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September  2018, 8(3&4): 829-854. doi: 10.3934/mcrf.2018037

Controllability and observability of some coupled stochastic parabolic systems

1. 

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

2. 

Key Laboratory of Applied Statistics of MOE, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

* Corresponding author: Xu Liu

Received  October 2017 Revised  May 2018 Published  September 2018

Fund Project: The second author is supported by the NSF of China under grant 11871142, by the Fok Ying Tong Education Foundation under grant 141001, and by the Shanghai Key Laboratory for Contemporary Applied Mathematics at the invitation of Professor Qi Zhang.

This paper is devoted to a study of controllability and observability problems for some stochastic coupled linear parabolic systems only by one control and through an observer, respectively. In order to get a null controllability result, the Lebeau-Robbiano technique is adopted. The key point is to prove an observability inequality for certain stochastic coupled backward parabolic system by an iteration, when terminal values belong to a finite dimensional space. Different from deterministic systems, Kalman-type rank conditions for the controllability of stochastic coupled parabolic systems do not hold any more. Meanwhile, based on the Carleman estimates method, an observability inequality and unique continuation property for general stochastic linear coupled parabolic systems through an observer are derived.

Citation: Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037
References:
[1]

F. Ammar-KhodjaA. BenabdallahC. Dupaix and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differ. Equ. Appl., 1 (2009), 427-457.  doi: 10.7153/dea-01-24.  Google Scholar

[2]

F. Ammar-KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[3]

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

[4]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[5]

X. Fu, Null controllability for the parabolic equation with a complex principal part, J. Funct. Anal., 257 (2009), 1333-1354.  doi: 10.1016/j.jfa.2009.05.024.  Google Scholar

[6]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Seoul, Korea, 1996.  Google Scholar

[7]

M. González-Burgos and L. de Teresa, Controllability results for cascade systems of m coupled parabolic PDEs by one control force, Port. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.  Google Scholar

[8]

N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations, J. Sov. Math., 16 (1981), 1233-1277.   Google Scholar

[9]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[10]

H. Li and Q. Lü, Null controllability for some systems of two backward stochastic heat equations with one control force, Chin. Ann. Math., 33 (2012), 909-918.  doi: 10.1007/s11401-012-0743-y.  Google Scholar

[11]

X. Liu, Controllability of some coupled stochastic parabolic systems with fractional order spatial differential operators by one control in the drift, SIAM J. Control Optim., 52 (2014), 836-860.  doi: 10.1137/130926791.  Google Scholar

[12]

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

[13]

Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.  doi: 10.1051/cocv/2012008.  Google Scholar

[14]

Q. Lü, Some results on the controllability of forward stochastic heat equations with control on the drift, J. Funct. Anal., 260 (2011), 832-851.  doi: 10.1016/j.jfa.2010.10.018.  Google Scholar

[15]

G. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[16]

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

[17]

G. Wang, L-null controllability for the heat equation and its consequence for the time optimal control problem, SIAM J. Control Optim., 47 (2008), 1701-1720.  doi: 10.1137/060678191.  Google Scholar

[18]

X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, roceedings of the International Congress of Mathematicians, Hyderabad, India, 4 (2010), 3008-3034.  doi: 10.1007/978-0-387-89488-1.  Google Scholar

[19]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Differential Equations, Elsevier Science, 3 (2007), 527–621. doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

show all references

References:
[1]

F. Ammar-KhodjaA. BenabdallahC. Dupaix and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differ. Equ. Appl., 1 (2009), 427-457.  doi: 10.7153/dea-01-24.  Google Scholar

[2]

F. Ammar-KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[3]

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

[4]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[5]

X. Fu, Null controllability for the parabolic equation with a complex principal part, J. Funct. Anal., 257 (2009), 1333-1354.  doi: 10.1016/j.jfa.2009.05.024.  Google Scholar

[6]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Seoul, Korea, 1996.  Google Scholar

[7]

M. González-Burgos and L. de Teresa, Controllability results for cascade systems of m coupled parabolic PDEs by one control force, Port. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.  Google Scholar

[8]

N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations, J. Sov. Math., 16 (1981), 1233-1277.   Google Scholar

[9]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[10]

H. Li and Q. Lü, Null controllability for some systems of two backward stochastic heat equations with one control force, Chin. Ann. Math., 33 (2012), 909-918.  doi: 10.1007/s11401-012-0743-y.  Google Scholar

[11]

X. Liu, Controllability of some coupled stochastic parabolic systems with fractional order spatial differential operators by one control in the drift, SIAM J. Control Optim., 52 (2014), 836-860.  doi: 10.1137/130926791.  Google Scholar

[12]

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

[13]

Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.  doi: 10.1051/cocv/2012008.  Google Scholar

[14]

Q. Lü, Some results on the controllability of forward stochastic heat equations with control on the drift, J. Funct. Anal., 260 (2011), 832-851.  doi: 10.1016/j.jfa.2010.10.018.  Google Scholar

[15]

G. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[16]

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

[17]

G. Wang, L-null controllability for the heat equation and its consequence for the time optimal control problem, SIAM J. Control Optim., 47 (2008), 1701-1720.  doi: 10.1137/060678191.  Google Scholar

[18]

X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, roceedings of the International Congress of Mathematicians, Hyderabad, India, 4 (2010), 3008-3034.  doi: 10.1007/978-0-387-89488-1.  Google Scholar

[19]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Differential Equations, Elsevier Science, 3 (2007), 527–621. doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

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