American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement
  • MCRF Home
  • This Issue
  • Next Article
    On the nonuniqueness and instability of solutions of tracking-type optimal control problems
doi: 10.3934/mcrf.2021020
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

A concise introduction to control theory for stochastic partial differential equations

Qi Lü , and  Xu Zhang

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.

Keywords: stochastic partial differential equation, controllability, observability, optimal control, linear quadratic control problem, Pontryagin type maximum principle.
Mathematics Subject Classification: Primary: 93E20, 60H15, 93B05, 93B07.
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:
[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

[6]

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.  Google Scholar

[7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[8]

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

[9]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Dumitrescu, B. 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]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[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.  Google Scholar

[17]

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.  Google Scholar

[18]

M. Fuhrman, Y. 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.  Google Scholar

[19]

T. Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math. J., 89 (1983), 129-193.  doi: 10.1017/S0027763000020298.  Google Scholar

[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. Gao, M. 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. Hafizoglu, I. Lasiecka, T. Levajković, H. 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

[25] K. Itô, Introduction to Probability Theory, Cambridge University Press, Cambridge, 1984.   Google Scholar
[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

[30]

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

[31]

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

[32]

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]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

Q. Lü, J. 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.  Google Scholar

[45]

Q. Lü, J. 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.  Google Scholar

[46]

Q. Lü, H. Zhang and X. Zhang, Second order optimality conditions for optimal control problems of stochastic evolution equations, preprint, arXiv: 1811.07337. Google Scholar

[47]

Q. Lü and X. Zhang, Well-posedness of backward stochastic differential equations with general filtration, J. Differential Equations, 254 (2013), 3200-3227.  doi: 10.1016/j.jde.2013.01.010.  Google Scholar

[48]

Q. Lü and X. Zhang, General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, SpringerBriefs in Mathematics. Springer, Cham, 2014. doi: 10.1007/978-3-319-06632-5.  Google Scholar

[49]

Q. Lü and X. Zhang, Transposition method for backward stochastic evolution equations revisited, and its application, Math. Control Relat. Fields, 5 (2015), 529-555.  doi: 10.3934/mcrf.2015.5.529.  Google Scholar

[50]

Q. Lü and X. Zhang, Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns, Comm. Pure Appl. Math., 68 (2015), 948-963.  doi: 10.1002/cpa.21503.  Google Scholar

[51]

Q. Lü and X. Zhang, Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application, Math. Control Relat. Fields, 8 (2018), 337-381.  doi: 10.3934/mcrf.2018014.  Google Scholar

[52]

Q. Lü and X. Zhang, A mini-course on stochastic control, Control and Inverse Problems for Partial Differential Equations, Ser. Contemp. Appl. Math. CAM, Higher Education Press, Beijing, 22 (2019), 171-254.  Google Scholar

[53]

Q. Lü and X. Zhang, Mathematical Control Theory for Stochastic Partial Differential Equations, Springer-Verlag, in press. Google Scholar

[54]

Q. Lü and X. Zhang, Optimal feedback for stochastic linear quadratic control and backward stochastic Riccati equations in infinite dimensions, preprint, arXiv: 1901.00978. Google Scholar

[55]

Q. Lü and X. Zhang, Exact controllability for a refined stochastic wave equation, preprint, arXiv: 1901.06074. Google Scholar

[56]

Q. Lü and X. Zhang, Control theory for stochastic distributed parameter systems, an engineering perspective, in submission. Google Scholar

[57]

R. S. Manning, J. H. Maddocks and J. D. Kahn, A continuum rod model of sequence-dependent DNA structure, J. Chem. Phys., 105 (1996), 5626-5646.  doi: 10.1063/1.472373.  Google Scholar

[58]

P.-A. Meyer, Probability and Potentials, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966.  Google Scholar

[59]

R. M. Murray and et al, Control in an Information Rich World. Report of the Panel on Future Directions in Control, Dynamics, and Systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718010.  Google Scholar

[60]

E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, N.J. 1967.  Google Scholar

[61]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[62]

S.-G. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.  doi: 10.1137/0328054.  Google Scholar

[63]

S.-G. Peng, Backward stochastic differential equation and exact controllability of stochastic control systems, Progr. Natur. Sci. (English Ed.)., 4 (1994), 274-284.   Google Scholar

[64]

D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Appl. Math., 52 (1973), 189-211.  doi: 10.1002/sapm1973523189.  Google Scholar

[65]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open problems, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar

[66]

D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar

[67] K. Stowe, An Introduction to Thermodynamics and Statistical Mechanics, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511801570.  Google Scholar
[68]

