# American Institute of Mathematical Sciences

November  2013, 33(11&12): 5217-5252. doi: 10.3934/dcds.2013.33.5217

## Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace

 1 Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki 2 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States

Received  February 2012 Revised  September 2012 Published  May 2013

We consider a second-order hyperbolic equation defined on an open bounded domain $\Omega$ in $\mathbb{R}^n$ for $n \geq 2$, with $C^2$-boundary $\Gamma = \partial \Omega = \overline{\Gamma_0 \cup \Gamma_1}$, $\Gamma_0 \cap \Gamma_1 = \emptyset$, subject to non-homogeneous Dirichlet boundary conditions for the entire boundary $\Gamma$. We then study the inverse problem of determining both the damping and the potential (source) coefficients simultaneously, in one shot, by means of an additional measurement of the Neumann boundary trace of the solution, in a suitable, explicit sub-portion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T > 0$. Under sharp conditions on the complementary part $\Gamma_0 = \Gamma \backslash \Gamma_1$, $T > 0$, and under sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and (ii) Lipschitz-stability. The latter (ii) is the main result of the paper. Our proof relies on a few main ingredients: (a) sharp Carleman estimates at the $H^1(\Omega) \times L^2(\Omega)$-level for second-order hyperbolic equations [23], originally introduced for control theory issues; (b) a correspondingly implied continuous observability inequality at the same energy level [23]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Dirichlet boundary data [14,15,16]. The proof of the linear uniqueness result (Section 3) also takes advantage of a convenient tactical route post-Carleman estimates" proposed by V. Isakov in [8, Thm. 8.2.2, p. 231]. Expressing the final results for the nonlinear inverse problem directly in terms of the data offers an additional challenge.
Citation: Shitao Liu, Roberto Triggiani. Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5217-5252. doi: 10.3934/dcds.2013.33.5217
##### References:
 [1] A. Bukhgeim, J. Cheng, V. Isakov and M. Yamamoto, Uniqueness in determining damping coefficients in hyperbolic equations,, Analytic Extension Formulas and their Applications, (2001), 27.   Google Scholar [2] A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse problem,, Sov., 24 (1981), 244.   Google Scholar [3] T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables independantes,, Ark. Mat. Astr. Fys., 2B (1939), 1.   Google Scholar [4] L.F.Ho, Observabilite frontiere de l'equation des ondes,, Comptes Rendus de l'Academie des Sciences de Paris, 302 (1986), 443.   Google Scholar [5] O. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations,, Inverse Problems, 17 (2001), 717.  doi: 10.1088/0266-5611/17/4/310.  Google Scholar [6] O. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations,, Comm. Partial Differential Equations, 26 (2001), 1409.  doi: 10.1081/PDE-100106139.  Google Scholar [7] V. Isakov, "Inverse Problems for Partial Differential Equations,", First Edition, (1998).   Google Scholar [8] V. Isakov, "Inverse Problems for Partial Differential Equations,", Second Edition, (2006).   Google Scholar [9] V. Isakov, "Inverse Source Problems,", American Mathematical Society, (2000).   Google Scholar [10] V. Isakov and M. Yamamoto, Carleman estimate with the Neumann boundary condition and its application to the observability inequality and inverse hyperbolic problems,, Contemp. Math., 268 (2000), 191.  doi: 10.1090/conm/268/04314.  Google Scholar [11] V. Isakov and M. Yamamoto, Stability in a wave source problem by Dirichlet data on subboundary,, J. of Inverse & Ill-Posed Problems, 11 (2003), 399.  doi: 10.1515/156939403770862802.  Google Scholar [12] M. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575.  doi: 10.1088/0266-5611/8/4/009.  Google Scholar [13] M. Klibanov and A. Timonov, "Carleman Estimates For Coefficient Inverse Problems and Numerical Applications,", VSP, (2004).  doi: 10.1515/9783110915549.  Google Scholar [14] I. Lasiecka, J. L. Lions and R. Triggiani, Non homogeneous boundary value problems for second-order hyperbolic operators,, J. Math. Pures Appl., 65 (1986), 149.   Google Scholar [15] I. Lasiecka and R. Triggiani, A cosine operator approach to modeling $L_2(0,T;L_2(\Omega))$ boundary input hyperbolic equations,, Appl. Math. & Optimiz., 7 (1981), 35.  doi: 10.1007/BF01442108.  Google Scholar [16] I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_2(0,T;L_2(\Gamma))$-Dirichlet boundary terms,, Appl. Math. & Optimiz., 10 (1983), 275.  doi: 10.1007/BF01448390.  