Carleman Estimates for Second Order Hyperbolic Operators and Applications, a Unified Approach

  • Xiaoyu FuEmail author
  • Qi Lü
  • Xu Zhang
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


In this chapter, we establish three Carleman estimates for second order hyperbolic operators. The first one is Theorem 4.1, which is used to solve an inverse hyperbolic problem. The second one is Theorem 4.2, a Carleman estimate in \(H^1\)-norm, and based on it, we further derive the third Carleman estimate in \(L^2\)-norm (see Theorem 4.3). As the applications of the later, we obtain the exact controllability of semilinear hyperbolic equations and the exponential decay of locally damped hyperbolic equations.


Carleman estimate Second order hyperbolic operator Exact controllability Exponential decay Inverse hyperbolic problem 


  1. 1.
    Alabau-Boussouira, F.: On some recent advances on stabilization for hyperbolic equations. In: Control of Partial Differential Equations. Lecture Notes in Mathematics, vol. 2048, pp. 1–100. Fond. CIME/CIME Found. Subser. Springer, Heidelberg (2012)Google Scholar
  2. 2.
    Alabau-Boussouira, F., Ammari, K.: Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system. J. Funct. Anal. 260, 2424–2450 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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, 1024–1065 (1992)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bellassoued, M., Yamamoto, M.: Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation. J. Math. Pures Appl. 85, 193–224 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bukhgeim, A.L., Klibanov, M.V.: Global uniqueness of class of multidimensional inverse problems. Soviet Math. Dokl. 24, 244–247 (1981)zbMATHGoogle Scholar
  6. 6.
    Cazenave, T., Haraux, A.: Equations d’évolution avec non-linéarité logarithmique. Ann. Fac. Sci. Toulouse. 2, 21–51 (1980)CrossRefGoogle Scholar
  7. 7.
    Dehman, B., Lebeau, G., Zuazua, E.: Stabilization and control for the subcritical semilinear wave equation. Ann. Sci. École Norm. Sup. 36, 525–551 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dos Santos Ferreira, D.: Sharp \(L^p\) Carleman estimates and unique continuation. Duke Math. J. 129, 503–550 (2005)Google Scholar
  9. 9.
    Duyckaerts, T., Zhang, X., Zuazua, E.: On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire. 25, 1–41 (2008)Google Scholar
  10. 10.
    Fattorini, H.O.: Local controllability of a nonlinear wave equations. Math. Syst. Theory 9, 30–45 (1975)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fattorini, H.O., Russell, D.L.: Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43, 272–292 (1971)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fu, X., Yong, J., Zhang, X.: Exact controllability for the multidimensional semilinear hyperbolic equations. SIAM J. Control Optim. 46, 1578–1614 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fu, X., Liu, X., Lü, Q., Zhang, X.: An internal observability estimate for stochastic hyperbolic equations. ESAIM Control Optim. Calc. Var. 22, 1382–1411 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Imanuvilov, OYu.: On Carlerman estimates for hyperbolic equations. Asymptot. Anal. 32, 185–220 (2002)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Imanuvilov, OYu., Yamamoto, M.: Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Probl. 17, 717–728 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kenig, C.E., Ruiz, A., Sogge, C.D.: Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. 55, 329–347 (1987)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Klibanov, M.V.: Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J. Inverse Ill-Posed Probl. 21, 477–560 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Koch, H., Tataru, D.: Dispersive estimates for principally normal pseudodifferential operators. Commun. Pure Appl. Math. 58, 217–284 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lebeau, G.: Un probléme d’unicité forte pour l’équation des ondes. Commun. Partial Differ. Equ. 24, 777–783 (1999)CrossRefGoogle Scholar
  20. 20.
    Li, W., Zhang, X.: Controllability of parabolic and hyperbolic equations: toward a unified theory. In: Control Theory of Partial Differential Equations. Lecture Notes in Pure and Applied Mathematics, vol. 242, pp. 157–174. Chapan & Hall/CRC, Boca Raton (2005)Google Scholar
  21. 21.
    Li, T.T.: Controllability and Observability for Quasilinear Hyperbolic Systems. AIMS Series on Applied Mathematics. vol. 3, American Institute of Mathematical Sciences (AIMS). Springfield (2010)Google Scholar
  22. 22.
    Li, X., Yong, J.: Optimal Control Theory for Infinite-Dimensional Systems. Birkhäuser Boston Inc, Boston (1995)CrossRefGoogle Scholar
  23. 23.
    Lions, J.L.: Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes distribués, Tome 1, Contrôlabilité exacte. Recherches en Mathématiques Appliquées. vol. 8, Masson, Paris (1988)Google Scholar
  24. 24.
    Liu, Y.: Some sufficient conditions for the controllability of wave equations with variable coefficients. Acta Appl. Math. 128, 181–191 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lü, Q., Yin, Z.: Unique continuation for stochastic hyperbolic equations. arXiv:1701.03599
  26. 26.
    Lü, Q.: Some results on the controllability of forward stochastic heat equations with control on the drift. J. Funct. Anal. 260, 832–851 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lü, Q., Zhang, X.: Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns. Commun. Pure Appl. Math. 68, 948–963 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Russell, D.L.: Controllability and stabilizability theory for linear partial differential equations: recent progress and open problems. SIAM Rev. 20, 639–739 (1978)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tebou, L.: A Carleman estimate based approach for the stabilization of some locally damped semilinear hyperbolic equations. ESAIM: Control Optim. Calc. Var. 14, 561–574 (2008)Google Scholar
  30. 30.
    Vessella, S.: Quantitative estimates of strong unique continuation for wave equations. Math. Ann. 367, 135–164 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Yuan, G.: Determination of two kinds of sources simultaneously for a stochastic wave equation. Inverse Probl. 31, 085003 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Zhang, X.: Explicit observability estimate for the wave equation with potential and its application. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456, 1101–1115 (2000)Google Scholar
  33. 33.
    Zhang, X.: Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities. SIAM J. Control Optim. 39, 812–834 (2001)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhang, X.: Carleman and observability estimates for stochastic wave equations. SIAM J. Math. Anal. 40, 851–868 (2008)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Zuazua, E.: Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire. 10, 109–129 (1993)Google Scholar
  36. 36.
    Zuazua, E.: Exact controllability for the semilinear wave equation. J. Math. Pures Appl. 69, 1–31 (1990)MathSciNetzbMATHGoogle Scholar

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsSichuan UniversityChengduChina

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