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doi: 10.3934/dcdsb.2021182
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Null controllability for a class of stochastic singular parabolic equations with the convection term

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author: Bin Wu

Received  September 2020 Revised  May 2021 Early access July 2021

Fund Project: This work is supported by NSFC (No.11661004, No.11601240)

This paper concerns the null controllability for a class of stochastic singular parabolic equations with the convection term in one dimensional space. Due to the singularity, we first transfer to study an approximate nonsingular system. Next we establish a new Carleman estimate for the backward stochastic singular parabolic equation with convection term and then an observability inequality for the adjoint system of the approximate system. Based on this observability inequality and an approximate argument, we obtain the null controllability result.

Citation: Lin Yan, Bin Wu. Null controllability for a class of stochastic singular parabolic equations with the convection term. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021182
References:
[1]

F. Alabau-BoussouiraP. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.  Google Scholar

[2]

V. BarbuA. Răşcanu 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

[3]

U. Biccari and E. Zuazua, Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function, J. Differential Equations, 261 (2016), 2809-2853.  doi: 10.1016/j.jde.2016.05.019.  Google Scholar

[4]

P. CannarsaP. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.  Google Scholar

[5]

P. CannarsaP. Martinez and J. Vancostenoble, Carleman estimates and null controllability for boundary-degenerate parabolic operators, C. R. Math. Acad. Sci. Paris., 347 (2009), 147-152.  doi: 10.1016/j.crma.2008.12.011.  Google Scholar

[6]

P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Probl., 26 (2010), 105003, 20pp. doi: 10.1088/0266-5611/26/10/105003.  Google Scholar

[7]

C. Cazacu, Controllability of the heat equation with an inverse-square potential localized on the boundary, SIAM J. Control Optim., 52 (2014), 2055-2089.  doi: 10.1137/120862557.  Google Scholar

[8]

E. CerpaP. Guzmán and A. Mercado, On the control of the linear Kuramoto-Sivashinsky equation, ESAIM Control Optim. Calc. Var., 23 (2017), 165-194.  doi: 10.1051/cocv/2015044.  Google Scholar

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[10]

R. Du, Null controllability for a class of degenerate parabolic equations with gradient terms, J. Evol. Equ., 19 (2019), 585-613.  doi: 10.1007/s00028-019-00487-8.  Google Scholar

[11]

R. DuJ. EichhornQ. Liu and C. Wang, Carleman estimates and null controllability of a class of singular parabolic equations, Adv. Nonlinear Anal., 8 (2019), 1057-1082.  doi: 10.1515/anona-2016-0266.  Google Scholar

[12]

S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, 33 (2008), 1996-2019.  doi: 10.1080/03605300802402633.  Google Scholar

[13]

J. C. Flores and L. de Teresa, Null controllability of one dimensional degenerate parabolic equations with first order terms, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 3963-3981.  doi: 10.3934/dcdsb.2020136.  Google Scholar

[14]

G. Fragnelli, Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 687-701.  doi: 10.3934/dcdss.2013.6.687.  Google Scholar

[15]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[16]

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

[17]

O. Glass and S. Guerrero, Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition, Systems Control Lett., 59 (2010), 390-395.  doi: 10.1016/j.sysconle.2010.05.001.  Google Scholar

[18]

V. Hern$\acute{a}$ndez-Santamaría, K. L. Balc'h and L. Peralta, Global null-controllability for stochastic semilinear parabolic equations, preprint, arXiv: 2010.08854v1 (2020). Google Scholar

[19]

O. Y. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.  Google Scholar

[20]

S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math., 34 (1981), 831-852.  doi: 10.1002/cpa.3160340605.  Google Scholar

[21]

J. -L. Lions, Optimal Control of System Governed by Partial Differential Equation, vol. 170, Springer-Verlag, New York, 1971.  Google Scholar

[22]

X. Liu, Global Carleman estimates for stochastic parabolic equations and its application, ESAIM Control Optim. Calc. Var., 20 (2014), 823-839.  doi: 10.1051/cocv/2013085.  Google Scholar

[23]

X. Liu and Y. Yu, Carleman estimates of some stochastic degenerate parabolic equations and application, SIAM J. Control Optim., 57 (2019), 3527-3552.  doi: 10.1137/18M1221448.  Google Scholar

[24]

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

[25]

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

[26]

Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM J. Control Optim., 51 (2013), 121-144.  doi: 10.1137/110830964.  Google Scholar

[27]

H. Ockendon, Channel flow with temperature-dependent viscosity and internal viscous dissipation, J. Fluid Mech., 93 (1979), 737-746.  doi: 10.1017/S0022112079002007.  Google Scholar

[28]

P. Rosenau and S. Kamin, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127.  doi: 10.1002/cpa.3160350106.  Google Scholar

[29]

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

[30]

G. Tessitore, Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE, Stochastic Analysis and Applications, 14 (1996), 461-486.  doi: 10.1080/07362999608809451.  Google Scholar

[31]

J. Vancostenoblec and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902.  doi: 10.1016/j.jfa.2007.12.015.  Google Scholar

[32]

C. Wang and R. Du, Carleman estimates and null controllability for a class of degenerate parabolic equations with convection terms, SIAM J. Control Optim., 52 (2014), 1457-1480.  doi: 10.1137/110820592.  Google Scholar

[33]

C. WangY. ZhouR. Du and Q. Liu, Carleman estimate for solutions to a degenerate convection-diffusion equation, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4207-4222.  doi: 10.3934/dcdsb.2018133.  Google Scholar

[34]

B. Wu, Q. Chen and Z. Wang, Carleman estimates for a stochastic degenerate parabolic equation and applications to null controllability and an inverse random source problem, Inverse Probl., 36 (2020), 075014, 38pp. doi: 10.1088/1361-6420/ab89c3.  Google Scholar

[35]

B. WuY. GaoZ. Wang and Q. Chen, Unique continuation for a reaction-diffusion system with cross diffusion, J. Inverse III-Posed Probl., 27 (2019), 511-525.  doi: 10.1515/jiip-2017-0094.  Google Scholar

[36]

B. Wu and J. Yu, Hölder stability of an inverse problem for a strongly coupled reaction-diffusion system, IMA J. Appl. Math., 82 (2017), 424-444.  doi: 10.1093/imamat/hxw058.  Google Scholar

[37]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Probl., 25 (2009), 123013, 75pp. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar

[38]

Y. Yan, Carleman estimates for stochastic parabolic equations with Neumann boundary conditions and applications, J. Math. Anal. Appl., 457 (2018), 248-272.  doi: 10.1016/j.jmaa.2017.08.003.  Google Scholar

[39]

G. Yuan, Determination of two kinds of sources simultaneously for a stochastic wave equation, Inverse Probl. 31 (2015), 085003, 13pp. doi: 10.1088/0266-5611/31/8/085003.  Google Scholar

[40]

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

show all references

References:
[1]

F. Alabau-BoussouiraP. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.  Google Scholar

[2]

V. BarbuA. Răşcanu 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

[3]

U. Biccari and E. Zuazua, Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function, J. Differential Equations, 261 (2016), 2809-2853.  doi: 10.1016/j.jde.2016.05.019.  Google Scholar

[4]

P. CannarsaP. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.  Google Scholar

[5]

P. CannarsaP. Martinez and J. Vancostenoble, Carleman estimates and null controllability for boundary-degenerate parabolic operators, C. R. Math. Acad. Sci. Paris., 347 (2009), 147-152.  doi: 10.1016/j.crma.2008.12.011.  Google Scholar

[6]

P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Probl., 26 (2010), 105003, 20pp. doi: 10.1088/0266-5611/26/10/105003.  Google Scholar

[7]

C. Cazacu, Controllability of the heat equation with an inverse-square potential localized on the boundary, SIAM J. Control Optim., 52 (2014), 2055-2089.  doi: 10.1137/120862557.  Google Scholar

[8]

E. CerpaP. Guzmán and A. Mercado, On the control of the linear Kuramoto-Sivashinsky equation, ESAIM Control Optim. Calc. Var., 23 (2017), 165-194.  doi: 10.1051/cocv/2015044.  Google Scholar

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar
[10]

R. Du, Null controllability for a class of degenerate parabolic equations with gradient terms, J. Evol. Equ., 19 (2019), 585-613.  doi: 10.1007/s00028-019-00487-8.  Google Scholar

[11]

R. DuJ. EichhornQ. Liu and C. Wang, Carleman estimates and null controllability of a class of singular parabolic equations, Adv. Nonlinear Anal., 8 (2019), 1057-1082.  doi: 10.1515/anona-2016-0266.  Google Scholar

