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A quantitative internal unique continuation for stochastic parabolic equations
Inverse problems for the fourth order Schrödinger equation on a finite domain
1. | School of Mathematics, Laboratory of Mathematics and Complex Systems, Beijing Normal University, 100875 Beijing, China |
References:
[1] |
L. Baudouin and J. P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation,, Inverse Problems, 18 (2002), 1537.
doi: 10.1088/0266-5611/18/6/307. |
[2] |
M. Bellassoued and M. Choulli, Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map,, J. Funct. Anal., 258 (2010), 161.
doi: 10.1016/j.jfa.2009.06.010. |
[3] |
F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications,, SIAM J. Control Optim., 48 (2010), 5357.
doi: 10.1137/100784278. |
[4] |
L. Cardoulis, M. Cristofol and P. Gaitan, Inverse problem for the Schrödinger operator in an unbounded strip,, J. Inverse Ill-Posed Probl., 16 (2008), 127.
doi: 10.1515/JIIP.2008.009. |
[5] |
K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Mathematics, (2000).
|
[6] |
S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems,, J. Funct. Anal., 254 (2008), 3037.
doi: 10.1016/j.jfa.2008.03.005. |
[7] |
G. Eskin, Inverse problems for the Schrödinger operators with electromagnetic potentials in domains with obstacles,, Inverse Problems, 19 (2003), 985.
doi: 10.1088/0266-5611/19/4/313. |
[8] |
E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case,, Adv. Differential Equations, 5 (2000), 465.
|
[9] |
X. Fu, Sharp observability inequalities for the 1-D plate equation with a potential,, Chin. Ann. Math. Ser. B., 33 (2012), 91.
doi: 10.1007/s11401-011-0689-5. |
[10] |
C. Hao, L. Hsiao and B. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations,, J. Math. Anal. Appl., 320 (2006), 246.
doi: 10.1016/j.jmaa.2005.06.091. |
[11] |
L. Ignat, A. F. Pazoto and L. Rosier, Inverse problem for the heat equation and the Schrödinger equation on a tree,, Inverse Problems, 28 (2012).
doi: 10.1088/0266-5611/28/1/015011. |
[12] |
V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations,, Phys. Rev. E, 53 (1996).
doi: 10.1103/PhysRevE.53.R1336. |
[13] |
V. I. Karpman and A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion,, Phys. Rev. D, 144 (2000), 194.
doi: 10.1016/S0167-2789(00)00078-6. |
[14] |
I. Lasiecka, R. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. I. $H^1(\Omega)$-estimates,, J. Inverse Ill-Posed Probl., 12 (2004), 43.
doi: 10.1163/156939404773972761. |
[15] |
E. Machtyngier, Exact controllability for the Schrödinger equation,, SIAM J. Control Optim., 32 (1994), 24.
doi: 10.1137/S0363012991223145. |
[16] |
N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435.
doi: 10.1088/0266-5611/17/5/313. |
[17] |
A. Mercado, A. Osses and L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights,, Inverse Problems, 24 (2008).
doi: 10.1088/0266-5611/24/1/015017. |
[18] |
B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case,, Dyn. Partial Differ. Equ., 4 (2007), 197.
doi: 10.4310/DPDE.2007.v4.n3.a1. |
[19] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[20] |
B. Pausader, The cubic fourth-order Schrödinger equation,, J. Funct. Anal., 256 (2009), 2473.
doi: 10.1016/j.jfa.2008.11.009. |
[21] |
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. |
[22] |
G. Yuan and M. Yamamoto, Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality,, Chin. Ann. Math. Ser. B., 31 (2010), 555.
doi: 10.1007/s11401-010-0585-4. |
[23] |
X. Zhang, Exact controllability of semilinear plate equations,, Asympt. Anal., 27 (2001), 95.
|
[24] |
C. Zheng and Z. Zhou, Exact controllability for the fourth order Schrödinger equation,, Chin. Ann. Math. Ser. B., 33 (2012), 395.
