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.

[2]

A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse problem,, Sov., 24 (1981), 244.

[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.

[4]

L.F.Ho, Observabilite frontiere de l'equation des ondes,, Comptes Rendus de l'Academie des Sciences de Paris, 302 (1986), 443.

[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.

[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.

[7]

V. Isakov, "Inverse Problems for Partial Differential Equations,", First Edition, (1998).

[8]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second Edition, (2006).

[9]

V. Isakov, "Inverse Source Problems,", American Mathematical Society, (2000).

[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.

[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.

[12]

M. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009.

[13]

M. Klibanov and A. Timonov, "Carleman Estimates For Coefficient Inverse Problems and Numerical Applications,", VSP, (2004). doi: 10.1515/9783110915549.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories,", 2, 2 (2000).

[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.

[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.

[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.

[24]

M. M. Lavrentev, V. G. Romanov and S. P. Shishataskii, "Ill-Posed Problems of Mathematical Physics and Analysis,", Amer. Math. Soc., 64 (1986).

[25]

J. L. Lions, "Controlabilite Exacte,", Perturbations et Stabilisation de Systemes Distribues, 1 (1988).

[26]

J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications,", I, I (1972).

[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.

[28]

S. Liu, Inverse problem for a structural acoustic interaction,, Nonlinear Anal., 74 (2011), 2647. doi: 10.1016/j.na.2010.12.020.

[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.

[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.

[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.

[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.

[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.

[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.

[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, ().

[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.

[37]

D. Tataru, A-priori estimates of Carleman's type in domains with boundary,, J. Math. Pures. et Appl., 73 (1994), 355.

[38]

D. Tataru, Boundary controllability for conservative PDE's,, Appl. Math. & Optimiz., 31 (1995), 257. doi: 10.1007/BF01215993.

[39]

D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems,, J. Math. Pures Appl., 75 (1996), 367.

[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.

[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.

[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.

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.

[2]

A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse problem,, Sov., 24 (1981), 244.

[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.

[4]

L.F.Ho, Observabilite frontiere de l'equation des ondes,, Comptes Rendus de l'Academie des Sciences de Paris, 302 (1986), 443.

[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.

[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.

[7]

V. Isakov, "Inverse Problems for Partial Differential Equations,", First Edition, (1998).

[8]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Second Edition, (2006).

[9]

V. Isakov, "Inverse Source Problems,", American Mathematical Society, (2000).

[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.

[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.

[12]

M. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009.

[13]

M. Klibanov and A. Timonov, "Carleman Estimates For Coefficient Inverse Problems and Numerical Applications,", VSP, (2004). doi: 10.1515/9783110915549.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories,", 2, 2 (2000).

[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.

[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.

[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.

[24]

M. M. Lavrentev, V. G. Romanov and S. P. Shishataskii, "Ill-Posed Problems of Mathematical Physics and Analysis,", Amer. Math. Soc., 64 (1986).

[25]

J. L. Lions, "Controlabilite Exacte,", Perturbations et Stabilisation de Systemes Distribues, 1 (1988).

[26]

J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications,", I, I (1972).

[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.

[28]

S. Liu, Inverse problem for a structural acoustic interaction,, Nonlinear Anal., 74 (2011), 2647. doi: 10.1016/j.na.2010.12.020.

[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.

[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.

[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.

[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.

[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.

[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.

[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, ().

[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.

[37]

D. Tataru, A-priori estimates of Carleman's type in domains with boundary,, J. Math. Pures. et Appl., 73 (1994), 355.

[38]

D. Tataru, Boundary controllability for conservative PDE's,, Appl. Math. & Optimiz., 31 (1995), 257. doi: 10.1007/BF01215993.

[39]

D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems,, J. Math. Pures Appl., 75 (1996), 367.

[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.

[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.

[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.

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