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doi: 10.3934/ipi.2022016
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A Carleman estimate and an energy method for a first-order symmetric hyperbolic system

1. 

Mediterranean University of Reggio Calabria, Department PAU, Via dell'Università 25 89124 Reggio Calabria, Italy

2. 

INdAM Unit, University of Catania, Italy

3. 

Institute of Mathematics for Industry, Kyushu University, Motooka, Nishi-ku, Fukuoka 819-0395, Japan

4. 

Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8914, Japan

5. 

Honorary Member of Academy of Romanian Scientists, Ilfov, nr. 3, Bucuresti, Romania

6. 

Correspondence member of Accademia Peloritana dei Pericolanti, Piazza S. Pugliatti 1, 98122 Messina, Italy

*Corresponding author: Masahiro Yamamoto

The first author was supported by Istituto Nazionale di Alta Matematica (INdAM), through the GNAMPA Research Project 2020, titled "Problemi inversi e di controllo per equazioni di evoluzione e loro applicazioni", coordinated by G. Floridia.

Received  October 2021 Revised  February 2022 Early access April 2022

Fund Project: The second author was supported by Grant-in-Aid for JSPS Fellows JP20J11497 of Japan Society for the Promotion of Science. The third author was supported by Grant-in-Aid for Scientific Research (A) 20H00117 of Japan Society for the Promotion of Science

For a symmetric hyperbolic system of the first order, we prove a Carleman estimate under some positivity condition concerning the coefficient matrices. Next, applying the Carleman estimate, we prove an observability $ L^2 $-estimate for initial values by boundary data.

Citation: Giuseppe Floridia, Hiroshi Takase, Masahiro Yamamoto. A Carleman estimate and an energy method for a first-order symmetric hyperbolic system. Inverse Problems and Imaging, doi: 10.3934/ipi.2022016
References:
[1]

L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer-Verlag, New York, 2012. doi: 10.1007/978-1-4419-7805-9.

[2]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.

[3]

A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of a class of multidimensional inverse problems, Sov. Math.-Dokl., 24 (1981), 244-247. 

[4]

P. Cannarsa, G. Floridia, F. Gögeleyen and M. Yamamoto, Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Problems, 35 (2019), 105013, 22 pp. doi: 10.1088/1361-6420/ab1c69.

[5]

P. CannarsaG. Floridia and M. Yamamoto, Observability inequalities for transport equations through Carleman estimates, Trends in Control Theory and Partial Differential Equations, Springer INdAM Ser., Springer, 32 (2019), 69-87.  doi: 10.1007/978-3-030-17949-6_4.

[6]

T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables independentes, Ark. Mat. Astr. Fys., 26 (1939), 9 pp.

[7]

O. Y. Émanuvilov, Controllability of parabolic equations, Sbornik Math., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.

[8]

G. Floridia and H. Takase, Observability inequalities for degenerate transport equations, Journal of Evolution Equations, 21 (2021), 5037-5053.  doi: 10.1007/s00028-021-00740-z.

[9]

G. Floridia and H. Takase, Inverse problems for first-order hyperbolic equations with time-dependent coefficients, Journal of Differential Equations, 305 (2021), 45-71.  doi: 10.1016/j.jde.2021.10.007.

[10]

P. Gaitan and H. Ouzzane, Inverse problem for a free transport equation using Carleman estimates, Appl. Anal., 93 (2014), 1073-1086.  doi: 10.1080/00036811.2013.816686.

[11]

F. Gölgeleyen and M. Yamamoto, Stability for some inverse problems for transport equations, SIAM J. Math. Anal., 48 (2016), 2319-2344.  doi: 10.1137/15M1038128.

[12]

L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin-New York, 1976.

[13]

X. Huang, O. Y. Imanuvilov and M. Yamamoto, Stability for inverse source problems by Carleman estimates, Inverse Problems, 36 (2020), 125006, 20 pp doi: 10.1088/1361-6420/aba892.

[14]

O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.

[15]

O. Y. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Problems, 17 (2001), 717-728.  doi: 10.1088/0266-5611/17/4/310.

[16]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lame system and the application to an inverse problem, ESAIM Control Optim. Calc. Var., 11 (2005), 1-56.  doi: 10.1051/cocv:2004030.

[17]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the Lame system with stress boundary condition, Publ. Res. Inst. Math. Sci., 43 (2007), 1023-1093.  doi: 10.2977/prims/1201012379.

[18]

V. Isakov, Inverse Problems for Partial Differential Equations, Second edition, Applied Mathematical Sciences, 127. Springer, New York, 2006.

[19]

M. A. Kazemi and M. V. Klibanov, Stability estimates for ill-posed Cauchy problem involving hyperbolic equations and inequalities, Appl. Anal., 50 (1993), 93-102.  doi: 10.1080/00036819308840186.

[20]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.

[21]

M. V. Klibanov and S. E. Pamyatnykh, Lipschitz stability of a non-standard problem for the non-stationary transport equation via a Carleman estimate, Inverse Problems, 22 (2006), 881-890.  doi: 10.1088/0266-5611/22/3/009.

[22]

M. V. Klibanov and S. E. Pamyatnykh, Global uniqueness for a coeffcient inverse problem for the non-stationary transport equation via Carleman estimate, J. Math. Anal. Appl., 343 (2008), 352-365.  doi: 10.1016/j.jmaa.2008.01.071.

[23]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coeffcient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. doi: 10.1515/9783110915549.

[24]

M. V. Klibanov and M. Yamamoto, Exact controllability for the time dependent transport equation, SIAM J. Control Optim., 46 (2007), 2071-2195.  doi: 10.1137/060652804.