doi: 10.1007/s11401-012-0711-6. |
[25] |
Z. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation,, Taiwanese J. Math., 16 (2012), 1991. Google Scholar |
show all references
References:
[1] |
L. Baudouin and J. P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation,, Inverse Problems, 18 (2002), 1537.
doi: 10.1088/0266-5611/18/6/307. |
[2] |
M. Bellassoued and M. Choulli, Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map,, J. Funct. Anal., 258 (2010), 161.
doi: 10.1016/j.jfa.2009.06.010. |
[3] |
F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications,, SIAM J. Control Optim., 48 (2010), 5357.
doi: 10.1137/100784278. |
[4] |
L. Cardoulis, M. Cristofol and P. Gaitan, Inverse problem for the Schrödinger operator in an unbounded strip,, J. Inverse Ill-Posed Probl., 16 (2008), 127.
doi: 10.1515/JIIP.2008.009. |
[5] |
K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Mathematics, (2000).
|
[6] |
S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems,, J. Funct. Anal., 254 (2008), 3037.
doi: 10.1016/j.jfa.2008.03.005. |
[7] |
G. Eskin, Inverse problems for the Schrödinger operators with electromagnetic potentials in domains with obstacles,, Inverse Problems, 19 (2003), 985.
doi: 10.1088/0266-5611/19/4/313. |
[8] |
E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case,, Adv. Differential Equations, 5 (2000), 465.
|
[9] |
X. Fu, Sharp observability inequalities for the 1-D plate equation with a potential,, Chin. Ann. Math. Ser. B., 33 (2012), 91.
doi: 10.1007/s11401-011-0689-5. |
[10] |
C. Hao, L. Hsiao and B. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations,, J. Math. Anal. Appl., 320 (2006), 246.
doi: 10.1016/j.jmaa.2005.06.091. |
[11] |
L. Ignat, A. F. Pazoto and L. Rosier, Inverse problem for the heat equation and the Schrödinger equation on a tree,, Inverse Problems, 28 (2012).
doi: 10.1088/0266-5611/28/1/015011. |
[12] |
V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations,, Phys. Rev. E, 53 (1996).
doi: 10.1103/PhysRevE.53.R1336. |
[13] |
V. I. Karpman and A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion,, Phys. Rev. D, 144 (2000), 194.
doi: 10.1016/S0167-2789(00)00078-6. |
[14] |
I. Lasiecka, R. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. I. $H^1(\Omega)$-estimates,, J. Inverse Ill-Posed Probl., 12 (2004), 43.
doi: 10.1163/156939404773972761. |
[15] |
E. Machtyngier, Exact controllability for the Schrödinger equation,, SIAM J. Control Optim., 32 (1994), 24.
doi: 10.1137/S0363012991223145. |
[16] |
N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435.
doi: 10.1088/0266-5611/17/5/313. |
[17] |
A. Mercado, A. Osses and L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights,, Inverse Problems, 24 (2008).
doi: 10.1088/0266-5611/24/1/015017. |
[18] |
B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case,, Dyn. Partial Differ. Equ., 4 (2007), 197.
doi: 10.4310/DPDE.2007.v4.n3.a1. |
[19] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[20] |
B. Pausader, The cubic fourth-order Schrödinger equation,, J. Funct. Anal., 256 (2009), 2473.
doi: 10.1016/j.jfa.2008.11.009. |
[21] |
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. |
[22] |
G. Yuan and M. Yamamoto, Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality,, Chin. Ann. Math. Ser. B., 31 (2010), 555.
doi: 10.1007/s11401-010-0585-4. |
[23] |
X. Zhang, Exact controllability of semilinear plate equations,, Asympt. Anal., 27 (2001), 95.
|
[24] |
C. Zheng and Z. Zhou, Exact controllability for the fourth order Schrödinger equation,, Chin. Ann. Math. Ser. B., 33 (2012), 395.
doi: 10.1007/s11401-012-0711-6. |
[25] |
Z. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation,, Taiwanese J. Math., 16 (2012), 1991. Google Scholar |
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