Google Scholar [17] I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control,, Appl. Math. & Optimiz., 19 (1989), 243.  doi: 10.1007/BF01448201.  Google Scholar [18] I. Lasiecka and R. Triggiani, Recent advances in regularity of second-order hyperbolic mixed problems and applications,, Dynamics Reported, 3 (1994), 104.  doi: 10.1007/978-3-642-78234-3_3.  Google Scholar [19] I. Lasiecka and R. Triggiani, Carleman estimates and exact controllability for a system of coupled, nonconservative second-order hyperbolic equations,, Marcel Dekker Lectures Notes Pure Appl. Math., 188 (1997), 215.   Google Scholar [20] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories,", 2, 2 (2000).   Google Scholar [21] 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 [22] I. Lasiecka, R. Triggiani and P. F. Yao, An observability estimate in $L_2(\Omega)\times H^{-1}(\Omega)$ for second order hyperbolic equations with variable coefficients,, Control of Distributed Parameter and Stochastic Systems, (1999), 71.   Google Scholar [23] I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot,, Contemp. Math., 268 (2000), 227.  doi: 10.1090/conm/268/04315.  Google Scholar [24] M. M. Lavrentev, V. G. Romanov and S. P. Shishataskii, "Ill-Posed Problems of Mathematical Physics and Analysis,", Amer. Math. Soc., 64 (1986).   Google Scholar [25] J. L. Lions, "Controlabilite Exacte,", Perturbations et Stabilisation de Systemes Distribues, 1 (1988).   Google Scholar [26] J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications,", I, I (1972).   Google Scholar [27] W. Littman, "Near Optimal Time Boundary Controllability for a Class of Hyperbolic Equations,", Lecture Notes in Control and Inform. Sci. 97, 97 (1987), 307.  doi: 10.1007/BFb0038763.  Google Scholar [28] S. Liu, Inverse problem for a structural acoustic interaction,, Nonlinear Anal., 74 (2011), 2647.  doi: 10.1016/j.na.2010.12.020.  Google Scholar [29] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping and potential coefficients of an inverse hyperbolic problem,, Nonlinear Anal. Real World Appl., 12 (2011), 1562.  doi: 10.1016/j.nonrwa.2010.10.014.  Google Scholar [30] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Neumann B.C. through an additional Dirichlet boundary trace,, SIAM J. Math. Anal., 43 (2011), 1631.  doi: 10.1137/100808988.  Google Scholar [31] S. Liu and R. Triggiani, Global uniqueness in determining electric potentials for a system of strongly coupled Schrödinger equations with magnetic potential terms,, J. Inverse Ill-Posed Probl., 19 (2011), 223.  doi: 10.1515/JIIP.2011.030.  Google Scholar [32] S. Liu and R. Triggiani, Recovering the damping coefficients for a system of coupled wave equations with Neumann BC: Uniqueness and stability,, Chin. Ann. Math. Ser B, 32 (2011), 669.  doi: 10.1007/s11401-011-0672-1.  Google Scholar [33] S. Liu and R. Triggiani, Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness,, Dynamical Systems and Differential Equations, (2011), 1001.   Google Scholar [34] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Dirichlet B.C. through an additional localized Neumann boundary trace,, Applicable Analysis, 91 (2012), 1551.  doi: 10.1080/00036811.2011.618125.  Google Scholar [35] S. Liu and R. Triggiani, "Boundary Control and Boundary Inverse Theory for Non-Homogeneous Second-Order Hyperbolic Equations: A Common Carleman Estimates Approach,", Special Volume in Book Series of American Institute of Mathematical Sciences, ().   Google Scholar [36] V. G. Mazya and T. O. Shaposhnikova, "Theory of Multipliers in Spaces of Differentiable Functions,", Monographs and Studies in Mathematics, 23 (1985).  doi: 10.1070/RM1983v038n03ABEH003484.  Google Scholar [37] D. Tataru, A-priori estimates of Carleman's type in domains with boundary,, J. Math. Pures. et Appl., 73 (1994), 355.   Google Scholar [38] D. Tataru, Boundary controllability for conservative PDE's,, Appl. Math. & Optimiz., 31 (1995), 257.  doi: 10.1007/BF01215993.  Google Scholar [39] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems,, J. Math. Pures Appl., 75 (1996), 367.   Google Scholar [40] R. Triggiani, Exact boundary controllability of $L_2(\Omega) \times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary and related problems,, Appl. Math. & Optimiz., 18 (1988), 241.  doi: 10.1007/BF01443625.  Google Scholar [41] R. Triggiani and P. F. Yao, Carleman estimates with no lower order terms for general Riemannian wave equations: Global uniqueness and observability in one shot,, Appl. Math. & Optimiz., 46 (2002), 331.  doi: 10.1007/s00245-002-0751-5.  Google Scholar [42] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl., 78 (1999), 65.  doi: 10.1016/S0021-7824(99)80010-5.  Google Scholar

