Stochastic linear quadratic optimal control problems for mean-field stochastic evolution equations

Lü, Qi

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 127.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We study a linear quadratic optimal control problem for mean-field stochastic evolution equation with the assumption that all the coefficients concerned in the problem are deterministic. We show that the existence of optimal feedback operators is equivalent to that of regular solution to the system which is coupled by two Riccati equations and an explicit formula of the optimal feedback control operator is given via the regular solution. We also show that the mentioned Riccati equations admit a unique strongly regular solution when the cost functional is uniformly convex.

[1] B. Acciaio, J. Backhoff-Veraguas and R. Carmona, Extended mean field control problems: stochastic maximum principle and transport perspective. SIAM J. Control Optim. 57 (2019) 3666–3693. | DOI | MR | Zbl

[2] N. Agram and B. Øksendal, Stochastic control of memory mean-field processes. Appl. Math. Optim. 79 (2019) 181–204. | DOI | MR | Zbl

[3] B.D.O. Anderson and J.B. Moore, Optimal Control: Linear Quadratic Methods. Prentice Hall, Englewood Cliffs, NJ (1989). | Zbl

[4] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite-dimensional systems. Vol. II. Birkhäuser Boston, Inc., Boston, MA (1993). | DOI | MR | Zbl

[5] A. Bensoussan, J. Frehse and P. Yam, Mean field games and mean field type control theory. Springer, New York (2013). | DOI | MR | Zbl

[6] F.J. Beutler, The operator theory of the pseudo-inverse. II. Unbounded operators with arbitrary range. J. Math. Anal. Appl. 10 (1965) 471–493. | DOI | MR | Zbl

[7] P. Cardaliaguet, F. Delarue, J.-M. Lasry and P.-L. Lions, The master equation and the convergence problem in mean field games. Princeton University Press, Princeton, NJ (2019). | MR

[8] R. Carmona and F. Delarue, Probabilistic theory of mean field games with applications. I. Mean field FBSDEs, control, and games. Springer, Cham (2018). | MR

[9] R. Carmona and F. Delarue, Probabilistic theory of mean field games with applications. II. Mean field games with common noise and master equations. Springer, Cham (2018). | DOI | MR

[10] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992). | DOI | MR | Zbl

[11] D.A. Dawson, Stochastic evolution equations. Math. Biosci. 15 (1972) 287–316. | DOI | MR | Zbl

[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 | MR | Zbl

[13] G. Lance, E. Trélat and E. Zuazua, Shape turnpike for linear parabolic PDE models. Systems Control Lett. 142 (2020) 104733. | DOI | MR | Zbl

[14] X. Li, J. Sun and J. Yong, Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability. To appear in: Probab. Uncertain. Quant. Risk doi: (2020). | DOI | Zbl

[15] X. Li and J. Yong, Optimal control theory for infinite-dimensional systems. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA (1995). | MR | Zbl

[16] Q. Lü, Well-posedness of stochastic Riccati equations and closed-loop solvability for stochastic linear quadratic optimal control problems. J. Differ. Equ. 267 (2019) 180–227. | DOI | MR | Zbl

[17] Q. Lü and X. Zhang, General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions. Springer Briefs in Mathematics. Springer, Cham (2014). | DOI | MR | Zbl

[18] A. Porretta and E. Zuazua, Long time versus steady state optimal control. SIAM J. Control Optim. 51 (2013) 4242–4273. | DOI | MR | Zbl

[19] J. Sun, Mean-field stochastic linear quadratic optimal control problems: open-loop solvabilities. ESAIM: COCV 23 (2017) 1099–1127. | Numdam | MR | Zbl

[20] J. Sun, X. Li and J. Yong, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems. SIAM J. Control Optim. 54 (2016) 2274–2308. | DOI | MR | Zbl

[21] T. Suzuki, Mean field theories and dual variation. Atlantis Press, Paris (2015). | MR | Zbl

[22] 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 | MR | Zbl

[23] J. B. Walsh, An introduction to stochastic partial differential equations. École d’été de probabilités de Saint-Flour, XIV–1984, 265–439, Lecture Notes in Math., 1180. Springer, Berlin (1986). | MR | Zbl

[24] T. Wang, Necessary conditions in stochastic linear quadratic problems and their applications. J. Math. Anal. Appl. 469 (2019) 280–297. | DOI | MR | Zbl

[25] T. Wang, On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems. ESAIM: COCV 26 (2020) 41. | Numdam | MR | Zbl

[26] Q. Wei, J. Yong and Z. Yu, Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions. ESAIM: COCV 25 (2019) 17. | Numdam | MR | Zbl

[27] J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations. SIAM J. Control Optim. 51 (2013) 2809–2838. | DOI | MR | Zbl

[28] J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations–time-consistent solutions. Trans. Amer. Math. Soc. 369 (2017) 5467–5523. | DOI | MR | Zbl

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