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Investigation of Inverse Problems for the Heat equation Based on the Theory of Stochastic Control

基于随机控制理论的热方程反问题研究

基本信息

批准号：
16540100
负责人：
TSUCHIYA Masaaki
金额：
$ 1.79万
依托单位：
Kanazawa University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540100/
关键词：
heat equation inverse problem stochastic control jump process Jacobi series Cesaro operator spectral spreading sequence discretized Markov transform 超離散力学系時系列モデルマルコフ連鎖

项目摘要

We study the inverse problem determining the shape of some unknown portion of the boundary of a domain based on parabolic equations and also study the one determining the heat conduction coefficients of a heat equation based on the theory of stochastic control. The former is treated through a suitably linearized equation by analytical method. The shape of deforming unknown portion is allowed depending on time and is assumed only to be Lipschitz continuous. The latter provides us with a new type of stochastic control. That is, the running cost is driven by the local time at the measurement place with respect to the controlled diffusion process. Therefore the corresponding HJB equation has singular source term involving the Dirac function supported on the measurement place.In the framework of stochastic control, we also consider some jump process concerned with common property resource and obtain an optimal control variable under suitable conditions.Related to the subject, we need to study some property of systems of orthogonal functions. In particular, the classical Hardy's inequality is extended to the case of Jacobi series and the boundedness of the transplantation operators and Cesaro operators are obtained.Finally, from a viewpoint of numerical analysis, we study some random sequences. An ultradiscrete dynamical system is constructed under consideration of discretized Markov transforms and bit error probabilities of certain communication systems are discussed by using spreading sequences of Markov chains.

研究了基于抛物型方程确定区域边界未知部分形状的反问题和基于随机控制理论确定热传导系数的反问题。前者用解析法通过适当线性化的方程来处理。变形未知部分的形状随时间变化是允许的，且仅假定为Lipschitz连续。后者为我们提供了一种新型的随机控制。也就是说，相对于受控扩散过程，运行成本由测量地点的当地时间驱动。在随机控制的框架下，我们还考虑了一些与公共属性资源有关的跳跃过程，并在适当的条件下得到了一个最优控制变量。与此相关，我们需要研究正交函数系的一些性质。特别地，将经典的Hardy不等式推广到Jacobi级数的情形，得到了移植算子和Cesaro算子的有界性。最后，从数值分析的角度研究了一些随机序列。在考虑离散化马尔可夫变换的情况下，构造了一个超离散动力系统，并利用马尔可夫链的扩展序列讨论了某些通信系统的误码率。