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June  2018, 8(2): 451-473. doi: 10.3934/mcrf.2018018

A second-order stochastic maximum principle for generalized mean-field singular control problem

Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, 999078, China

* Corresponding author: Hancheng Guo

Received  April 2017 Revised  October 2017 Published  March 2018

Fund Project: Research supported partially by FDCT 025/2016/A1.

In this paper, we study the generalized mean-field stochastic control problem when the usual stochastic maximum principle (SMP) is not applicable due to the singularity of the Hamiltonian function. In this case, we derive a second order SMP. We introduce the adjoint process by the generalized mean-field backward stochastic differential equation. The keys in the proofs are the expansion of the cost functional in terms of a perturbation parameter, and the use of the range theorem for vector-valued measures.

Citation: 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
References:
[1]

V. Arkin and I. Saksonov, Necessary optimality conditions of optimality in the problems of control of stochastic differential-equations, Doklady Akademii Nauk SSSR., 244 (1979), 11-15.   Google Scholar

[2]

D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems, Vol. 117. Elsevier, 1975.  Google Scholar

[3]

A. Bensoussan, Lectures on stochastic control, Nonlinear Filtering and Stochastic Control, 972 (1982), 1-62.  doi: 10.1007/BFb0064859.  Google Scholar

[4]

J. M. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Review, 20 (1978), 62-78.  doi: 10.1137/1020004.  Google Scholar

[5]

R. BuckdahnJ. Li and J. Ma, A stochastic maximum principle for general mean-field systems, Applied Mathematics and Optimization, 74 (2016), 507-534.  doi: 10.1007/s00245-016-9394-9.  Google Scholar

[6]

R. BuckdahnJ. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stoch. Proc. App., 19 (2009), 3133-3154.  doi: 10.1016/j.spa.2009.05.002.  Google Scholar

[7]

R. BuckdahnJ. LiS. Peng and C. Rainer, Mean-field stochastic differential equations and associated PDEs, Ann. Probab., 45 (2017), 824-878.  doi: 10.1214/15-AOP1076.  Google Scholar

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P. Cardaliaguet, Weak solutions for first order mean field games with local coupling, Analysis and Geometry in Control Theory and its Applications, 11 (2015), 111-158.  doi: 10.1007/978-3-319-06917-3_5.  Google Scholar

[9]

R. Gabasov and F. M. Kirillova, High order necessary conditions for optimality, SIAM J. Control, 10 (1972), 127-168.  doi: 10.1137/0310012.  Google Scholar

[10]

U. G. Haussmann, A Stochastic Maximum Principle for Optimal Control of Diffusions, Essex, UK: Longman Scientific and Technical, 1986. doi: 10. 1007/BF00047571.  Google Scholar

[11]

U. G. Haussmann, General necessary conditions for optimal control of stochastic systems, Math. Program. Study, 6 (1976), 30-48.  doi: 10.1007/BFb0120743.  Google Scholar

[12]

M. A. Kazemi-Dehkordi, Necessary conditions for optimality of singular controls, J. Optim. Theor. Appl., 43 (1984), 629-637.  doi: 10.1007/BF00935010.  Google Scholar

[13]

A. J. Krener, The high-order maximum principle and its application to singular extremals, SIAM J. Control, 15 (1977), 256-293.  doi: 10.1137/0315019.  Google Scholar

[14]

H. J. Kushner, On the stochastic maximum principle: Fixed time of control, Journal of Mathematical Analysis and Applications, 11 (1965), 78-92.  doi: 10.1016/0022-247X(65)90070-3.  Google Scholar

[15]

H. J. Kushner, Necessary conditions for continuous parameter stochastic optimization problems, SIAM Journal of Control, 10 (1972), 550-565.  doi: 10.1137/0310041.  Google Scholar

[16]

J. Li, Stochastic maximum principle in the mean-field controls, Automatica, 48 (2012), 366-373.  doi: 10.1016/j.automatica.2011.11.006.  Google Scholar

[17]

Q. Lü, Second order necessary conditions for optimal control problems of stochastic evolution equations, Control Conference (CCC), 2016 35th Chinese. IEEE, (2016), 2620-2625.  doi: 10.1109/ChiCC.2016.7553759.  Google Scholar

[18]

K. Mizukami and H. Wu, New necessary conditions for optimality of singular controls in optimal control problems, Int. J. Systems Sci., 23 (1992), 1335-1345.  doi: 10.1080/00207729208949387.  Google Scholar

[19]

L. Mou and J. Yong, A variational formula for stochastic controls and some applications, Pure Appl. Math. Q, 3 (2007), 539-567.  doi: 10.4310/PAMQ.2007.v3.n2.a7.  Google Scholar

[20]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, System Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[21]

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

[22]

L. S. Pontrvagin, Mathematical Theory of Optimal Processes, CRC Press, 1987. Google Scholar

[23]

S. J. Tang, A second-order maximum principle for singular optimal stochastic controls, Discrete and Continuous Dynamical System Series B, 14 (2010), 1581-1599.  doi: 10.3934/dcdsb.2010.14.1581.  Google Scholar

