# American Institute of Mathematical Sciences

September  2011, 1(3): 307-330. doi: 10.3934/mcrf.2011.1.307

## Global Carleman estimate on a network for the wave equation and application to an inverse problem

 1 CNRS; LAAS; 7 avenue du colonel Roche, F-31077 Toulouse Cedex 4, France 2 Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin en Yvelines, F-78035 Versailles, France 3 Institut Elie Cartan de Nancy & INRIA (Project-Team CORIDA), BP 239, F-54506 Vandœuvre-les-Nancy Cedex, France

Received  March 2011 Revised  May 2011 Published  September 2011

We are interested in an inverse problem for the wave equation with potential on a star-shaped network. We prove the Lipschitz stability of the inverse problem consisting in the determination of the potential on each string of the network with Neumann boundary measurements at all but one external vertices. Our main tool, proved in this article, is a global Carleman estimate for the network.
Citation: Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307
##### References:
 [1] K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Differential Integral Equations, 17 (2004), 1395.   Google Scholar [2] S. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings,, ZAMM Z. Angew. Math. Mech., 90 (2010), 136.  doi: 10.1002/zamm.200900295.  Google Scholar [3] L. Baudouin, A. Mercado and A. Osses, A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem,, Inverse Problems, 23 (2007), 257.  doi: 10.1088/0266-5611/23/1/014.  Google Scholar [4] M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method,, Inverse Problems, 20 (2004), 647.  doi: 10.1088/0266-5611/20/3/002.  Google Scholar [5] M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients,, Appl. Anal., 83 (2004), 983.  doi: 10.1080/0003681042000221678.  Google Scholar [6] A. Benabdallah, Y. Dermenjian and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem,, J. Math. Anal. Appl., 336 (2007), 865.  doi: 10.1016/j.jmaa.2007.03.024.  Google Scholar [7] A. L. Bukhgeim, "Volterra Equations and Inverse Problems,", Inverse and Ill-Posed Problems Series, (1999).   Google Scholar [8] A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems,, Dokl. Akad. Nauk SSSR, 260 (1981), 269.   Google Scholar [9] R. Dáger, Observation and control of vibrations in tree-shaped networks of strings,, SIAM J. Control Optim., 43 (2004), 590.  doi: 10.1137/S0363012903421844.  Google Scholar [10] R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings,, In, (2000), 1006.   Google Scholar [11] R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in $1\text{-}d$ Flexible Multi-Structures," Mathématiques & Applications (Berlin), 50,, Springer-Verlag, (2006).   Google Scholar [12] O. Y. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement,, Inverse Problems, 19 (2003), 157.  doi: 10.1088/0266-5611/19/1/309.  Google Scholar [13] Oleg Yu. Imanuvilov and Masahiro Yamamoto, Global uniqueness and stability in determining coefficients of wave equations,, Comm. Partial Differential Equations, 26 (2001), 1409.   Google Scholar [14] V. Isakov, "Inverse Problems for Partial Differential Equations," second edition, Applied Mathematical Sciences, 127,, Springer, (2006).   Google Scholar [15] M. V. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575.  doi: 10.1088/0266-5611/8/4/009.  Google Scholar [16] M. V. Klibanov and J. Malinsky, Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data,, Inverse Problems, 7 (1991), 577.  doi: 10.1088/0266-5611/7/4/007.  Google Scholar [17] M. V. Klibanov and A. Timonov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications,", Inverse and Ill-Posed Problems Series, (2004).   Google Scholar [18] J. E. Lagnese, G. Leugering, and E. J. P. G. Schmidt, "Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures,", Systems & Control: Foundations & Applications, (1994).   Google Scholar [19] I. Lasiecka, R. Triggiani and P.-F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13.  doi: 10.1006/jmaa.1999.6348.  Google Scholar [20] J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité Exacte," Recherches en Mathématiques Appliquées, 8,, Masson, (1988).   Google Scholar [21] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Translated from the French by P. Kenneth, (1972).   Google Scholar [22] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, Netw. Heterog. Media, 2 (2007), 425.  doi: 10.3934/nhm.2007.2.425.  Google Scholar [23] S. Nicaise and O. Zaïr, Identifiability, stability and reconstruction results of point sources by boundary measurements in heteregeneous trees,, Rev. Mat. Complut., 16 (2003), 151.   Google Scholar [24] J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional hyperbolic inverse problem,, J. Inverse Ill-Posed Probl., 5 (1997), 55.  doi: 10.1515/jiip.1997.5.1.55.  Google Scholar [25] E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings,, SIAM J. Control Optim., 30 (1992), 229.  doi: 10.1137/0330015.  Google Scholar [26] J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks,, SIAM J. Control Optim., 48 (2009), 2771.  doi: 10.1137/080733590.  Google Scholar [27] F. Visco-Comandini, M. Mirrahimi and M. Sorine, Some inverse scattering problems on star-shaped graphs,, J. Math. Anal. Appl., 378 (2011), 343.   Google Scholar [28] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl. (9), 78 (1999), 65.  doi: 10.1016/S0021-7824(99)80010-5.  Google Scholar [29] M. Yamamoto and X. Zhang, Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns,, Appl. Math. Optim., 48 (2003), 211.  doi: 10.1007/s00245-003-0775-5.  Google Scholar [30] X. Zhang, Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities,, SIAM J. Control Optim., 39 (2000), 812.  doi: 10.1137/S0363012999350298.  Google Scholar [31] E. Zuazua, Control and stabilization of waves on 1-d networks,, in, (2011).   Google Scholar

