Identification of a Reflection Boundary Coefficient in an Acoustic Wave Equation by Optimal Control Techniques

  • Suzanne Lenhart
  • Vladimir Protopopescu
  • Jiongmin Yong
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 5)


We apply optimal control techniques to find approximate solutions to an inverse problem for the acoustic wave equation. The inverse problem (assumed here to have a solution) is to determine the boundary refection coefficient from partial measurements of the acoustic signal. The sought reflection coefficient is treated as a control and the goal — quantified by an objective functional — is to drive the model solution close to the experimental data by adjusting this coefficient. The problem is solved by finding the optimal control that minimizes the objective functional. Then by driving the “cost of the control” to zero one proves that the sequence of optimal controls represents a converging sequence of estimates for the solution of the inverse problem. Compared to classical regularization methods (e.g. Tikhonov coupled with optimization schemes), our approach yields: (i) a systematic procedure to solve inverse problems of identification type and (ii) an explicit expression for the approximations of the solution.


Inverse Problem Weak Solution Optimal Control Problem Parameter Estimation Problem Boundary Optimal Control 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhäuser, Boston, 1989.zbMATHCrossRefGoogle Scholar
  2. 2.
    V. Barbu and N. H. Pavel, Determining the acoustic impedance in the 1-D wave equation via an optimal control problem, SIAM J. Optimal Control 35 (1997), 1544–1556.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    B.P. Belinskiy, J.P. Dauer, C. Martin, and M.A. Shubov, On controllability of an oscillating continuum with a viscous damping,preprint.Google Scholar
  4. 4.
    J. Chavent, On parameter idenitifiability, Proceedings of the 7th IFAC Symposium on Identification and System Parameter Estimations, Pergamon Press, York, 1985, pp. 531–36.Google Scholar
  5. 5.
    J. D. Craig and C. Brown, Inverse Problems in Astronomy, Adam Hilger Ltd., Bristol and Boston, 1986.zbMATHGoogle Scholar
  6. 6.
    L.C. Evans, Partial Differential Equations Vol. 3B (1993), Berkeley Math Lecture Notes, Berkeley.Google Scholar
  7. 7.
    K.D. Graham and D.L. Russell, Boundary value control of the wave equation in a spherical region, SIAM Journal on Control 13 (1975), 174–196.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    C.W. Groetsch, Inverse Problems in the Mathematical Sciences, Vieweg, Braunschweig, Wiesbaden, 1993.Google Scholar
  9. 9.
    F. James and M. Sepulveda, Parameter identification for a model of chromatographic column, Inverse Problems 10 (1994), 367–385.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Suzanne Lenhart
    • 1
    • 2
  • Vladimir Protopopescu
    • 2
  • Jiongmin Yong
    • 3
  1. 1.Mathematics DepartmentUniversity of TennesseeKnoxvilleUSA
  2. 2.Computer Science and Mathematics DivisionOak Ridge National LaboratoryOak RidgeUSA
  3. 3.Department of MathematicsFudan UniversityShanghaiChina

Personalised recommendations