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Identification of the Potential Coefficient in the Wave Equation with Incomplete Data: A Sentinel Method

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Abstract

In this paper, we consider a wave equation with incomplete data, where we do not know the potential coefficient and the initial conditions. From observing the system in the boundary, we want to get information on the potential coefficient independently of the initial conditions. This can be obtained using the sentinel method of Lions, which is a functional insensitive to certain parameters. Shows us through the adjoint system that the existence of the sentinel is equivalent to an optimal control problem. We solve this optimal control problem by using the Hilbert uniqueness method (HUM).

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REFERENCES

  1. M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monogrphs in Mathematics (Springer, Tokyo, 2017). https://doi.org/10.1007/978-4-431-56600-7

  2. M. Yamamoto, “Uniqueness and stability in multidimensional hyperbolic inverse problems,” J. Math. Pures Appl. 78, 65–98 (1999). https://doi.org/10.1016/S0021-7824(99)80010-5

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Rakesh, “Reconstruction for an inverse problem for the wave equation with constant velocity,” Inverse Probl. 6, 91–98 (1990). https://doi.org/10.1088/0266-5611/6/1/009

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Khaĭdarov, “Carleman estimates and inverse problems for second order hyperbolic equations,” Math. USSR Sb. 58, 267–277 (1987). https://doi.org/10.1070/SM1987v058n01ABEH003103

    Article  MathSciNet  Google Scholar 

  5. J.-P. Puel and M. Yamamoto, “On a global estimate in a linear inverse hyperbolic problem,” Inverse Probl. 12, 995–1002 (1996). https://doi.org/10.1088/0266-5611/12/6/013

    Article  MathSciNet  MATH  Google Scholar 

  6. Rakesh and W. W. Symes, “Uniqueness for an inverse problem for the wave equation: Inverse problem for the wave equation,” Commun. Partial Differ. Equations 13, 87–96 (1988). https://doi.org/10.1080/03605308808820539

    Article  MATH  Google Scholar 

  7. O. Yu. Imanuvilov and M. Yamamoto, “Global Lipschitz stability in an inverse hyperbolic problem by interior observations,” Inverse Probl. 17, 717–728 (2001). https://doi.org/10.1088/0266-5611/17/4/310

    Article  MathSciNet  MATH  Google Scholar 

  8. O. Yu. Imanuvilov and M. Yamamoto, “Global uniqueness and stability in determining coe?cient of wave equations,” Commun. Partial Differ. Equtaions 26, 1409–1425 (2001). https://doi.org/10.1081/PDE-100106139

    Article  MATH  Google Scholar 

  9. J.-L. Lions, Sentinelles pour les systèmes distribués à données incomplètes, Recherches en Mathématiques Appliquées, Vol. 21 (Masson, Paris, 1992).

  10. A. Merabet, A. Ayadi, and A. Omrane, “Detection of pollution terms in nonlinear second order wave systems,” Int. J. Parallel, Emergent Distrib. Syst. 34, 13–20 (2019). https://doi.org/10.1080/17445760.2017.1318134

    Article  Google Scholar 

  11. J.-P. Kernévez, The Sentinel Method and Its Application to Environmental Pollution Problems (CRC Press, Boca Raton, Fla., 1997).

    MATH  Google Scholar 

  12. F. Molinet, “Simulation numérique de problèmes d’écosystèmes. Sentinelles pour la detection d’origine de pollution,” PhD Thesis (Paris 11, 1994).

  13. R. Mosé, M. E. Stoeckel, C. Poulard, P. Ackerer, and F. Lehmann, “Transport parameters identification: application of the Sentinel method,” Comput. Geosci. 4, 251–273 (2000). https://doi.org/10.1023/A:1011516101288

    Article  MATH  Google Scholar 

  14. L. Chafia, A. Abdelhamid, and H. Abdelhak, “Identification problem of a fractional thermoelastic deformation system with incomplete data: A sentinel method,” Nonlinear Stud. 29, 399–410 (2022).

    MathSciNet  Google Scholar 

  15. B. Amel and R. Imad, “Identification of the source term in Navier–Stokes system with incomplete data,” AIMS Math. 4, 516–526 (2019). https://doi.org/10.3934/math.2019.3.516

    Article  MathSciNet  MATH  Google Scholar 

  16. Y. Miloudi, O. Nakoulima, and A. Omrane, “On the instantaneous sentinels in pollution problems of incomplete data,” Inverse Probl. Sci. Eng. 17, 451–459 (2009). https://doi.org/10.1080/17415970802015948

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Omrane, “Some aspects of the sentinel method for pollution problems, in Air Quality-Monitoring and Modeling (InTech, 2012), pp. 185–204. https://doi.org/10.5772/33173

    Book  Google Scholar 

  18. S. Sandel and A. Ayadi, “Boundary sentinels for the resolution of a geometrical problem,” Turk. J. Math. 42, 548–556 (2018). https://doi.org/10.3906/mat-1607-5

    Article  MathSciNet  MATH  Google Scholar 

  19. J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systemes distribués : Perturbations, Vol. 1 : Contrôlabilité exacte, Recherches en Mathématiques Appliquées, Vol. 8 (Masson, Paris, 1988).

  20. J. L. Lions, “Exact controllability, stabilization and perturbations for distributed systems,” SIAM Rev. 30, 1–68 (1988). https://doi.org/10.1137/1030001

    Article  MathSciNet  MATH  Google Scholar 

  21. B. Dehman and A. Omrane, “On the controllability under constraints on the control for hyperbolic equations,” Appl. Math. E-Notes 10, 36–39 (2010).

    MathSciNet  MATH  Google Scholar 

  22. E. Zuazua, “Controllability and observability of partial differential equations: some results and open problems,” in Handbook of Differential Equations: Evolutionary Equations, Vol. 3, Ed. by C. M. Dafermos and E. Feireisl (North-Holland, Amsterdam, 2007), vol. 3, pp. 527–621. https://doi.org/10.1016/S1874-5717(07)80010-7

    Book  Google Scholar 

  23. V. Komornik, “A new method of exact controllability in short time and applications,” Ann. Faculté Sci. Toulouse: Math., Ser. 5 10, 415–464 (1989).

    Article  MATH  Google Scholar 

  24. V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Recherches en Mathématiques Appliquées, Vol. 36 (Wiley-Masson, Paris, 1994).

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Authors and Affiliations

  1. Laboratory of Mathematics, Informatics and Systems (LAMIS), Department of Mathematics and Computer Sciences, Echahid Cheikh Larbi Tebessi University, 12002, Tebessa, Algeria

    Billal Elhamza &  Abdelhak Hafdallah

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  1. Billal Elhamza
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  2. Abdelhak Hafdallah
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Correspondence to Billal Elhamza or Abdelhak Hafdallah.

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Billal Elhamza, Abdelhak Hafdallah Identification of the Potential Coefficient in the Wave Equation with Incomplete Data: A Sentinel Method. Russ Math. 66, 102–111 (2022). https://doi.org/10.3103/S1066369X22120027

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