[25]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[26] T. Li and T. Qin, Physics and Partial Differential Equations, Vol. Ⅰ, Higher Education Press, Beijing, 2013. 
[27]

S. Li and M. Yamamoto, An inverse source problem for Maxwell's equations in anisotropic media, Appl. Anal., 84 (2005), 1051-1067.  doi: 10.1080/00036810500047725.

[28]

M. Machida and M. Yamamoto, Global Lipschitz stability in determining coefficients of the radiative transport equation, Inverse Problems, 30 (2014), 035010, 16 pp doi: 10.1088/0266-5611/30/3/035010.

[29] S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, London, 1973. 
[30]

I.G. Petrovsky, Lectures on Partial Differential Equations, Dover Publications, Inc., New York, 1991.

[31]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75 pp. doi: 10.1088/0266-5611/25/12/123013.

show all references

References:
[1]

L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer-Verlag, New York, 2012. doi: 10.1007/978-1-4419-7805-9.

[2]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.

[3]

A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of a class of multidimensional inverse problems, Sov. Math.-Dokl., 24 (1981), 244-247. 

[4]

P. Cannarsa, G. Floridia, F. Gögeleyen and M. Yamamoto, Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Problems, 35 (2019), 105013, 22 pp. doi: 10.1088/1361-6420/ab1c69.

[5]

P. CannarsaG. Floridia and M. Yamamoto, Observability inequalities for transport equations through Carleman estimates, Trends in Control Theory and Partial Differential Equations, Springer INdAM Ser., Springer, 32 (2019), 69-87.  doi: 10.1007/978-3-030-17949-6_4.

[6]

T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables independentes, Ark. Mat. Astr. Fys., 26 (1939), 9 pp.

[7]

O. Y. Émanuvilov, Controllability of parabolic equations, Sbornik Math., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.

[8]

G. Floridia and H. Takase, Observability inequalities for degenerate transport equations, Journal of Evolution Equations, 21 (2021), 5037-5053.  doi: 10.1007/s00028-021-00740-z.

[9]

G. Floridia and H. Takase, Inverse problems for first-order hyperbolic equations with time-dependent coefficients, Journal of Differential Equations, 305 (2021), 45-71.  doi: 10.1016/j.jde.2021.10.007.

[10]

P. Gaitan and H. Ouzzane, Inverse problem for a free transport equation using Carleman estimates, Appl. Anal., 93 (2014), 1073-1086.  doi: 10.1080/00036811.2013.816686.

[11]

F. Gölgeleyen and M. Yamamoto, Stability for some inverse problems for transport equations, SIAM J. Math. Anal., 48 (2016), 2319-2344.  doi: 10.1137/15M1038128.

[12]

L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin-New York, 1976.

[13]

X. Huang, O. Y. Imanuvilov and M. Yamamoto, Stability for inverse source problems by Carleman estimates, Inverse Problems, 36 (2020), 125006, 20 pp doi: 10.1088/1361-6420/aba892.

[14]

O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.

[15]

O. Y. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Problems, 17 (2001), 717-728.  doi: 10.1088/0266-5611/17/4/310.

[16]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lame system and the application to an inverse problem, ESAIM Control Optim. Calc. Var., 11 (2005), 1-56.  doi: 10.1051/cocv:2004030.

[17]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the Lame system with stress boundary condition, Publ. Res. Inst. Math. Sci., 43 (2007), 1023-1093.  doi: 10.2977/prims/1201012379.

[18]

V. Isakov, Inverse Problems for Partial Differential Equations, Second edition, Applied Mathematical Sciences, 127. Springer, New York, 2006.

[19]

M. A. Kazemi and M. V. Klibanov, Stability estimates for ill-posed Cauchy problem involving hyperbolic equations and inequalities, Appl. Anal., 50 (1993), 93-102.  doi: 10.1080/00036819308840186.

[20]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.

[21]

M. V. Klibanov and S. E. Pamyatnykh, Lipschitz stability of a non-standard problem for the non-stationary transport equation via a Carleman estimate, Inverse Problems, 22 (2006), 881-890.  doi: 10.1088/0266-5611/22/3/009.

[22]

M. V. Klibanov and S. E. Pamyatnykh, Global uniqueness for a coeffcient inverse problem for the non-stationary transport equation via Carleman estimate, J. Math. Anal. Appl., 343 (2008), 352-365.  doi: 10.1016/j.jmaa.2008.01.071.

[23]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coeffcient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. doi: 10.1515/9783110915549.

[24]

M. V. Klibanov and M. Yamamoto, Exact controllability for the time dependent transport equation, SIAM J. Control Optim., 46 (2007), 2071-2195.  doi: 10.1137/060652804.

[25]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[26] T. Li and T. Qin, Physics and Partial Differential Equations, Vol. Ⅰ, Higher Education Press, Beijing, 2013. 
[27]

S. Li and M. Yamamoto, An inverse source problem for Maxwell's equations in anisotropic media, Appl. Anal., 84 (2005), 1051-1067.  doi: 10.1080/00036810500047725.

[28]

M. Machida and M. Yamamoto, Global Lipschitz stability in determining coefficients of the radiative transport equation, Inverse Problems, 30 (2014), 035010, 16 pp doi: 10.1088/0266-5611/30/3/035010.

[29] S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, London, 1973. 
[30]

I.G. Petrovsky, Lectures on Partial Differential Equations, Dover Publications, Inc., New York, 1991.

[31]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75 pp. doi: 10.1088/0266-5611/25/12/123013.

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