J. Sun and J. Yong, Stochastic Linear-Quadratic Optimal Control Theory: Open-Loop and Closed-Loop Solutions, SpringerBriefs in Mathematics. Springer, Cham, 2019. doi: 10.1007/978-3-030-20922-3.  Google Scholar

[69]

M. Tang, Q. Meng and M. Wang, Forward and backward mean-field stochastic partial differential equation and optimal control, Chin. Ann. Math. Ser. B, 40 (2019), 515-540.  doi: 10.1007/s11401-019-0149-1.  Google Scholar

[70]

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

[71]

J. van Neerven, ${\gamma}$-radonifying operators - A survey, The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, Proc. Centre Math. Appl. Austral. Nat. Univ., Austral. Nat. Univ., Canberra, 44 (2010), 1-61.  Google Scholar

[72]

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 Problems, 36 (2020), 075014, 38 pp. doi: 10.1088/1361-6420/ab89c3.  Google Scholar

[73]

D. Yang and J. Zhong, Observability inequality of backward stochastic heat equations for measurable sets and its applications, SIAM J. Control Optim., 54 (2016), 1157-1175.  doi: 10.1137/15M1033289.  Google Scholar

[74]

J. Yong, Time-inconsistent optimal control problems, Proceedings of the International Congress of Mathematicians-Seoul 2014, Seoul, Korea, 4 (2014), 947-969.  Google Scholar

[75]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[76]

K. Yosida, Functional Analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8.  Google Scholar

[77]

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

[78]

G. Yuan, Conditional stability in determination of initial data for stochastic parabolic equations, Inverse Problems, 33 (2017), 035014, 26 pp. doi: 10.1088/1361-6420/aa5d7a.  Google Scholar

[79]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

[80]

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

[81]

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

[82]

X. Zhou, Sufficient conditions of optimality for stochastic systems with controllable diffusions, IEEE Trans. Auto. Control, 41 (1996), 1176-1179.  doi: 10.1109/9.533678.  Google Scholar

[83]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Eequations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 3 (2006), 527-621.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

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

[6]

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.  Google Scholar

[7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[8]

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

[9]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Dumitrescu, B. 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]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[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.  Google Scholar

[17]

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.  Google Scholar

[18]

M. Fuhrman, Y. 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.  Google Scholar

[19]

T. Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math. J., 89 (1983), 129-193.  doi: 10.1017/S0027763000020298.  Google Scholar

[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. Gao, M. 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. Hafizoglu, I. Lasiecka, T. Levajković, H. 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

[25] K. Itô, Introduction to Probability Theory, Cambridge University Press, Cambridge, 1984.   Google Scholar
[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

[30]

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

[31]

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

[32]

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]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

Q. Lü, J. 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.  Google Scholar

[45]

Q. Lü, J. 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.  Google Scholar

[46]

Q. Lü, H. Zhang and X. Zhang, Second order optimality conditions for optimal control problems of stochastic evolution equations, preprint, arXiv: 1811.07337. Google Scholar

[47]

Q. Lü and X. Zhang, Well-posedness of backward stochastic differential equations with general filtration, J. Differential Equations, 254 (2013), 3200-3227.  doi: 10.1016/j.jde.2013.01.010.  Google Scholar

[48]

Q. Lü and X. Zhang, General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, SpringerBriefs in Mathematics. Springer, Cham, 2014. doi: 10.1007/978-3-319-06632-5.  Google Scholar

[49]

Q. Lü and X. Zhang, Transposition method for backward stochastic evolution equations revisited, and its application, Math. Control Relat. Fields, 5 (2015), 529-555.  doi: 10.3934/mcrf.2015.5.529.  Google Scholar

[50]

Q. Lü and X. Zhang, Global uniqueness for an inverse stochastic hyperbolic problem with three unknowns, Comm. Pure Appl. Math., 68 (2015), 948-963.  doi: 10.1002/cpa.21503.  Google Scholar

[51]

Q. Lü and X. Zhang, Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application, Math. Control Relat. Fields, 8 (2018), 337-381.  doi: 10.3934/mcrf.2018014.  Google Scholar

[52]

Q. Lü and X. Zhang, A mini-course on stochastic control, Control and Inverse Problems for Partial Differential Equations, Ser. Contemp. Appl. Math. CAM, Higher Education Press, Beijing, 22 (2019), 171-254.  Google Scholar

[53]

Q. Lü and X. Zhang, Mathematical Control Theory for Stochastic Partial Differential Equations, Springer-Verlag, in press. Google Scholar

[54]

Q. Lü and X. Zhang, Optimal feedback for stochastic linear quadratic control and backward stochastic Riccati equations in infinite dimensions, preprint, arXiv: 1901.00978. Google Scholar

[55]

