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: |
[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. Cannarsa, G. 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.![]() ![]() ![]() |