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Applications of Observability Inequalities

  • Jone ApraizEmail author
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 18)

Abstract

This article presents two observability inequalities for the heat equation over Ω × (0, T). In the first one, the observation is from a subset of positive measure in Ω × (0, T), while in the second, the observation is from a subset of positive surface measure on ∂Ω × (0, T). We will provide some applications for the above-mentioned observability inequalities, the bang-bang property for the minimal time control problems and the bang-bang property for the minimal norm control problems, and also establish new open problems related to observability inequalities and the aforementioned applications.

Keywords

Parabolic equations Control theory Controllability Observability inequalities Bang-Bang properties 

AMS 2010 Codes:

49J20 49J30 58E25 93B05 93B07 35K05 

References

  1. 1.
    Apraiz, J., Escauriaza, L.: Null-control and measurable sets. ESAIM Control Optim. Calc. Var. 19, 239–254 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Apraiz, J., Escauriaza, L., Wang, G., Zhang, C.: Observability inequalities and measurable sets. J. Eur. Math. Soc. 16, 2433–2475 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brown, R.M.: The method of layer potentials for the heat equation in Lipschitz cylinders. Am. J. Math. 111, 339–379 (1989)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fabes, E.B., Salsa, S.: Estimates of caloric measure and the initial-Dirichlet problem for the heat equation in Lipschitz cylinders. Trans. Am. Math. Soc. 279, 635–650 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fursikov, A.V., Yu Imanuvilov, O.: Controllability of Evolution Equations. Lecture Notes Series, vol. 34. Seoul National University, Seoul (1996)Google Scholar
  6. 6.
    Lebeau, G., Robbiano, L.: Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20, 335–356 (1995)CrossRefGoogle Scholar
  7. 7.
    Lions, J.L.: Optimal Control for Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefGoogle Scholar
  8. 8.
    Phung, K.D., Wang, G.: An observability estimate for parabolic equations from a measurable set in time and its applications. J. Eur. Math. Soc. 15, 681–703 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Phung, K.D., Wang, G., Zhang, X.: On the existence of time optimal controls for linear evolution equations. Discrete Contin. Dynam. Syst. Ser. B 8, 925–941 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Vessella, S.: A continuous dependence result in the analytic continuation problem. Forum Math. 11, 695–703 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wang, G.: L -null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim. 47, 1701–1720 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zhang, C.: An observability estimate for the heat equation from a product of two measurable sets. J. Math. Anal. Appl. 396, 7–12 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad del País VascoLeioaSpain

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