show all references

##### References:
 [1] K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings,, Differential Integral Equations, 17 (2004), 1395.   Google Scholar [2] S. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings,, ZAMM Z. Angew. Math. Mech., 90 (2010), 136.  doi: 10.1002/zamm.200900295.  Google Scholar [3] L. Baudouin, A. Mercado and A. Osses, A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem,, Inverse Problems, 23 (2007), 257.  doi: 10.1088/0266-5611/23/1/014.  Google Scholar [4] M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method,, Inverse Problems, 20 (2004), 647.  doi: 10.1088/0266-5611/20/3/002.  Google Scholar [5] M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients,, Appl. Anal., 83 (2004), 983.  doi: 10.1080/0003681042000221678.  Google Scholar [6] A. Benabdallah, Y. Dermenjian and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem,, J. Math. Anal. Appl., 336 (2007), 865.  doi: 10.1016/j.jmaa.2007.03.024.  Google Scholar [7] A. L. Bukhgeim, "Volterra Equations and Inverse Problems,", Inverse and Ill-Posed Problems Series, (1999).   Google Scholar [8] A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems,, Dokl. Akad. Nauk SSSR, 260 (1981), 269.   Google Scholar [9] R. Dáger, Observation and control of vibrations in tree-shaped networks of strings,, SIAM J. Control Optim., 43 (2004), 590.  doi: 10.1137/S0363012903421844.  Google Scholar [10] R. Dáger and E. Zuazua, Controllability of star-shaped networks of strings,, In, (2000), 1006.   Google Scholar [11] R. Dáger and E. Zuazua, "Wave Propagation, Observation and Control in $1\text{-}d$ Flexible Multi-Structures," Mathématiques & Applications (Berlin), 50,, Springer-Verlag, (2006).   Google Scholar [12] O. Y. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement,, Inverse Problems, 19 (2003), 157.  doi: 10.1088/0266-5611/19/1/309.  Google Scholar [13] Oleg Yu. Imanuvilov and Masahiro Yamamoto, Global uniqueness and stability in determining coefficients of wave equations,, Comm. Partial Differential Equations, 26 (2001), 1409.   Google Scholar [14] V. Isakov, "Inverse Problems for Partial Differential Equations," second edition, Applied Mathematical Sciences, 127,, Springer, (2006).   Google Scholar [15] M. V. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575.  doi: 10.1088/0266-5611/8/4/009.  Google Scholar [16] M. V. Klibanov and J. Malinsky, Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data,, Inverse Problems, 7 (1991), 577.  doi: 10.1088/0266-5611/7/4/007.  Google Scholar [17] M. V. Klibanov and A. Timonov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications,", Inverse and Ill-Posed Problems Series, (2004).   Google Scholar [18] J. E. Lagnese, G. Leugering, and E. J. P. G. Schmidt, "Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures,", Systems & Control: Foundations & Applications, (1994).   Google Scholar [19] I. Lasiecka, R. Triggiani and P.-F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13.  doi: 10.1006/jmaa.1999.6348.  Google Scholar [20] J.-L. Lions, "Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité Exacte," Recherches en Mathématiques Appliquées, 8,, Masson, (1988).   Google Scholar [21] J.-L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Translated from the French by P. Kenneth, (1972).   Google Scholar [22] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, Netw. Heterog. Media, 2 (2007), 425.  doi: 10.3934/nhm.2007.2.425.  Google Scholar [23] S. Nicaise and O. Zaïr, Identifiability, stability and reconstruction results of point sources by boundary measurements in heteregeneous trees,, Rev. Mat. Complut., 16 (2003), 151.   Google Scholar [24] J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional hyperbolic inverse problem,, J. Inverse Ill-Posed Probl., 5 (1997), 55.  doi: 10.1515/jiip.1997.5.1.55.  Google Scholar [25] E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings,, SIAM J. Control Optim., 30 (1992), 229.  doi: 10.1137/0330015.  Google Scholar [26] J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks,, SIAM J. Control Optim., 48 (2009), 2771.  doi: 10.1137/080733590.  Google Scholar [27] F. Visco-Comandini, M. Mirrahimi and M. Sorine, Some inverse scattering problems on star-shaped graphs,, J. Math. Anal. Appl., 378 (2011), 343.   Google Scholar [28] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl. (9), 78 (1999), 65.  doi: 10.1016/S0021-7824(99)80010-5.  Google Scholar [29] M. Yamamoto and X. Zhang, Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns,, Appl. Math. Optim., 48 (2003), 211.  doi: 10.1007/s00245-003-0775-5.  Google Scholar [30] X. Zhang, Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities,, SIAM J. Control Optim., 39 (2000), 812.  doi: 10.1137/S0363012999350298.  Google Scholar [31] E. Zuazua, Control and stabilization of waves on 1-d networks,, in, (2011).   Google Scholar
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