This paper is concerned with a mean-field linear quadratic (LQ, for short) optimal control problem with deterministic coefficients. It is shown that convexity of the cost functional is necessary for the finiteness of the mean-field LQ problem, whereas uniform convexity of the cost functional is sufficient for the open-loop solvability of the problem. By considering a family of uniformly convex cost functionals, a characterization of the finiteness of the problem is derived and a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Then, it is proved that the uniform convexity of the cost functional is equivalent to the solvability of two coupled differential Riccati equations and the unique open-loop optimal control admits a state feedback representation in the case that the cost functional is uniformly convex. Finally, some examples are presented to illustrate the theory developed.

Accepted:

DOI: 10.1051/cocv/2016023

Keywords: Mean-field stochastic differential equation, linear quadratic optimal control, Riccati equation, finiteness, open-loop solvability, feedback representation

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@article{COCV_2017__23_3_1099_0, author = {Sun, Jingrui}, title = {Mean-Field stochastic {Linear} {Quadratic} optimal control problems: {Open-loop} solvabilities}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1099--1127}, publisher = {EDP-Sciences}, volume = {23}, number = {3}, year = {2017}, doi = {10.1051/cocv/2016023}, mrnumber = {3660461}, zbl = {1393.49024}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016023/} }

TY - JOUR AU - Sun, Jingrui TI - Mean-Field stochastic Linear Quadratic optimal control problems: Open-loop solvabilities JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1099 EP - 1127 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016023/ DO - 10.1051/cocv/2016023 LA - en ID - COCV_2017__23_3_1099_0 ER -

%0 Journal Article %A Sun, Jingrui %T Mean-Field stochastic Linear Quadratic optimal control problems: Open-loop solvabilities %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1099-1127 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016023/ %R 10.1051/cocv/2016023 %G en %F COCV_2017__23_3_1099_0

Sun, Jingrui. Mean-Field stochastic Linear Quadratic optimal control problems: Open-loop solvabilities. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 1099-1127. doi : 10.1051/cocv/2016023. http://archive.numdam.org/articles/10.1051/cocv/2016023/

Nonlinear diffusion governed by McKean−Vlasov equation on Hilbert space and optimal control. SIAM J. Control Optim. 46 (2007) 356–378. | DOI | MR | Zbl

,A semilinear McKean−Vlasov stochastic evolution equation in Hilbert space. Stoch. Proc. Appl. 60 (1995) 65–85. | DOI | MR | Zbl

and ,Controlled McKean−Vlasov equations. Commun. Appl. Anal. 5 (2001) 183–206. | MR | Zbl

and ,A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63 (2011) 341–356. | DOI | MR | Zbl

and ,McKean−Vlasov limit in portfolio optimization. Stoch. Anal. Appl. 28 (2010) 884–906. | DOI | MR | Zbl

and ,A general maximum principle for SDEs of mean-field type. Appl. Math. Optim. 64 (2011) 197–216. | DOI | MR | Zbl

, and ,Mean-field backward stochastic differential equations: A limit approach. Ann. Probab. 37 (2009) 1524–1565. | DOI | MR | Zbl

, , and ,Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Process. Appl. 119 (2009) 3133–3154. | DOI | MR | Zbl

, and ,Dynamics of the McKean−Vlasov equation. Ann. Probab. 22 (1994) 431–441. | DOI | MR | Zbl

,McKean−Vlasov equations with discontinuous coefficients. Soochow J. Math. 20 (1994) 507–526. | MR | Zbl

,Approximate McKean−Vlasov representations for a class of SPDEs. Stochastics 82 (2010) 53–68. | DOI | MR | Zbl

and ,Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Statist. Phys. 31 (1983) 29–85. | DOI | MR

,Large deviations from the McKean−Vlasov limit for weakly interacting diffusions. Stochastics 20 (1987) 247–308. | DOI | MR | Zbl

and ,On the McKean−Vlasov limit for interacting diffusions. Math. Nachr. 137 (1988) 197–248. | DOI | MR | Zbl

,McKean−Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets. Stoch. Proc. Appl. 40 (1992) 69–82. | DOI | MR | Zbl

,A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Math. Control Relat. Fields 5 (2015) 97–139. | DOI | MR | Zbl

, , and ,Large population stochastic dynamic games: Closed-loop McKean−Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6 (2006) 221–251. | DOI | MR | Zbl

, , and ,Foundations of kinetic theory. Proc. of 3rd Berkeley Sympos. Math. Statist. Prob. 3 (1956) 171–197. | MR | Zbl

,Macroscopic limit for stochastic partial differential equations of McKean−Vlasov type. Prob. Theory Relat. Fields 146 (2010) 189–222. | DOI | MR | Zbl

and ,On a class of backward McKean−Vlasov stochastic equations in Hilbert space: Existence and convergence properties. Dynam. Syst. Appl. 16 (2007) 643–664. | MR | Zbl

and ,A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56 (1966) 1907–1911. | DOI | MR | Zbl

,B. $\xd8$ksendal and X. Y. Zhou, A mean-field stochastic maximum principle via Malliavin calculus. Stochastics 84 (2012) 643–666. | DOI | MR | Zbl

,Controllability of McKean−Vlasov stochastic integrodifferential evolution equation in Hilbert spaces. Numer. Funct. Anal. Optim. 29 (2008) 1328–1346. | DOI | MR | Zbl

, and ,Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations. J. Austral. Math. Soc., Ser. A 43 (1987) 246–256. | DOI | MR | Zbl

,Linear Quadratic Stochastic Differential Games: Open-Loop and Closed-Loop Saddle Points. SIAM J. Control Optim. 52 (2014) 4082–4121. | DOI | MR | Zbl

and ,Open-Loop and Closed-Loop Solvabilities for Stochastic Linear Quadratic Optimal Control Problems. SIAM J. Control Optim. 54 (2016) 2274–2308. | DOI | MR | Zbl

, and ,A. Yu. Veretennikov, On ergodic measures for McKean−Vlasov stochastic equations, in Monte Carlo and quasi-Monte Carlo methods 2004. Springer, Berlin (2006) 471–486. | MR | Zbl

Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations. SIAM J. Control Optim. 51 (2013) 2809–2838. | DOI | MR | Zbl

,J. Yong and X.Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999). | MR | Zbl

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