若干偏微分方程控制系统的适定正则性及稳定性分析

项目名称： 若干偏微分方程控制系统的适定正则性及稳定性分析

项目编号： No.61503230

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 自动化技术、计算机技术

项目作者： 温瑞丽

作者单位： 山西大学

项目金额： 21万元

中文摘要： 偏微分方程控制系统与集中参数系统的显著差别在于控制器的空间位置可以选择，边界控制由于实际需要成为自然选择。但边界的控制与测量必然导致控制算子和观测算子无界，于是在何种最优状态空间求解，就成了重要问题。由此催生了在Salamon-Weiss意义下的适定正则性问题。在抽象的适定性和正则性框架下许多有穷维的结果被平行推广到分布参数系统。但究竟何种偏微分系统，特别是高维的系统是适定正则的就成了分布参数系统控制理论一个新的研究方向。本项目首先将几类具有实际意义的偏微分方程控制系统纳入到抽象系统框架中，应用偏微分方程、分布参数系统控制、算子半群以及黎曼几何等数学理论获得所考虑系统的适定性和正则性。在控制系统为正则时，给出直接传输算子的解析表达式。进而，对适定系统讨论其稳定性，给出能量的某种衰减率。本项目属于偏微分方程控制的新题目，是数学与控制理论密切结合的典型事例，因此具有重要的理论意义和应用价值。

中文关键词： 偏微分方程控制系统；适定性；正则性；稳定性；可控性

英文摘要： The significant difference of the control systems of partial differential equations and lumped parameter systems is that the spatial location of the controller can be selected, boundary controls become the result of natural selection because of the practical needs. However, boundary controls and measurements inevitably lead to the unboundedness of the control operator and observation operator, so it becomes an important problem that looking for the solution in what kind of optimal state space. All which bring about the well-posedness and regularity in the sense of Salamon-Weiss. In the abstract framework of well-posedness and regularity, many finite-dimensional results are parallel extended to the distributed parameter systems. So it becomes a new research direction in the control theory of distributed parameter systems that what kinds of partial differential systems, especially high-dimensional systems, are well-posed and regular. The project, firstly, forms some control systems of partial differential equations with practical significance into the abstract frame of system, and we will get the well-posedness and regularity of the discussed systems by making use of the theory of partial differential equations, the control theory of distributed parameter systems, the theory of operator semi-group, the theory of Riemannian geometry and other mathematical theories. And we give the analytic expressions of the feedthrough operators when the control systems are regular. Furthermore, we study the stabilization for well-posed systems, and give the decay rates of the energy. The project is a new topic of the control of partial differential equations, and a typical example of the close combination of mathematics and control theory, therefore, the study has important theoretical significance and practical values.

英文关键词： Control systems for partial differential equations;Well-posedness;Regularity;Stabilization;Controllability