Robust mean field linear quadratic social control: open-loop and closed-loop strategies. (English) Zbl 1494.49026
Summary: This paper investigates the robust social optimum problem for linear quadratic mean field control systems by the direct approach, where model uncertainty appears in both drift and diffusion terms of each agent. We take a zero-sum game approach by considering local disturbance as the control of an adversarial player. Under centralized information structure, we first obtain the necessary and sufficient condition for the existence of open-loop and closed-loop saddle points, which are characterized by the solvability of forward-backward stochastic differential equations (FBSDEs) and two coupled Riccati equations, respectively. By considering the infinite system, we next design a set of decentralized open-loop strategies based on mean field FBSDEs and obtain closed-loop strategies in terms of two uncoupled Riccati equations. Finally, the performance of the proposed decentralized strategies is analyzed and the efficiency is verified by numerical simulation.
MSC:
| 49N80 | Mean field games and control |
| 91A16 | Mean field games (aspects of game theory) |
| 93E03 | Stochastic systems in control theory (general) |
| 93E20 | Optimal stochastic control |
| 49N10 | Linear-quadratic optimal control problems |
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