Partial derivative with respect to the measure and its application to general controlled mean-field systems
Partial derivative with respect to the measure and its application to general controlled mean-field systems
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测量的偏导数及其在一般受控平均场系统中的应用
DOI：
10.1016/j.spa.2021.01.003
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发表时间：
2021-01
影响因子：
1.4
通讯作者：
Juan Li
中科院分区：
文献类型：
--
作者：
Rainer Buckdahn;Yajie Chen;Juan Li
令 (E, E) 为任意可测空间。本文首先重点研究函数 f 的偏导数：P 2, 0 (R d× E)→ R，定义在概率测度空间 μ (R d× E, B (R d)⊗ E) 上，其第一边际 μ 1≔ μ (⋅× E) 具有有限二阶矩。该偏导数是关于 q (d x, z) 求的，其中 μ 对其第二边际 μ 2 (⋅)= μ (R d×⋅) 具有分解 μ (d x d z)= q (d x, z) μ 2 (d z)。简化语言，我们将讨论关于以其第二边际为条件的定律 μ 的导数。我们的结果将函数 g 的导数：P 2 (R d)→ R 扩展到 PL Lions 的有限二阶矩的概率测度空间（参见 Lions (2013)），但也涵盖了考虑 E= R k 并假设 f 在 P 2 (R d× R k) 上的可微性的最新方法，以便使用导数 ∂ μ f 定义偏导数 (∂ μ f) 1。论文的第二部分重点研究随机极大值原理，其中受控状态过程由具有部分信息的一般平均场随机微分方程驱动。控制集假设是一个可测空间，受控系统的系数，即动力学系数和成本函数系数，取决于受控状态过程X、控制v、X的部分信息以及(X, v)的联合律。通过考虑一个新的二阶变分方程和相应的二阶伴随方程，以及一种全新的方法来证明一阶变分方程解的估计，通过最优控制的尖峰变分并借助特制的二阶展开形式来证明最优原理。我们强调，在我们的假设中，我们不需要控制变量中的系数或控制过程定律中的系数的任何规律性。
Let (E, E) be an arbitrary measurable space. The paper first focuses on studying the partial derivative of a function f: P 2, 0 (R d× E)→ R defined on the space of probability measures μ over (R d× E, B (R d)⊗ E) whose first marginal μ 1≔ μ (⋅× E) has a finite second order moment. This partial derivative is taken with respect to q (d x, z), where μ has the disintegration μ (d x d z)= q (d x, z) μ 2 (d z) with respect to its second marginal μ 2 (⋅)= μ (R d×⋅). Simplifying the language, we will speak of the derivative with respect to the law μ conditioned to its second marginal. Our results extend those of the derivative of a function g: P 2 (R d)→ R over the space of probability measures with finite second order moment by PL Lions (see Lions (2013)) but cover also as a particular case recent approaches considering E= R k and supposing the differentiability of f over P 2 (R d× R k), in order to use the derivative∂ μ f to define the partial derivative (∂ μ f) 1. The second part of the paper focuses on investigating a stochastic maximum principle, where the controlled state process is driven by a general mean-field stochastic differential equation with partial information. The control set is just supposed to be a measurable space, and the coefficients of the controlled system, ie, those of the dynamics as well as of the cost functional, depend on the controlled state process X, the control v, a partial information on X, as well as on the joint law of (X, v). Through considering a new second-order variational equation and the corresponding second-order adjoint equation, and a totally new method to prove the estimate for the solution of the first-order variational equation, the optimal principle is proved through spike variation of an optimal control and with the help of the tailor-made form of second-order expansion. We emphasize that in our assumptions we do not need any regularity of the coefficients neither in the control variable nor with respect to the law of the control process.
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DOI：
10.4310/cis.2006.v6.n3.a5
发表时间：
2006-10
期刊：
Commun. Inf. Syst.
影响因子：
--
作者：
P. Caines;Minyi Huang;R. Malhamé
通讯作者：
P. Caines;Minyi Huang;R. Malhamé
影响因子：
--
作者：
Y. Achdou;P. Cardaliaguet;François Delarue;A. Porretta;Filippo Santambrogio
通讯作者：
Y. Achdou;P. Cardaliaguet;François Delarue;A. Porretta;Filippo Santambrogio
DOI：
10.1137/080738465
发表时间：
2009-06
期刊：
SIAM J. Control. Optim.
影响因子：
--
作者：
Jianhui Huang;Guangchen Wang;J. Xiong
通讯作者：
Jianhui Huang;Guangchen Wang;J. Xiong
影响因子：
1.8
作者：
R. Buckdahn;Juan Li;Jin Ma
通讯作者：
R. Buckdahn;Juan Li;Jin Ma
影响因子：
1.4
作者：
Juan Li
通讯作者：
Juan Li