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

本文研究了基于抛物型方程的确定区域边界未知部分形状的反问题，以及基于随机控制理论的确定热传导系数的反问题。前者是通过一个适当的线性化方程的解析方法处理。允许变形未知部分的形状取决于时间，并且仅假设为Lipschitz连续。后者为我们提供了一种新型的随机控制。也就是说，运行成本由相对于受控扩散过程的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性质源有关的跳过程，并在适当的条件下得到了一个最优控制变量，与此相关，我们还需要研究正交函数系的一些性质。特别地，将经典的哈代不等式推广到Jacobi级数的情形，并得到了移植算子和塞萨罗算子的有界性。最后，从数值分析的角度研究了一些随机序列。在考虑离散马尔可夫变换的基础上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.

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.