show all references

##### References:
 [1] A. Bukhgeim, J. Cheng, V. Isakov and M. Yamamoto, Uniqueness in determining damping coefficients in hyperbolic equations,, Analytic Extension Formulas and their Applications, (2001), 27.   Google Scholar [2] A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse problem,, Sov., 24 (1981), 244.   Google Scholar [3] T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables independantes,, Ark. Mat. Astr. Fys., 2B (1939), 1.   Google Scholar [4] L.F.Ho, Observabilite frontiere de l'equation des ondes,, Comptes Rendus de l'Academie des Sciences de Paris, 302 (1986), 443.   Google Scholar [5] O. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations,, Inverse Problems, 17 (2001), 717.  doi: 10.1088/0266-5611/17/4/310.  Google Scholar [6] O. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations,, Comm. Partial Differential Equations, 26 (2001), 1409.  doi: 10.1081/PDE-100106139.  Google Scholar [7] V. Isakov, "Inverse Problems for Partial Differential Equations,", First Edition, (1998).   Google Scholar [8] V. Isakov, "Inverse Problems for Partial Differential Equations,", Second Edition, (2006).   Google Scholar [9] V. Isakov, "Inverse Source Problems,", American Mathematical Society, (2000).   Google Scholar [10] V. Isakov and M. Yamamoto, Carleman estimate with the Neumann boundary condition and its application to the observability inequality and inverse hyperbolic problems,, Contemp. Math., 268 (2000), 191.  doi: 10.1090/conm/268/04314.  Google Scholar [11] V. Isakov and M. Yamamoto, Stability in a wave source problem by Dirichlet data on subboundary,, J. of Inverse & Ill-Posed Problems, 11 (2003), 399.  doi: 10.1515/156939403770862802.  Google Scholar [12] M. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575.  doi: 10.1088/0266-5611/8/4/009.  Google Scholar [13] M. Klibanov and A. Timonov, "Carleman Estimates For Coefficient Inverse Problems and Numerical Applications,", VSP, (2004).  doi: 10.1515/9783110915549.  Google Scholar [14] I. Lasiecka, J. L. Lions and R. Triggiani, Non homogeneous boundary value problems for second-order hyperbolic operators,, J. Math. Pures Appl., 65 (1986), 149.   Google Scholar [15] I. Lasiecka and R. Triggiani, A cosine operator approach to modeling $L_2(0,T;L_2(\Omega))$ boundary input hyperbolic equations,, Appl. Math. & Optimiz., 7 (1981), 35.  doi: 10.1007/BF01442108.  Google Scholar [16] I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_2(0,T;L_2(\Gamma))$-Dirichlet boundary terms,, Appl. Math. & Optimiz., 10 (1983), 275.  doi: 10.1007/BF01448390.  Google Scholar [17] I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control,, Appl. Math. & Optimiz., 19 (1989), 243.  doi: 10.1007/BF01448201.  Google Scholar [18] I. Lasiecka and R. Triggiani, Recent advances in regularity of second-order hyperbolic mixed problems and applications,, Dynamics Reported, 3 (1994), 104.  doi: 10.1007/978-3-642-78234-3_3.  Google Scholar [19] I. Lasiecka and R. Triggiani, Carleman estimates and exact controllability for a system of coupled, nonconservative second-order hyperbolic equations,, Marcel Dekker Lectures Notes Pure Appl. Math., 188 (1997), 215.   Google Scholar [20] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories,", 2, 2 (2000).   Google Scholar [21] 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 [22] I. Lasiecka, R. Triggiani and P. F. Yao, An observability estimate in $L_2(\Omega)\times H^{-1}(\Omega)$ for second order hyperbolic equations with variable coefficients,, Control of Distributed Parameter and Stochastic Systems, (1999), 71.   Google Scholar [23] I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot,, Contemp. Math., 268 (2000), 227.  