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September 2018, 7(3): 465-499. doi: 10.3934/eect.2018023

Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications

School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

* Corresponding author: Peng Gao

Received  September 2017 Revised  February 2018 Published  July 2018

Fund Project: The author is supported by NSFC Grant (11601073)

In this paper, we establish the Carleman estimates for forward and backward stochastic fourth order Schrödinger equations, on basis of which, we can obtain the observability, unique continuation property and the exact controllability for the forward and backward stochastic fourth order Schrödinger equations.

Citation: Peng Gao. Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications. Evolution Equations & Control Theory, 2018, 7 (3) : 465-499. doi: 10.3934/eect.2018023
References:
[1]

V. BarbuA. Răscanu and G. Tessitore, Carleman estimate and controllability of linear stochastic heat equations, Appl. Math. Optim., 47 (2003), 97-120. doi: 10.1007/s00245-002-0757-z.

[2]

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.

[3]

J. Bergh and J. Löfström, Interpolation Spaces, An introduction, 1976.

[4]

J. L. BonaS. M. Sun and B. Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Communications in Partial Differential Equations, 28 (2003), 1391-1436. doi: 10.1081/PDE-120024373.

[5]

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), 1-9.

[6]

F. Flandoli, Regularity Theory and Stochastic Flows for Parabolic SPDEs, CRC Press, 1995.

[7]

P. Gao, Carleman estimate and unique continuation property for the linear stochastic Korteweg-de Vries equation, Bulletin of the Australian Mathematical Society, 90 (2014), 283-294. doi: 10.1017/S0004972714000276.

[8]

P. GaoM. Chen and Y. Li, Observability estimates and null controllability for forward and backward linear stochastic Kuramoto-Sivashinsky equations, SIAM Journal on Control and Optimization, 53 (2015), 475-500. doi: 10.1137/130943820.

[9]

P. Gao, Global Carleman estimates for linear stochastic Kawahara equation and their applications, Mathematics of Control, Signals, and Systems, 28 (2016), Art. 21, 22 pp. doi: 10.1007/s00498-016-0173-6.

[10]

W. Grecksch and H. Lisei, Stochastic nonlinear equations of Schrödinger type, Stochastic Analysis and Applications, 29 (2011), 631-653. doi: 10.1080/07362994.2011.581091.

[11]

C. HaoL. Hsiao and B. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations, Journal of Mathematical Analysis and Applications, 320 (2006), 246-265. doi: 10.1016/j.jmaa.2005.06.091.

[12]

C. HaoL. Hsiao and B. Wang, Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces, Journal of Mathematical Analysis and Applications, 328 (2007), 58-83. doi: 10.1016/j.jmaa.2006.05.031.

[13]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.

[14]

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.

[15]

J. U. Kim, Approximate controllability of a stochastic wave equation, Applied Mathematic Optimization, 49 (2004), 81-98. doi: 10.1007/s00245-003-0781-7.

[16]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. I, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, translated fromthe French by P. Kenneth, 1972.

[17]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. II, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, translated fromthe French by P. Kenneth, 1972.

[18]

Q. Lü, Exact controllability for stochastic transport equations, SIAM Journal on Control and Optimization, 52 (2014), 397-419. doi: 10.1137/130910373.

[19]

Q. Lü, Exact controllability for stochastic Schrödinger equations, Journal of Differential Equations, 255 (2013), 2484-2504. doi: 10.1016/j.jde.2013.06.021.

[20]

Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM Journal on Control and Optimization, 51 (2013), 121-144. doi: 10.1137/110830964.

[21]

Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems, Inverse Problems, 28 (2012), 045008, 18 pp. doi: 10.1088/0266-5611/28/4/045008.

[22]

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.

[23]

B. Pausader, The cubic fourth-order Schrödinger equation, Journal of Functional Analysis, 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009.

[24]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edn, Texts in Applied Mathematics, Vol. 13, Springer-Verlag, New York, 2004.

[25]

L. Rosier and B. Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956. doi: 10.1137/050631409.

[26]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim., 48 (2009), 2191-2216. doi: 10.1137/050641508.

[27]

R. WenS. Chai and B. Z. Guo, Well-posedness and exact controllability of fourth order Schrödinger equation with boundary control and collocated observation, SIAM Journal on Control and Optimization, 52 (2014), 365-396. doi: 10.1137/120902744.

[28]

C. Zheng and Z. Zhou, Exact controllability for the fourth order Schrödinger equation, Chinese Annals of Mathematics, Series B, 33 (2012), 395-404. doi: 10.1007/s11401-012-0711-6.

