Mathematical Sciences: Topics in Stochastic Control and Random Media

本文讨论了随机控制理论和波在随机介质中的多次散射的数学问题。它是该领域两位知名专家周教授和梅纳迪教授共同努力的成果。主要的研究工作将投入到随机控制问题的各种理论问题的研究。假设动态过程受随机信号的影响。目标是确定一个控制函数，以优化某个目标，表示为过程轨迹上的函数。控制应该具有脉冲特性。这是一种理想的物理情况，在这种情况下，人们可以观察到过程轨迹的突然变化，这可以被认为是不连续或跳跃。例如，仓库中的库存可以用这种方式表示，因为它们在新库存交付时经历了价值的跳跃。对脉冲性质的控制通常用狄拉克分布在数学上表示。这些脉冲的时间和质量是人们希望以最优方式选择的控制参数。

This proposal is concerned with the mathematical problems of stochastic control theory and multiple scattering of waves in random media. It is a product of joint efforts of two known specialists in this area, Professors Chow and Menaldi. The main research effort will be invested in the study of various theoretical issues in the stochastic control problems. It is assumed that dynamical process is subject to influence of random signals. The goal is to determine a control function that optimizes a certain objective, expressed as a function on process trajectories. The control is supposed to have an impulse character. This is an idealized version of a physical situation where one can observe abrupt changes inprocess trajectories, which can be thought of as discontinuities, or jumps. Fo example, the inventory stocks in a warehouse may be represented in this way, as they experience a jump in the value at the delivery of new stock. The control of an impulse nature is usually represented mathematically by Dirac distributions. The timings and masses of such impulses are the control parameters that one wants to select in an optimal fashion.