American Institute of Mathematical Sciences

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March  2015, 5(1): 177-189. doi: 10.3934/mcrf.2015.5.177

Inverse problems for the fourth order Schrödinger equation on a finite domain

Chuang Zheng 1,

1. 

School of Mathematics, Laboratory of Mathematics and Complex Systems, Beijing Normal University, 100875 Beijing, China

Received  April 2014 Revised  September 2014 Published  January 2015

In this paper we establish a global Carleman estimate for the fourth order Schrödinger equation with potential posed on a $1-d$ finite domain. The Carleman estimate is used to prove the Lipschitz stability for an inverse problem consisting in recovering a stationary potential in the Schrödinger equation from boundary measurements.
Keywords: fourth order Schrödinger equation, Carleman estimate., Inverse problem.
Mathematics Subject Classification: Primary: 35R30, 35Q40; Secondary: 35L6.
Citation: Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control and Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177
References:
[1]

L. Baudouin and J. P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 18 (2002), 1537-1554. 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), 161-195. 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), 5357-5397. 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), 127-146. doi: 10.1515/JIIP.2008.009.

[5]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.

[6]

S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems, J. Funct. Anal., 254 (2008), 3037-3078. 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), 985-996. 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), 465-514.

[9]

X. Fu, Sharp observability inequalities for the 1-D plate equation with a potential, Chin. Ann. Math. Ser. B., 33 (2012), 91-106. 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), 246-265. 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), 015011, 30 pp. 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), R1336-R1339. 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), 194-210. 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), 43-123. doi: 10.1163/156939404773972761.

[15]

E. Machtyngier, Exact controllability for the Schrödinger equation, SIAM J. Control Optim., 32 (1994), 24-34. doi: 10.1137/S0363012991223145.

[16]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. 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), 015017, 18 pp. 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), 197-225. doi: 10.4310/DPDE.2007.v4.n3.a1.

[19]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[20]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. 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), 65-98. 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), 555-578. doi: 10.1007/s11401-010-0585-4.

[23]

X. Zhang, Exact controllability of semilinear plate equations, Asympt. Anal., 27 (2001), 95-125.

[24]

C. Zheng and Z. Zhou, Exact controllability for the fourth order Schrödinger equation, Chin. Ann. Math. Ser. B., 33 (2012), 395-404. 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), 1991-2017.

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), 1537-1554. 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), 161-195. 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), 5357-5397. 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), 127-146. doi: 10.1515/JIIP.2008.009.

[5]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.

[6]

S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems, J. Funct. Anal., 254 (2008), 3037-3078. 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), 985-996. 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), 465-514.

[9]

X. Fu, Sharp observability inequalities for the 1-D plate equation with a potential, Chin. Ann. Math. Ser. B., 33 (2012), 91-106. 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), 246-265. 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), 015011, 30 pp. 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), R1336-R1339. 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), 194-210. 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), 43-123. doi: 10.1163/156939404773972761.

[15]

E. Machtyngier, Exact controllability for the Schrödinger equation, SIAM J. Control Optim., 32 (1994), 24-34. doi: 10.1137/S0363012991223145.

[16]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. 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), 015017, 18 pp. 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), 197-225. doi: 10.4310/DPDE.2007.v4.n3.a1.

[19]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[20]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. 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), 65-98. 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), 555-578. doi: 10.1007/s11401-010-0585-4.

[23]

X. Zhang, Exact controllability of semilinear plate equations, Asympt. Anal., 27 (2001), 95-125.

[24]

C. Zheng and Z. Zhou, Exact controllability for the fourth order Schrödinger equation, Chin. Ann. Math. Ser. B., 33 (2012), 395-404. 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), 1991-2017.

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