[12]

S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, 33 (2008), 1996-2019.  doi: 10.1080/03605300802402633.  Google Scholar

[13]

J. C. Flores and L. de Teresa, Null controllability of one dimensional degenerate parabolic equations with first order terms, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 3963-3981.  doi: 10.3934/dcdsb.2020136.  Google Scholar

[14]

G. Fragnelli, Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 687-701.  doi: 10.3934/dcdss.2013.6.687.  Google Scholar

[15]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[16]

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

[17]

O. Glass and S. Guerrero, Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition, Systems Control Lett., 59 (2010), 390-395.  doi: 10.1016/j.sysconle.2010.05.001.  Google Scholar

[18]

V. Hern$\acute{a}$ndez-Santamaría, K. L. Balc'h and L. Peralta, Global null-controllability for stochastic semilinear parabolic equations, preprint, arXiv: 2010.08854v1 (2020). Google Scholar

[19]

O. Y. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.  Google Scholar

[20]

S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math., 34 (1981), 831-852.  doi: 10.1002/cpa.3160340605.  Google Scholar

[21]

J. -L. Lions, Optimal Control of System Governed by Partial Differential Equation, vol. 170, Springer-Verlag, New York, 1971.  Google Scholar

[22]

X. Liu, Global Carleman estimates for stochastic parabolic equations and its application, ESAIM Control Optim. Calc. Var., 20 (2014), 823-839.  doi: 10.1051/cocv/2013085.  Google Scholar

[23]

X. Liu and Y. Yu, Carleman estimates of some stochastic degenerate parabolic equations and application, SIAM J. Control Optim., 57 (2019), 3527-3552.  doi: 10.1137/18M1221448.  Google Scholar

[24]

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

[25]

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

[26]

Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM J. Control Optim., 51 (2013), 121-144.  doi: 10.1137/110830964.  Google Scholar

[27]

H. Ockendon, Channel flow with temperature-dependent viscosity and internal viscous dissipation, J. Fluid Mech., 93 (1979), 737-746.  doi: 10.1017/S0022112079002007.  Google Scholar

[28]

P. Rosenau and S. Kamin, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127.  doi: 10.1002/cpa.3160350106.  Google Scholar

[29]

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

[30]

G. Tessitore, Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE, Stochastic Analysis and Applications, 14 (1996), 461-486.  doi: 10.1080/07362999608809451.  Google Scholar

[31]

J. Vancostenoblec and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902.  doi: 10.1016/j.jfa.2007.12.015.  Google Scholar

[32]

C. Wang and R. Du, Carleman estimates and null controllability for a class of degenerate parabolic equations with convection terms, SIAM J. Control Optim., 52 (2014), 1457-1480.  doi: 10.1137/110820592.  Google Scholar

[33]

C. WangY. ZhouR. Du and Q. Liu, Carleman estimate for solutions to a degenerate convection-diffusion equation, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4207-4222.  doi: 10.3934/dcdsb.2018133.  Google Scholar

[34]

B. Wu, Q. Chen and Z. Wang, Carleman estimates for a stochastic degenerate parabolic equation and applications to null controllability and an inverse random source problem, Inverse Probl., 36 (2020), 075014, 38pp. doi: 10.1088/1361-6420/ab89c3.  Google Scholar

[35]

B. WuY. GaoZ. Wang and Q. Chen, Unique continuation for a reaction-diffusion system with cross diffusion, J. Inverse III-Posed Probl., 27 (2019), 511-525.  doi: 10.1515/jiip-2017-0094.  Google Scholar

[36]

B. Wu and J. Yu, Hölder stability of an inverse problem for a strongly coupled reaction-diffusion system, IMA J. Appl. Math., 82 (2017), 424-444.  doi: 10.1093/imamat/hxw058.  Google Scholar

[37]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Probl., 25 (2009), 123013, 75pp. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar

[38]

Y. Yan, Carleman estimates for stochastic parabolic equations with Neumann boundary conditions and applications, J. Math. Anal. Appl., 457 (2018), 248-272.  doi: 10.1016/j.jmaa.2017.08.003.  Google Scholar

[39]

G. Yuan, Determination of two kinds of sources simultaneously for a stochastic wave equation, Inverse Probl. 31 (2015), 085003, 13pp. doi: 10.1088/0266-5611/31/8/085003.  Google Scholar

[40]

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

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