[24]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.  Google Scholar

[25]

H. Zhang and X. Zhang, Pointwise second-order necessary conditions for stochastic optimal controls, Part Ⅰ: The case of convex control constraint, SIAM Journal on Control and Optimization, 53 (2015), 2267-2296.  doi: 10.1137/14098627X.  Google Scholar

show all references

References:
[1]

V. Arkin and I. Saksonov, Necessary optimality conditions of optimality in the problems of control of stochastic differential-equations, Doklady Akademii Nauk SSSR., 244 (1979), 11-15.   Google Scholar

[2]

D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems, Vol. 117. Elsevier, 1975.  Google Scholar

[3]

A. Bensoussan, Lectures on stochastic control, Nonlinear Filtering and Stochastic Control, 972 (1982), 1-62.  doi: 10.1007/BFb0064859.  Google Scholar

[4]

J. M. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Review, 20 (1978), 62-78.  doi: 10.1137/1020004.  Google Scholar

[5]

R. BuckdahnJ. Li and J. Ma, A stochastic maximum principle for general mean-field systems, Applied Mathematics and Optimization, 74 (2016), 507-534.  doi: 10.1007/s00245-016-9394-9.  Google Scholar

[6]

R. BuckdahnJ. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stoch. Proc. App., 19 (2009), 3133-3154.  doi: 10.1016/j.spa.2009.05.002.  Google Scholar

[7]

R. BuckdahnJ. LiS. Peng and C. Rainer, Mean-field stochastic differential equations and associated PDEs, Ann. Probab., 45 (2017), 824-878.  doi: 10.1214/15-AOP1076.  Google Scholar

[8]

P. Cardaliaguet, Weak solutions for first order mean field games with local coupling, Analysis and Geometry in Control Theory and its Applications, 11 (2015), 111-158.  doi: 10.1007/978-3-319-06917-3_5.  Google Scholar

[9]

R. Gabasov and F. M. Kirillova, High order necessary conditions for optimality, SIAM J. Control, 10 (1972), 127-168.  doi: 10.1137/0310012.  Google Scholar

[10]

U. G. Haussmann, A Stochastic Maximum Principle for Optimal Control of Diffusions, Essex, UK: Longman Scientific and Technical, 1986. doi: 10. 1007/BF00047571.  Google Scholar

[11]

U. G. Haussmann, General necessary conditions for optimal control of stochastic systems, Math. Program. Study, 6 (1976), 30-48.  doi: 10.1007/BFb0120743.  Google Scholar

[12]

M. A. Kazemi-Dehkordi, Necessary conditions for optimality of singular controls, J. Optim. Theor. Appl., 43 (1984), 629-637.  doi: 10.1007/BF00935010.  Google Scholar

[13]

A. J. Krener, The high-order maximum principle and its application to singular extremals, SIAM J. Control, 15 (1977), 256-293.  doi: 10.1137/0315019.  Google Scholar

[14]

H. J. Kushner, On the stochastic maximum principle: Fixed time of control, Journal of Mathematical Analysis and Applications, 11 (1965), 78-92.  doi: 10.1016/0022-247X(65)90070-3.  Google Scholar

[15]

H. J. Kushner, Necessary conditions for continuous parameter stochastic optimization problems, SIAM Journal of Control, 10 (1972), 550-565.  doi: 10.1137/0310041.  Google Scholar

[16]

J. Li, Stochastic maximum principle in the mean-field controls, Automatica, 48 (2012), 366-373.  doi: 10.1016/j.automatica.2011.11.006.  Google Scholar

[17]

Q. Lü, Second order necessary conditions for optimal control problems of stochastic evolution equations, Control Conference (CCC), 2016 35th Chinese. IEEE, (2016), 2620-2625.  doi: 10.1109/ChiCC.2016.7553759.  Google Scholar

[18]

K. Mizukami and H. Wu, New necessary conditions for optimality of singular controls in optimal control problems, Int. J. Systems Sci., 23 (1992), 1335-1345.  doi: 10.1080/00207729208949387.  Google Scholar

[19]

L. Mou and J. Yong, A variational formula for stochastic controls and some applications, Pure Appl. Math. Q, 3 (2007), 539-567.  doi: 10.4310/PAMQ.2007.v3.n2.a7.  Google Scholar

[20]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, System Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[21]

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

[22]

L. S. Pontrvagin, Mathematical Theory of Optimal Processes, CRC Press, 1987. Google Scholar

[23]

S. J. Tang, A second-order maximum principle for singular optimal stochastic controls, Discrete and Continuous Dynamical System Series B, 14 (2010), 1581-1599.  doi: 10.3934/dcdsb.2010.14.1581.  Google Scholar

[24]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.  Google Scholar

[25]

H. Zhang and X. Zhang, Pointwise second-order necessary conditions for stochastic optimal controls, Part Ⅰ: The case of convex control constraint, SIAM Journal on Control and Optimization, 53 (2015), 2267-2296.  doi: 10.1137/14098627X.  Google Scholar

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