[29]

X. Zhang, Carleman and observability estimates for stochastic wave equations, SIAM J. Math. Anal., 40 (2008), 851-868. doi: 10.1137/070685786.

[30]

C. Zheng, Inverse problems for the fourth order Schrödinger equation on a finite domain, Math. Control Relat. Fields, 5 (2015), 177-189. doi: 10.3934/mcrf.2015.5.177.

show all references

References:
[1]

V. BarbuA. Răscanu and G. Tessitore, Carleman estimate and controllability of linear stochastic heat equations, Appl. Math. Optim., 47 (2003), 97-120. doi: 10.1007/s00245-002-0757-z.

[2]

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.

[3]

J. Bergh and J. Löfström, Interpolation Spaces, An introduction, 1976.

[4]

J. L. BonaS. M. Sun and B. Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Communications in Partial Differential Equations, 28 (2003), 1391-1436. doi: 10.1081/PDE-120024373.

[5]

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), 1-9.

[6]

F. Flandoli, Regularity Theory and Stochastic Flows for Parabolic SPDEs, CRC Press, 1995.

[7]

P. Gao, Carleman estimate and unique continuation property for the linear stochastic Korteweg-de Vries equation, Bulletin of the Australian Mathematical Society, 90 (2014), 283-294. doi: 10.1017/S0004972714000276.

[8]

P. GaoM. Chen and Y. Li, Observability estimates and null controllability for forward and backward linear stochastic Kuramoto-Sivashinsky equations, SIAM Journal on Control and Optimization, 53 (2015), 475-500. doi: 10.1137/130943820.

[9]

P. Gao, Global Carleman estimates for linear stochastic Kawahara equation and their applications, Mathematics of Control, Signals, and Systems, 28 (2016), Art. 21, 22 pp. doi: 10.1007/s00498-016-0173-6.

[10]

W. Grecksch and H. Lisei, Stochastic nonlinear equations of Schrödinger type, Stochastic Analysis and Applications, 29 (2011), 631-653. doi: 10.1080/07362994.2011.581091.

[11]

C. HaoL. Hsiao and B. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations, Journal of Mathematical Analysis and Applications, 320 (2006), 246-265. doi: 10.1016/j.jmaa.2005.06.091.

[12]

C. HaoL. Hsiao and B. Wang, Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces, Journal of Mathematical Analysis and Applications, 328 (2007), 58-83. doi: 10.1016/j.jmaa.2006.05.031.

[13]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.

[14]

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.

[15]

J. U. Kim, Approximate controllability of a stochastic wave equation, Applied Mathematic Optimization, 49 (2004), 81-98. doi: 10.1007/s00245-003-0781-7.

[16]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. I, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, translated fromthe French by P. Kenneth, 1972.

[17]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. II, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, translated fromthe French by P. Kenneth, 1972.

[18]

Q. Lü, Exact controllability for stochastic transport equations, SIAM Journal on Control and Optimization, 52 (2014), 397-419. doi: 10.1137/130910373.

[19]

Q. Lü, Exact controllability for stochastic Schrödinger equations, Journal of Differential Equations, 255 (2013), 2484-2504. doi: 10.1016/j.jde.2013.06.021.

[20]

Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM Journal on Control and Optimization, 51 (2013), 121-144. doi: 10.1137/110830964.

[21]

Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems, Inverse Problems, 28 (2012), 045008, 18 pp. doi: 10.1088/0266-5611/28/4/045008.

[22]

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.

[23]

B. Pausader, The cubic fourth-order Schrödinger equation, Journal of Functional Analysis, 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009.

[24]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edn, Texts in Applied Mathematics, Vol. 13, Springer-Verlag, New York, 2004.

[25]

L. Rosier and B. Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956. doi: 10.1137/050631409.

[26]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim., 48 (2009), 2191-2216. doi: 10.1137/050641508.

[27]

R. WenS. Chai and B. Z. Guo, Well-posedness and exact controllability of fourth order Schrödinger equation with boundary control and collocated observation, SIAM Journal on Control and Optimization, 52 (2014), 365-396. doi: 10.1137/120902744.

[28]

C. Zheng and Z. Zhou, Exact controllability for the fourth order Schrödinger equation, Chinese Annals of Mathematics, Series B, 33 (2012), 395-404. doi: 10.1007/s11401-012-0711-6.

[29]

X. Zhang, Carleman and observability estimates for stochastic wave equations, SIAM J. Math. Anal., 40 (2008), 851-868. doi: 10.1137/070685786.

[30]

C. Zheng, Inverse problems for the fourth order Schrödinger equation on a finite domain, Math. Control Relat. Fields, 5 (2015), 177-189. doi: 10.3934/mcrf.2015.5.177.

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