SIAM Journal on Control and Optimization, Volume 59, Issue 2, Page 977-1006, January 2021.
We derive a framework to compute optimal controls for problems with states in the space of probability measures. Since many optimal control problems constrained by a system of ordinary differential equations modeling interacting particles converge to optimal control problems constrained by a partial differential equation in the mean-field limit, it is interesting to have a calculus directly on the mesoscopic level of probability measures which allows us to derive the corresponding first-order optimality system. In addition to this new calculus, we provide relations for the resulting system to the first-order optimality system derived on the particle level and the first-order optimality system based on $L^2$-calculus under additional regularity assumptions. We further justify the use of the $L^2$-adjoint in numerical simulations by establishing a link between the adjoint in the space of probability measures and the adjoint corresponding to $L^2$-calculus. Moreover, we prove a convergence rate for the convergence of the optimal controls corresponding to the particle formulation to the optimal controls of the mean-field problem as the number of particles tends to infinity.

中文翻译：

---

概率测度空间中的均值最优控制和最优条件

SIAM控制与优化杂志，第59卷，第2期，第977-1006页，2021年1月。
我们推导了一个框架，用于计算概率测度空间中状态问题的最优控制。由于许多由建模共同粒子的常微分方程组约束的最优控制问题收敛于在均值场极限内由偏微分方程约束的最优控制问题，因此直接在概率测度的介观水平上进行微积分是很有趣的，允许我们推导相应的一阶最优系统。除了这种新的演算，我们还提供了所得系统与在粒子级上推导的一阶最优系统以及在其他规则假设下基于$ L ^ 2 $演算的一阶最优系统之间的关系。通过在概率度量空间中的伴随与对应于$ L ^ 2 $-演算的伴随之间建立联系，我们进一步证明了在数值模拟中使用$ L ^ 2 $伴随。此外，我们证明了随着粒子数趋于无穷大，对应于粒子配方的最优控制收敛到平均场问题的最优控制的收敛速度。