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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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.   Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.   Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

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