具有模型不确定性和公共噪声的平均场博弈与控制及应用研究

Due to the wide applications, the topic of mean field games has drawn much attention in many fields such as system control, mathematics and economics. At present, the previous work focused on the case that the model is well known, but there are still many challenging problems to be solved on mean field games and control with model uncertainty and common noise. This project is devoted to the research on the mean field games and social optimal control problems with model uncertainty and common noise. The key points are as follows. (1) The "soft control" approach is adopted to deal with the model uncertainty, and robust decentralized strategies are designed. The results are further applied to the opinion dynamics. (2) The social optimal strategies are constructed for the mean field models with volatility-uncertain common noise, where the difficulty lies in the derivation of the equivalent social optimal cost based on the person-by-person optimality principle and the dual theory of backward stochastic different equations. (3) By tackling an asymmetric Riccati equation, asymptotic optimal decentralized strategies is designed for the mean field production problem with the noisy price. The results can potentially promote the study of mean field game theory in a substantive way, and also have significant value in the fields of economics, engineering and sociology.

平均场博弈专题因其广泛应用得到了系统控制、数学以及经济学等诸多领域的关注和重视。目前已有研究结果集中在模型精确可知的情形，而存在模型不确定性和公共噪声的平均场博弈与控制还有许多挑战性问题尚待解决。本项目致力于研究具有模型不确定性和公共噪声的平均场博弈与社会最优控制问题，其关键在于：(1) 采用“软控制”的方式对付模型不确定性，设计鲁棒分散策略并应用结果到观点动力学中；(2) 构造具有方差不确定性公共噪声平均场模型的社会最优策略，其难点在于结合Person-by-Person最优性原理和正倒向随机方程的对偶理论导出等价的社会最优指标；(3) 通过处理非对称的Riccati方程，针对具有扰动价格的平均场产量调节控制问题，设计分散渐近最优策略。相关结果有望推动平均场博弈理论研究的实质性进展，并在经济、工程以及社会学等领域有重要应用价值。

本项目针对含有模型不确定性和公共噪声的平均场博弈与控制问题开展研究，具体包括(1)摆脱传统不动点方法的框架，发展“解耦正倒向方程方法”，得到平均场系统一致镇定的充要条件，(2)解决了含有模型和波动性不确定性的平均场社会控制问题；(3)建立了平均场产量调节模型。高影响力期刊IEEE TAC、Automatica和SIAM J. Control Optim.发表论文5篇，包括长文4篇。获得国家优秀青年基金资助。所获成果为平均场博弈研究提供了重要而有效的理论方法，相关应用延拓了垄断市场理论。