# American Institute of Mathematical Sciences

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

 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.
Citation: Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control & 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. 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.

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##### 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.
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