doi: 10.3934/mcrf.2021020
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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 Early access 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 and Related Fields, doi: 10.3934/mcrf.2021020
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F. Dou and Q. Lü, Time-inconsistent linear quadratic optimal control problems for stochastic evolution equations, SIAM J. Control Optim., 58 (2020), 485-509.  doi: 10.1137/19M1250339.

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show all references

References:
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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.

[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.

[3]

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

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

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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.

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F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, 264. Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.

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D. A. Dawson, Stochastic evolution equations, Math. Biosci., 15 (1972), 287-316.  doi: 10.1016/0025-5564(72)90039-9.

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F. Dou and Q. Lü, Partial approximate controllability for linear stochastic control systems, SIAM J. Control Optim., 57 (2019), 1209-1229.  doi: 10.1137/18M1164640.

[10]

F. Dou and Q. Lü, Time-inconsistent linear quadratic optimal control problems for stochastic evolution equations, SIAM J. Control Optim., 58 (2020), 485-509.  doi: 10.1137/19M1250339.

[11]

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.

[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.

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G. Fabbri, F. Gozzi and A. Świȩch, Stochastic Optimal Control in Infinite Dimension. Dynamic Programming and HJB Equations, Probability Theory and Stochastic Modelling, 82. Springer, Cham, 2017. doi: 10.1007/978-3-319-53067-3.

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H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.

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H. Frankowska and Q. Lü, First and second order necessary optimality conditions for controlled stochastic evolution equations with control and state constraints, J. Differential Equations, 268 (2020), 2949-3015.  doi: 10.1016/j.jde.2019.09.045.

[16]

X. Fu and X. Liu, Controllability and observability of some stochastic complex Ginzburg-Landau equations, SIAM J. Control Optim., 55 (2017), 1102-1127.  doi: 10.1137/15M1039961.

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X. Fu, Q. Lü and X. Zhang, Carleman Estimates for Second Order Partial Differential Operators and Applications, A Unified Approach, Springer, Cham, 2019. doi: 10.1007/978-3-030-29530-1.

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M. FuhrmanY. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, Appl. Math. Optim., 68 (2013), 181-217.  doi: 10.1007/s00245-013-9203-7.

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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.

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

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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.

[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.

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

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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.

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

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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.

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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.

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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.

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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.

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

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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.

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

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

[40]

Q. Lü, Stochastic well-posed systems and well-posedness of some stochastic partial differential equations with boundary control and observation, SIAM J. Control Optim., 53 (2015), 3457-3482.  doi: 10.1137/151002605.

[41]

Q. Lü, Well-posedness of stochastic Riccati equations and closed-loop solvability for stochastic linear quadratic optimal control problems, J. Differential Equations, 267 (2019), 180-227.  doi: 10.1016/j.jde.2019.01.008.

[42]

Q. Lü, Stochastic linear quadratic optimal control problems for mean-field stochastic evolution equations, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 127, 28 pp. doi: 10.1051/cocv/2020081.

[43]

Q. Lü, T. Wang and X. Zhang, Characterization of optimal feedback for stochastic linear quadratic control problems, Probab. Uncertain. Quant. Risk., 2 (2017), Paper no. 11, 20 pp. doi: 10.1186/s41546-017-0022-7.

[44]

Q. LüJ. Yong and X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals, J. Eur. Math. Soc., 14 (2012), 1795-1823.  doi: 10.4171/JEMS/347.

[45]

Q. LüJ. Yong and X. Zhang, Erratum to "Representation of Itô integrals by Lebesgue/ Bochner integrals", J. Eur. Math. Soc., 20 (2018), 259-260.  doi: 10.4171/JEMS/765.

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