbl 0836.35023

[27] Zhang X., Exact controllability of semilinear plate equations, Asymptotic Anal. 27 (2001) 95-125. | MR 1852002 | Zbl 1007.35008

[28] Zhang X., Zuazua E., Exact controllability of the semi-linear wave equation, in: Blondel V.D., Megretski A. (Eds.), Sixty Open Problems in the Mathematics of Systems and Control, Princeton University Press, 2004, pp. 173-178.

[29] Zuazua E., Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993) 109-129. | Numdam | MR 1212631 | Zbl 0769.93017

[30] Zuazua E., Remarks on the controllability of the Schrödinger equation, in: Bandrauk A., Delfour M.C., Le Bris C. (Eds.), Quantum Control: Mathematical and Numerical Challenges, CRM Proc. Lecture Notes, vol. 33, Amer. Math. Soc., Providence, RI, 2003, pp. 181-199. | MR 2043529

[31] Zuily C., Uniqueness and Nonuniqueness in the Cauchy Problem, Progress in Mathematics, vol. 33, Birkhäuser Boston, Boston, MA, 1983. | MR 701544 | Zbl 0521.35003