Research for State Estimation and Optimal Control of Stochastic Distributed Parameter Systems with Free Boundary

1.随机两相Stefan系统的状态和自由边界估计：在本研究中，我们考虑了随机两相Stefan系统的状态和自由边界估计问题。首先，利用关于时间和空间变量的有限差分方法讨论了具有白噪声系数的状态方程解的存在唯一性。其次，利用鞅表示理论推导了状态估计器和自由边界估值器的动力学表达式。最后，给出了通过数字仿真实验获得的具有代表性的样本运行，包括一种实现非线性估计器动态的简单逼近方法。具有障碍的随机分布参数系统的状态估计和自由边界估计：本研究涉及带有障碍的随机分布参数系统的状态和自由边界估计问题。将系统模型描述为一个随机变分不等式，利用惩罚方法研究了解的存在唯一性。给出了分布噪声观测下无状态边界估计器的动力学性质。为了支持本文提出的理论，文中给出了一个算例，包括数字仿真实验结果。

1. State and free boundary estimations for stochastic two-phase Stefan systems: In this study, we consider the state and free boundary estimation problems for stochastic two-phase Stefan systems. First, the existence and uniqueness properties of the solution to the state equation with white noise coefficients are discussed by using the finite difference method with respect to time and spatial variables. Secondly, the state and free boundary estimator dynamics are derived by using martingale representation theory. Finally, representative sample runs obtained by digital simulation experiments are shown, including a simple approximation method for realizing non-linear estimator dynamics.2. On the state and free boundary estimations for stochastic distributed parameter systems with obstacle: This study is concerned with the state and free boundary estimation problems for the stochastic distributed parameter systems with obstacle. Formulating the system model as a stochastic variational inequality, the existence and uniqueness properties of the solution are investigated by using the penalization method. The dynamics of state-free boundary estimator is given under a distributed noisy observation. For the purpose of supporting the theoretical aspects developed here, an illustrative example is shown including results of digital simulation experiments.