Q. Lü and X. Zhang, Exact controllability for a refined stochastic wave equation, preprint, arXiv: 1901.06074. Google Scholar

[56]

Q. Lü and X. Zhang, Control theory for stochastic distributed parameter systems, an engineering perspective, in submission. Google Scholar

[57]

R. S. Manning, J. H. Maddocks and J. D. Kahn, A continuum rod model of sequence-dependent DNA structure, J. Chem. Phys., 105 (1996), 5626-5646.  doi: 10.1063/1.472373.  Google Scholar

[58]

P.-A. Meyer, Probability and Potentials, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966.  Google Scholar

[59]

R. M. Murray and et al, Control in an Information Rich World. Report of the Panel on Future Directions in Control, Dynamics, and Systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718010.  Google Scholar

[60]

E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, N.J. 1967.  Google Scholar

[61]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[62]

S.-G. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.  doi: 10.1137/0328054.  Google Scholar

[63]

S.-G. Peng, Backward stochastic differential equation and exact controllability of stochastic control systems, Progr. Natur. Sci. (English Ed.)., 4 (1994), 274-284.   Google Scholar

[64]

D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Appl. Math., 52 (1973), 189-211.  doi: 10.1002/sapm1973523189.  Google Scholar

[65]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open problems, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar

[66]

D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar

[67] K. Stowe, An Introduction to Thermodynamics and Statistical Mechanics, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511801570.  Google Scholar
[68]

J. Sun and J. Yong, Stochastic Linear-Quadratic Optimal Control Theory: Open-Loop and Closed-Loop Solutions, SpringerBriefs in Mathematics. Springer, Cham, 2019. doi: 10.1007/978-3-030-20922-3.  Google Scholar

[69]

M. Tang, Q. Meng and M. Wang, Forward and backward mean-field stochastic partial differential equation and optimal control, Chin. Ann. Math. Ser. B, 40 (2019), 515-540.  doi: 10.1007/s11401-019-0149-1.  Google Scholar

[70]

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

[71]

J. van Neerven, ${\gamma}$-radonifying operators - A survey, The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, Proc. Centre Math. Appl. Austral. Nat. Univ., Austral. Nat. Univ., Canberra, 44 (2010), 1-61.  Google Scholar

[72]

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 Problems, 36 (2020), 075014, 38 pp. doi: 10.1088/1361-6420/ab89c3.  Google Scholar

[73]

D. Yang and J. Zhong, Observability inequality of backward stochastic heat equations for measurable sets and its applications, SIAM J. Control Optim., 54 (2016), 1157-1175.  doi: 10.1137/15M1033289.  Google Scholar

[74]

J. Yong, Time-inconsistent optimal control problems, Proceedings of the International Congress of Mathematicians-Seoul 2014, Seoul, Korea, 4 (2014), 947-969.  Google Scholar

[75]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[76]

K. Yosida, Functional Analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8.  Google Scholar

[77]

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

[78]

G. Yuan, Conditional stability in determination of initial data for stochastic parabolic equations, Inverse Problems, 33 (2017), 035014, 26 pp. doi: 10.1088/1361-6420/aa5d7a.  Google Scholar

[79]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

[80]

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

[81]

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

[82]

X. Zhou, Sufficient conditions of optimality for stochastic systems with controllable diffusions, IEEE Trans. Auto. Control, 41 (1996), 1176-1179.  doi: 10.1109/9.533678.  Google Scholar

[83]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Eequations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 3 (2006), 527-621.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

[1]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022
[2]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110
[3]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[4]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067
[5]

Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021031
[6]

Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040
[7]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021
[8]

Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004
[9]

Tijana Levajković, Hermann Mena, Amjad Tuffaha. The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach. Evolution Equations & Control Theory, 2016, 5 (1) : 105-134. doi: 10.3934/eect.2016.5.105
[10]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[11]

Ying Hu, Shanjian Tang. Mixed deterministic and random optimal control of linear stochastic systems with quadratic costs. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 1-. doi: 10.1186/s41546-018-0035-x
[12]

Qi Lü, Tianxiao Wang, Xu Zhang. Characterization of optimal feedback for stochastic linear quadratic control problems. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 11-. doi: 10.1186/s41546-017-0022-7
[13]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[14]

Jiongmin Yong. A deterministic linear quadratic time-inconsistent optimal control problem. Mathematical Control & Related Fields, 2011, 1 (1) : 83-118. doi: 10.3934/mcrf.2011.1.83
[15]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77
[16]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035
[17]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61
[18]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[19]

Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control & Related Fields, 2020, 10 (4) : 785-826. doi: 10.3934/mcrf.2020020
[20]

Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100

2020 Impact Factor: 1.284

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences