若干偏微分方程控制系统的适定正则性及稳定性分析

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.

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。

本项目研究了偏微分方程控制系统的适定正则性、精确能控性、以及带干扰的控制系统稳定性等问题。主要研究了四阶 Schrödinger 方程在铰支边界情形下分别带有 Dirichlet 边界控制和矩边界控制及同位观测的控制系统、变系数四阶 Schrödinger 方程在 Neumann 边界控制和同位观测下的控制系统，证明系统为适定正则的，并在控制系统为正则时给出了相应的直接传输算子，进一步证明了所考虑系统为精确能控的，因此在已得到的适定结果下可直接获得控制系统在比例反馈下为指数稳定的。研究了一类基尔霍夫型非线性梁的自适应镇定和参数估计问题，根据控制端的速度和不确定输入扰动的未知参数估计，设计了一种高增益自适应调节器，证明该系统是稳定的，估计的参数收敛于未知参数。另外，对边界带有干扰的控制系统稳定性问题做了一些研究，例如：波方程、热方程、Schrödinger 方程等控制系统，并对所得结果进行了数值模拟验证。本项目研究问题有较强的工程背景，获得的结果对相应工程技术问题起理论指导作用。