Idempotent Methods and Fundamental Solutions

幂等代数包括极大加、极小加半域和极小极大半环。这些代数与与Hamilton-Jacobi-Bellman(HJB)和Hamilton-Jacobi-Isaacs偏微分方程(PDE)相关的半群有很深的联系。在过去的十年中，人们发现了一类一阶HJB偏微分方程解的极大值无维灾方法。最重要的是，对于某些类型的HJB偏微分方程，这些方法可以解决比经典方法(即基于网格的方法)更高维度的问题，这些方法受到维度诅咒的影响。最近，人们发现幂等分布性质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。第一个是进一步发展无最大维灾的数值方法，特别是将适用范围扩展到扩散过程。第二，这些方法在具有随机输入的开放量子系统中的量子自旋控制中的应用。在第三方面，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，其中动态由偏微分方程组描述。第四部分讨论保守系统两点边值问题(TPBVP)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与终端数据相关的代价函数的幂等卷积，将两点边值问题转化为初值问题。此外，由于这些是基本的解，一旦计算出来，就可以很容易地将该系统和持续时间的一大类TPBVP中的任何一种转化为初值问题。研究人员还在扩展这一理论，以涵盖经典的n体问题。在欧几里得空间中，n-体TPBVP可以转化为一个微分对策，并且基本解可以在欧几里得空间中以集合的形式得到。由于半个世纪以来的维灾问题，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度的增加而指数增长。基于MAX-PLUS的无维度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投资组合优化。对两点边值问题基本解的研究将使动力系统问题的快速求解成为可能，人们可以在不同的初值/终值条件下寻求动力学问题的解。也就是说，基本解决方案允许对不同的初始数据和结束数据重复使用相同的对象。目前感兴趣的应用包括由线性无限维动力学控制的系统，如标准的热方程和波动方程，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。还将考虑将引力场以外的其他类型的势场扩展到其他类别。

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.