The aim of this paper is to provide a general framework for solving a class of inverse source problems by using exact observability of infinite dimensional systems. More precisely, we show that if a system is exactly observable, then a source term in this system can be identified by knowing its intensity and appropriate observations which often correspond to measurements of some boundary traces. This abstract theory is then applied to obtain new identifiability results for a system governed by the Euler–Bernoulli plate equation. Using a different methodology, we show that exact observability can be used to identify both the locations and the intensities of combinations of point sources in the plate equation.

MSC codes

  1. 93C25
  2. 93B07
  3. 93C20
  4. 35R30

MSC codes

  1. inverse source problem
  2. exact observability
  3. plate equation

Get full access to this article

View all available purchase options and get full access to this article.


M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1992.
C. J. S. Alves and A. L. Silvestre, On the determination of point-forces on a Stokes system, Math. Comput. Simulation, 66 (2004), pp. 385–397.
C. Bardos, G. Lebeau, and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Rend. Sem. Mat. Univ. Politec. Torino, 1988, Special Issue (1989), pp. 11–31.
C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), pp. 1024–1065.
J.-M. Bony, Cours d'analyse. Théorie des distributions et analyse de Fourier, Les Éditions de l'École Polytechnique, Palaiseau, France, 2001.
N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), pp. 157–191.
N. Burq, Contrôle de l'équation des ondes dans des ouverts comportant des coins, Bull. Soc. Math. France, 126 (1998), pp. 601–637.
S. Dolecki and D. L. Russell, A general theory of observation and control., SIAM J. Control Optim., 15 (1977), pp. 185–220.
R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966), pp. 413–415.
A. El Badia and T. Ha-Duong, Determination of point wave sources by boundary measurements, Inverse Problems, 17 (2001), pp. 1127–1139.
A. El Badia and T. Ha-Duong, On an inverse source problem for the heat equation. Application to a pollution detection problem, J. Inverse Ill-Posed Probl., 10 (2002), pp. 585–599.
M. Kitahara, Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates, Stud. Appl. Mech. 10, Elsevier Scientific, Amsterdam, 1985.
V. Komornik and M. Yamamoto, Upper and lower estimates in determining point sources in a wave equation, Inverse Problems, 18 (2002), pp. 319–329.
V. Komornik and M. Yamamoto, Estimation of point sources and applications to inverse problems, Inverse Problems, 21 (2005), pp. 2051–2070.
R. Kress, Linear Integral Equations, Appl. Math. Sci. 82, Springer-Verlag, Berlin, 1989.
G. Lebeau, Contrôle de l'équation de Schrödinger, J. Math. Pures Appl. (9), 71 (1992), pp. 267–291.
S. Nicaise and O. Zaïr, Identifiability, stability and reconstruction results of point sources by boundary measurements in heteregeneous trees, Rev. Mat. Complut., 16 (2003), pp. 151–178.
S. Nicaise and O. Zaïr, Determination of point sources in vibrating beams by boundary measurements: Identifiability, stability, and reconstruction results, Electron. J. Differential Equations, No. 20 (2004).
J.-P. Puel and M. Yamamoto, Applications de la contrôlabilité exacte à quelques problèmes inverses hyperboliques, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), pp. 1171–1176.
D. Salamon, Infinite–dimensional linear systems with unbounded control and observation: A functional analytical approach, Trans. Amer. Math. Soc., 300 (1987), pp. 383–431.
D. Salamon, Realization theory in Hilbert space, Math. Systems Theory, 21 (1989), pp. 147–164.
G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation, Trans. Amer. Math. Soc., 361 (2009), pp. 951–977.
M. Tucsnak and G. Weiss, Simultaneous exact controllability and some applications, SIAM J. Control Optim., 38 (2000), pp. 1408–1427.
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2009; also available online from http://www.iecn.u-nancy.fr/~tucsnak/obsbook.pdf.
Y. H. Wang, Global uniqueness and stability for an inverse plate problem, J. Optim. Theory Appl., 132 (2007), pp. 161–173.
G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems, 7 (1994), pp. 23–57.
G. Weiss, Transfer functions of regular linear systems, Part I: Characterizations of regularity, Trans. Amer. Math. Soc., 342 (1994), pp. 827–854.
J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1987.
M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems, 11 (1995), pp. 481–496.
M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl. (9), 78 (1999), pp. 65–98.

Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 1632 - 1659
ISSN (online): 1095-7138


Submitted: 29 May 2008
Accepted: 27 January 2009
Published online: 7 May 2009

MSC codes

  1. 93C25
  2. 93B07
  3. 93C20
  4. 35R30

MSC codes

  1. inverse source problem
  2. exact observability
  3. plate equation



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.







Copy the content Link

Share with email

Email a colleague

Share on social media