March  2015, 5(1): 97-139. doi: 10.3934/mcrf.2015.5.97

A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon

1. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China, China

2. 

Department of Mathematics, University of Central Florida, Orlando, FL 32816

Received  July 2013 Revised  November 2013 Published  January 2015

A linear-quadratic (LQ, for short) optimal control problem is considered for mean-field stochastic differential equations with constant coefficients in an infinite horizon. The stabilizability of the control system is studied followed by the discussion of the well-posedness of the LQ problem. The optimal control can be expressed as a linear state feedback involving the state and its mean, through the solutions of two algebraic Riccati equations. The solvability of such kind of Riccati equations is investigated by means of semi-definite programming method.
Citation: 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
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show all references

References:
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Stoch. Proc. Appl., 60 (1995), 65-85. doi: 10.1016/0304-4149(95)00050-X.  Google Scholar

[2]

Comm. Appl. Anal., 5 (2001), 183-206.  Google Scholar

[3]

SIAM J. Control Optim., 46 (2007), 356-378. doi: 10.1137/050645944.  Google Scholar

[4]

IEEE Transactions on Automatic Control, 45 (2000), 1131-1143. doi: 10.1109/9.863597.  Google Scholar

[5]

SIAM J. Appl. Math., 17 (1969), 434-440. doi: 10.1137/0117041.  Google Scholar

[6]

Appl. Math. Optim., 63 (2011), 341-356. doi: 10.1007/s00245-010-9123-8.  Google Scholar

[7]

Springer-Verlag, 2003.  Google Scholar

[8]

2nd edition, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[9]

Stoch. Anal. Appl., 28 (2010), 884-906. doi: 10.1080/07362994.2010.482836.  Google Scholar

[10]

SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[11]

Applied Mathematics & Optimization, 64 (2011), 197-216. doi: 10.1007/s00245-011-9136-y.  Google Scholar

[12]

Ann. Probab., 37 (2009), 1524-1565. doi: 10.1214/08-AOP442.  Google Scholar

[13]

Stoch. Process. Appl., 119 (2009), 3133-3154. doi: 10.1016/j.spa.2009.05.002.  Google Scholar

[14]

Ann. Probab., 22 (1994), 431-441. doi: 10.1214/aop/1176988866.  Google Scholar

[15]

Soochow J. Math., 20 (1994), 507-526.  Google Scholar

[16]

Stochastics, 82 (2010), 53-68. doi: 10.1080/17442500902723575.  Google Scholar

[17]

J. Statist. Phys., 31 (1983), 29-85. doi: 10.1007/BF01010922.  Google Scholar

[18]

Stochastics, 20 (1987), 247-308. doi: 10.1080/17442508708833446.  Google Scholar

[19]

Int. J. Robust and Nonlinear Contr., 6 (1996), 1015-1022. doi: 10.1002/(SICI)1099-1239(199611)6:9/10<1015::AID-RNC266>3.0.CO;2-0.  Google Scholar

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[22]

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Comm. Inform. Systems, 6 (2006), 221-251. doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

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Stochastics, 84 (2012), 643-666. doi: 10.1080/17442508.2011.651619.  Google Scholar

[31]

Numer. Funct. Anal. Optim., 29 (2008), 1328-1346. doi: 10.1080/01630560802580679.  Google Scholar

[32]

Proc. Cambridge Philos. Soc., 51 (1955), 406-413. doi: 10.1017/S0305004100030401.  Google Scholar

[33]

J. Austral. Math. Soc., Ser. A, 43 (1987), 246-256. doi: 10.1017/S1446788700029384.  Google Scholar

[34]

Mathematical Programming Ser. B, 95 (2003), 189-217. doi: 10.1007/s10107-002-0347-5.  Google Scholar

[35]

SIAM Rev., 38 (1996), 49-95. doi: 10.1137/1038003.  Google Scholar

[36]

in Monte Carlo and quasi-Monte Carlo methods 2004, Springer, Berlin, 2006, 471-486. doi: 10.1007/3-540-31186-6_29.  Google Scholar

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SIAM J. Control Optim., 51 (2013), 2809-2838. doi: 10.1137/120892477.  Google Scholar

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Applications of Mathematics (New York), 43, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

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