Idempotent Methods and Fundamental Solutions

幂等方法和基本解决方案

• 批准号: 1312569
• 负责人: William McEneaney
• 金额: $ 25万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1312569&HistoricalAwards=false
• 关键词: Idempotent; Methods; Fundamental; Solutions

Idempotent algebras include the max-plus and min-plus semifields and the min-max semiring. There is a deep relation between these algebras and the semigroups associated with Hamilton-Jacobi-Bellman (HJB) and Hamilton-Jacobi-Isaacs partial differential equations (PDEs). During the past decade, the max-plus curse-of-dimensionality-free methods for solution of classes of first-order HJB PDEs were discovered. Most importantly, for some classes of HJB PDEs, these methods can solve problems in significantly higher dimensions than would be feasible with classical, i.e., grid-based, methods, which are subject to the curse of dimensionality. More recently, it has been discovered that idempotent distributive properties allow such methods to address stochastic control and dynamic game problems, and this is opening up interesting new domains. The project has four components. The first is the further development of max-plus curse-of-dimensionality-free numerical methods, specifically extending the domain of applicability to diffusion processes. Second, application of these methods to the control of quantum-spin in the case of open quantum systems where one has stochastic inputs. On the third front, the investigators are developing idempotent algebra-based methods for solution of linear, infinite-dimensional control and estimation problems where the dynamics are described by systems of PDEs. The fourth component regards fundamental solutions of two-point boundary value problems (TPBVPs) for conservative systems, in which case one may apply the Stationary Action Principle. The fundamental solutions convert two-point boundary value problems into initial value problems via an idempotent convolution of the fundamental solution with a cost function related to the terminal data. Moreover, as these are fundamental solutions, once computed, one can easily convert any of a large class of TPBVPs for that system and time duration into an initial value problem. The investigators are also expanding this theory to cover the classic n-body problem. There, it can be found that an n-body TPBVP, posed in terms of the stationary-action principle, can be converted into a differential game, and the fundamental solution can be obtained as a set in Euclidean space.Optimal control has not generally been feasible due to the half-century old problem of the curse of dimensionality, whereby with classical grid-based approaches, the computational complexity grows exponentially fast with the dimension of the system being controlled. Max-plus based curse-of-dimensionality-free methods are not necessarily subject to this exponential growth. Consequently, this research is opening up large application areas which were previously inaccessible. Specific applications of interest (among many possible application domains) include spacecraft and aircraft guidance and control, quantum spin control, signal amplification for long fiber-optic networks, UAV sensor tasking operations and portfolio optimization. The efforts on fundamental solutions for two-point boundary value problems will allow rapid solution of dynamical system problems where one seeks solutions of the dynamics for a variety of initial/terminal conditions. That is, the fundamental solution allows one to use the same object repeatedly for varying initial and terminal data. The applications of current interest include systems governed by linear, infinite-dimensional dynamics such as the standard heat and wave equations, as well as the n-body problem. In the latter case, applications could include rapid estimation of asteroid threats. Extensions to other classes of potential fields, beyond gravitational, will also be considered.

幂等代数包括极大加半域和极小加半域以及极小极大半环。这些代数与Hamilton-Jacobi-Bellman （HJB）和Hamilton-Jacobi-Isaacs偏微分方程（PDEs）相关的半群有很深的关系。在过去的十年中，人们发现了求解一类一阶HJB偏微分方程的最大加无维数方法。最重要的是，对于某些类别的HJB pde，这些方法可以解决比经典方法（即基于网格的方法）更高维的问题，后者受到维度的诅咒。最近，人们发现幂等分布特性允许这种方法解决随机控制和动态博弈问题，这开辟了有趣的新领域。该项目有四个组成部分。首先是进一步发展无最大加维数的数值方法，特别是扩展了扩散过程的适用范围。其次，将这些方法应用于具有随机输入的开放量子系统的量子自旋控制。在第三个方面，研究人员正在开发基于幂等代数的方法来解决线性，无限维控制和估计问题，其中动力学是由偏微分方程系统描述的。第四部分考虑保守系统两点边值问题（TPBVPs）的基本解，在这种情况下可以应用定常作用原理。基本解通过基本解与终端数据相关的成本函数的幂等卷积将两点边值问题转化为初值问题。此外，由于这些都是基本的解决方案，一旦计算出来，就可以很容易地将该系统和时间持续时间的任何一大类tpbvp转换为初始值问题。研究人员还将这一理论扩展到经典的n体问题。在那里，可以发现用定动原理构造的n体TPBVP可以转化为微分对策，其基本解可以在欧几里德空间中作为一个集合得到。由于半个世纪以来的维数诅咒问题，最优控制通常是不可行的，在经典的基于网格的方法中，计算复杂度随着被控制系统的维数呈指数级增长。基于Max-plus的无维诅咒方法不一定会受到这种指数增长的影响。因此，这项研究正在开辟以前无法进入的大型应用领域。感兴趣的具体应用（在许多可能的应用领域中）包括航天器和飞机制导和控制，量子自旋控制，长光纤网络的信号放大，无人机传感器任务操作和组合优化。对两点边值问题的基本解的努力将允许快速解决动力系统问题，其中寻求各种初始/终端条件的动力学解。也就是说，基本解决方案允许对不同的初始和终端数据重复使用相同的对象。当前感兴趣的应用包括由线性，无限维动力学控制的系统，如标准热和波方程，以及n体问题。在后一种情况下，应用可能包括对小行星威胁的快速估计。扩展到其他类别的势场，除了引力，也将考虑。