论文标题
通过拉格朗日方法，确定性平均场类型最佳控制问题的Pontryagin最大原理
Pontryagin maximum principle for the deterministic mean field type optimal control problem via the Lagrangian approach
论文作者
论文摘要
我们研究了确定性平均场类型自由点最佳控制问题的必要最佳条件。我们的研究依赖于拉格朗日方法，该方法将平均场类型控制系统视为一群无限的代理，这些代理被某些概率空间的元素标记。首先，我们以拉格朗日形式得出了Pontryagin的最大原理。此外，我们考虑了Kantorovich和Eulerian形式化，它们分别通过轨迹和非局部连续性方程的分布来描述平均场类型控制系统。我们证明，坎托维奇或欧拉尔式配方中的当地最小化器决定了拉格朗日方法中的当地最小化器。使用它，我们推断出Kantorovich和Eulerian形式中的Pontryagin最大原理。为了说明一般理论，我们检查了平均场类型线性二次调节器的模型系统。我们表明，在这种情况下，最佳策略是由线性反馈决定的。
We study necessary optimality conditions for the deterministic mean field type free-endpoint optimal control problem. Our study relies on the Lagrangian approach that treats the mean field type control system as a crowd of infinitely many agents who are labeled by elements of some probability space. First, we derive the Pontryagin maximum principle in the Lagrangian form. Furthermore, we consider the Kantorovich and Eulerian formalizations which describe mean field type control systems via distributions on the set of trajectories and nonlocal continuity equation respectively. We prove that local minimizers in the Kantorovich or Eulerian formulations determine local minimizers within the Lagrangian approach. Using this, we deduce the Pontryagin maximum principle in the Kantorovich and Eulerian forms. To illustrate the general theory, we examine a model system of mean field type linear quadratic regulator. We show that the optimal strategy in this case is determined by a linear feedback.
