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Null controllability for a class of stochastic singular parabolic equations with the convection term

  • * Corresponding author: Bin Wu

    * Corresponding author: Bin Wu

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

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  • 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.

    Mathematics Subject Classification: 93B05, 93B07, 35R60, 35K67.


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  • [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [9] G. Da Prato and  J. ZabczykStochastic Equations in Infinite Dimensions, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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).
    [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.
    [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.
    [21] J. -L. Lions, Optimal Control of System Governed by Partial Differential Equation, vol. 170, Springer-Verlag, New York, 1971.
    [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.
    [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.
    [24] Q. Lü, Exact controllability for stochastic transport equations, SIAM J. Control Optim., 52 (2014), 397-419.  doi: 10.1137/130910373.
    [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.
    [26] Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM J. Control Optim., 51 (2013), 121-144.  doi: 10.1137/110830964.
    [27] H. Ockendon, Channel flow with temperature-dependent viscosity and internal viscous dissipation, J. Fluid Mech., 93 (1979), 737-746.  doi: 10.1017/S0022112079002007.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [37] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Probl., 25 (2009), 123013, 75pp. doi: 10.1088/0266-5611/25/12/123013.
    [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.
    [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.
    [40] X. Zhang, Carleman and observability estimates for stochastic wave equations, SIAM J. Math. Anal., 40 (2008), 851-868.  doi: 10.1137/070685786.
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