Deterministic mean field games with control on the acceleration
Deterministic mean field games with control on the acceleration
复制标题
控制加速度的确定性平均场游戏
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
N. Tchou
中科院分区：
文献类型：
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作者：
Y. Achdou;Paola Mannucci;Claudio Marchi;N. Tchou
在目前的工作中，我们研究了有限时间范围内的确定性平均场博弈(MFGS)，其中通用智能体的动力学由加速度控制。它们由一个偏微分方程系统描述，该系统耦合了状态分布密度的连续性方程(时间向前)和代表代理的最佳值的哈密顿-雅各比方程(时间向后)。状态变量是Pair(x，v)∈RN×RNDocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$(x，v)in{mathbb{R}}^N imes{mathbb{R}}^N$$end{Document}其中x代表位置，v代表速度。动力学通常被称为二重积分器。在这种情况下，系统的哈密顿量既不是严格凸的，也不是强制的，因此关于MFGS的现有结果不能应用。此外，我们将假设哈密顿量是无界的。我们用消失粘性方法证明了MFG系统弱解的存在性，并通过与最优控制相关的流动将状态分布刻画为初始分布的映象。
In the present work, we study deterministic mean field games (MFGs) with finite time horizon in which the dynamics of a generic agent is controlled by the acceleration. They are described by a system of PDEs coupling a continuity equation for the density of the distribution of states (forward in time) and a Hamilton–Jacobi equation for the optimal value of a representative agent (backward in time). The state variable is the pair (x,v)∈RN×RNdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$(x,v)in {mathbb {R}}^N imes {mathbb {R}}^N$$end{document} where x stands for the position and v stands for the velocity. The dynamics is often referred to as the double integrator. In this case, the Hamiltonian of the system is neither strictly convex nor coercive, hence the available results on MFGs cannot be applied. Moreover, we will assume that the Hamiltonian is unbounded w.r.t. the velocity variable v. We prove the existence of a weak solution of the MFG system via a vanishing viscosity method and we characterize the distribution of states as the image of the initial distribution by the flow associated with the optimal control.