doi: 10.1090/conm/268/04315.  Google Scholar [24] M. M. Lavrentev, V. G. Romanov and S. P. Shishataskii, "Ill-Posed Problems of Mathematical Physics and Analysis,", Amer. Math. Soc., 64 (1986).   Google Scholar [25] J. L. Lions, "Controlabilite Exacte,", Perturbations et Stabilisation de Systemes Distribues, 1 (1988).   Google Scholar [26] J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications,", I, I (1972).   Google Scholar [27] W. Littman, "Near Optimal Time Boundary Controllability for a Class of Hyperbolic Equations,", Lecture Notes in Control and Inform. Sci. 97, 97 (1987), 307.  doi: 10.1007/BFb0038763.  Google Scholar [28] S. Liu, Inverse problem for a structural acoustic interaction,, Nonlinear Anal., 74 (2011), 2647.  doi: 10.1016/j.na.2010.12.020.  Google Scholar [29] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping and potential coefficients of an inverse hyperbolic problem,, Nonlinear Anal. Real World Appl., 12 (2011), 1562.  doi: 10.1016/j.nonrwa.2010.10.014.  Google Scholar [30] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Neumann B.C. through an additional Dirichlet boundary trace,, SIAM J. Math. Anal., 43 (2011), 1631.  doi: 10.1137/100808988.  Google Scholar [31] S. Liu and R. Triggiani, Global uniqueness in determining electric potentials for a system of strongly coupled Schrödinger equations with magnetic potential terms,, J. Inverse Ill-Posed Probl., 19 (2011), 223.  doi: 10.1515/JIIP.2011.030.  Google Scholar [32] S. Liu and R. Triggiani, Recovering the damping coefficients for a system of coupled wave equations with Neumann BC: Uniqueness and stability,, Chin. Ann. Math. Ser B, 32 (2011), 669.  doi: 10.1007/s11401-011-0672-1.  Google Scholar [33] S. Liu and R. Triggiani, Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness,, Dynamical Systems and Differential Equations, (2011), 1001.   Google Scholar [34] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Dirichlet B.C. through an additional localized Neumann boundary trace,, Applicable Analysis, 91 (2012), 1551.  doi: 10.1080/00036811.2011.618125.  Google Scholar [35] S. Liu and R. Triggiani, "Boundary Control and Boundary Inverse Theory for Non-Homogeneous Second-Order Hyperbolic Equations: A Common Carleman Estimates Approach,", Special Volume in Book Series of American Institute of Mathematical Sciences, ().   Google Scholar [36] V. G. Mazya and T. O. Shaposhnikova, "Theory of Multipliers in Spaces of Differentiable Functions,", Monographs and Studies in Mathematics, 23 (1985).  doi: 10.1070/RM1983v038n03ABEH003484.  Google Scholar [37] D. Tataru, A-priori estimates of Carleman's type in domains with boundary,, J. Math. Pures. et Appl., 73 (1994), 355.   Google Scholar [38] D. Tataru, Boundary controllability for conservative PDE's,, Appl. Math. & Optimiz., 31 (1995), 257.  doi: 10.1007/BF01215993.  Google Scholar [39] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems,, J. Math. Pures Appl., 75 (1996), 367.   Google Scholar [40] R. Triggiani, Exact boundary controllability of $L_2(\Omega) \times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary and related problems,, Appl. Math. & Optimiz., 18 (1988), 241.  doi: 10.1007/BF01443625.  Google Scholar [41] R. Triggiani and P. F. Yao, Carleman estimates with no lower order terms for general Riemannian wave equations: Global uniqueness and observability in one shot,, Appl. Math. & Optimiz., 46 (2002), 331.  doi: 10.1007/s00245-002-0751-5.  Google Scholar [42] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl., 78 (1999), 65.  doi: 10.1016/S0021-7824(99)80010-5.  Google Scholar
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