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doi: 10.3934/mcrf.2021020

A concise introduction to control theory for stochastic partial differential equations

School of Mathematics, Sichuan University, Chengdu 610064, Sichuan Province, China

* Corresponding author: Qi Lü

Received  June 2020 Published  March 2021

Fund Project: The first author is supported by NSF of China under grants 12025105, 11971334 and 11931011, by the Chang Jiang Scholars Program from the Chinese Education Ministry, and by the Science Development Project of Sichuan University under grants 2020SCUNL201. The second author is supported by the NSF of China under grants 11931011 and 11821001, and by the Science Development Project of Sichuan University under grant 2020SCUNL201

The aim of this notes is to give a concise introduction to control theory for systems governed by stochastic partial differential equations. We shall mainly focus on controllability and optimal control problems for these systems. For the first one, we present results for the exact controllability of stochastic transport equations, null and approximate controllability of stochastic parabolic equations and lack of exact controllability of stochastic hyperbolic equations. For the second one, we first introduce the stochastic linear quadratic optimal control problems and then the Pontryagin type maximum principle for general optimal control problems. It deserves mentioning that, in order to solve some difficult problems in this field, one has to develop new tools, say, the stochastic transposition method introduced in our previous works.

Citation: Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021020
References:
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show all references

References:
[1]

N. Agram and B. Øksendal, Stochastic control of memory mean-field processes, Appl. Math. Optim., 79 (2019), 181-204.  doi: 10.1007/s00245-017-9425-1.  Google Scholar

[2]

A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7.  Google Scholar

[3]

J.-M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14 (1976), 419-444.  doi: 10.1137/0314028.  Google Scholar

[4]

A. M. Bruckner, J. B. Bruckner and B. S. Thomson, Real Analysis, Prentice Hall (Pearson), Upper Saddle River, 1997. Google Scholar

[5]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications. I. Mean Field FBSDEs, Control, and Games, Probability Theory and Stochastic Modelling, 83. Springer, Cham, 2018.  Google Scholar

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K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362.  doi: 10.1137/120882433.  Google Scholar

[12]

R. DumitrescuB. Øksendal and A. Sulem, Stochastic control for mean-field stochastic partial differential equations with jumps, J. Optim. Theory Appl., 176 (2018), 559-584.  doi: 10.1007/s10957-018-1243-3.  Google Scholar

[13]

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[14]

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[15]

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[16]

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[17]

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[18]

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[19]

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[20]

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

[21]

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

[22]

P. R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, 1950.  Google Scholar

[23]

C. HafizogluI. LasieckaT. LevajkovićH. Mena and A. Tuffaha, The stochastic linear quadratic control problem with singular estimates, SIAM J. Control Optim., 55 (2017), 595-626.  doi: 10.1137/16M1056183.  Google Scholar

[24]

H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach, Second edition, Universitext. Springer, New York, 2010. doi: 10.1007/978-0-387-89488-1.  Google Scholar

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[26]

R. E. Kalman, On the general theory of control systems, Butterworth, London, 1 (1961), 481-492.   Google Scholar

[27]

M. V. Klibanov and M. Yamamoto, Exact controllability for the time dependent transport equation, SIAM J. Control Optim., 46 (2007), 2071-2195.  doi: 10.1137/060652804.  Google Scholar

[28]

T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems, American Institute of Mathematical Sciences (AIMS), Springfield, MO; Higher Education Press, Beijing, 2010.  Google Scholar

[29]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome $1$. Contrôlabilité Exacte, Recherches en Mathématiques Appliquées 8, Masson, Paris, 1988.  Google Scholar

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J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. II, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

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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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

Q. Lü, Observability estimate and state observation problems for stochastic hyperbolic equations, Inverse Problems, 29 (2013), 095011, 22 pp. doi: 10.1088/0266-5611/29/9/095011.  Google Scholar

[38]

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

[39]

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