变系数结构声学系统能控性和稳定性的研究

The structural acoustic system is a partial differential equation system with clear engineering background. In the practical application, the constituent materials of the structural acoustic system are often non-uniform, which results in the model of the description system having the characteristic of variable coefficients. Therefore, the study of the structural acoustic system with variable coefficients is of great theoretical and practical significance. The structural acoustic system with variable coefficients of this project is composed of two partial differential equations, one is to call the wave equation which is typically invoked in the modelling, so as to describe an interior acoustic field; On the other hand, either wave, plate, beam or shell equations, subject to various degrees of damping, have been utilized so as to model the structural component of the coupled PDE system. In this project, we focuses on the controllability and stabilization of the structural acoustic system with variable coefficients, the main research tool is Riemann geometry method, which combines the Riemann geometry method and the traditional method of partial differential equation control theory, overcome the difficulties caused by the main part of the variable coefficients, and establishing the corresponding observation inequality and the energy decay rate.

结构声学系统是有明确工程背景的偏微分方程系统. 在实际的应用中，结构声学系统的构成材料往往是不均匀的，这会导致描述系统的模型具有变系数的特点，因此，变系数结构声学系统的研究有着重要的理论意义和实际意义. 本项目研究的变系数结构声学系统是由两个偏微分方程耦合而成，其中一个是在模型中调用波动方程，以便描述内部声场；另一方面，利用波、板、梁或壳方程，在不同程度的阻尼作用或者不同的控制条件下，对耦合偏微分系统的结构部件进行建模。在本项目中，我们主要研究变系数结构声学系统的能控性和稳定性，主要的研究工具是黎曼几何方法，将黎曼几何方法和偏微分方程控制理论的传统方法相结合，克服变系数主部所带来的困难，建立相应的观测不等式和能量衰减估计。

结构声系统是偏微分方程和分布参数控制理论中一类重要的数学模型。由于智能材料技术方面的创新，而且这些创新的设计在控制工程的应用，大大提高了对结构声系统研究的兴趣。在本项目中，我们主要研究具有内部阻尼和边界阻尼或者边界记忆项的变系数结构声系统的能量衰减估计，以及具有Kirchoff 板的变系数结构声学系统的能控性。在项目研究过程中，我们的主要研究工具是黎曼几何方法，在适当的系数条件和区域边界条件下，得到变系数结构声系统的边界可控和能量衰减估计。