This paper discusses the control of coherent structures in turbulent flows, which has broad applications among complex systems in science and technology. Mean field games have been proved a powerful tool and are proposed here to control the stochastic Lagrangian tracers as players tracking the flow field. We derive optimal control solutions for general nonlinear fluid systems using mean field game models, and develop computational algorithms to efficiently solve the resulting coupled forward and backward mean field system. A precise link is established for the control of Lagrangian tracers and the scalar vorticity field based on the functional Hamilton-Jacobi equations derived from the mean field models. New iterative numerical strategy is then constructed to compute the optimal solution with fast convergence. We verify the skill of the mean field control models and illustrate their practical efficiency on a prototype model modified from the viscous Burger's equation under various cost functions in both deterministic and stochastic formulations. The good model performance implies potential effectiveness of the strategy for more general high-dimensional turbulent systems.

翻译：本文讨论了湍流中相干结构的控制问题，该问题在科学与技术的复杂系统中具有广泛应用。平均场博弈已被证明是一种强大工具，本文提出将其用于控制作为追踪流场的随机拉格朗日示踪粒子。我们通过平均场博弈模型推导了一般非线性流体系统的最优控制解，并开发了高效求解耦合正向与反向平均场系统的计算算法。基于平均场模型导出的泛函哈密顿-雅可比方程，建立了拉格朗日示踪粒子控制与标量涡度场之间的精确联系。随后构建了新的迭代数值策略，以快速收敛计算最优解。我们通过修改自粘性伯格斯方程的原型模型，在确定性和随机两种形式下验证了平均场控制模型的性能，并展示了其在实际成本函数下的效率。良好的模型性能表明该策略在更一般的高维湍流系统中具有潜在有效性。