Analysis and control of evolutionary plates and elastic structures

在这个项目中的努力将集中在数学分析和控制偏微分方程（PDE）模型，描述某些弹性动力学，这是在自然和人造世界。目前考虑的弹性进化偏微分方程模型也可能受到系统外部的影响;例如，在某些边界界面上受到阻尼力的弹性体。因此，我们将要分析的主导PDE模型可以想象地构成PDE动力学的耦合，这些PDE动力学在性质上是完全不同的;例如，一个vonKarman动力平板偏微分方程与一个热过程耦合将产生一个耦合的偏微分方程系统，它表现出双曲和抛物偏微分方程的唯象特征，但不能说是严格的双曲或抛物偏微分方程。对于这类线性和非线性的发展方程，我们主要研究以下问题：（1）描述弹性结构与周围流体介质相互作用的耦合PDE模型的精确能控性和一般能达性。 根据物理应用，弹性体的运动将通过流体边界控制的间接手段来控制。(2)结构声学系统的一致稳定性。在这种情况下，腔室几何形状内部的声波耦合到弹性方程，该弹性方程对腔室壁的一部分上的弯曲振动进行建模;这里的弹性分量将表现出一些可量化的阻尼测量，从弱粘性阻尼到超强Kelvin-Voight阻尼。 对于这些系统，并在适当的几何假设下，我们打算调查的可能性，弹性阻尼传播到整个复合系统，在某种程度上，每个组件-表面上无阻尼波以及结构阻尼弹性壁组件-以某种可辨别的速度衰减。(3)某些非线性发展板PDE系统解或流的渐近行为的结果。特别是，我们将集中在那些系统是“非梯度”，也就是说，没有可用的李雅普诺夫函数在相关的有限能量空间，可以用来跟踪给定的，可能是非耗散的，PDE的长时间行为。我们希望我们的工作将最终稳定这样的非线性过程的一个全球性的紧凑attractor.我们相信，这个项目的结果可以给好处远远超出其内在价值的数学学科的贡献.例如，在我们上述和预期的结构声学均匀衰减研究中，我们预计稳定性结果将严重依赖于正在起作用的特定腔室几何形状。因此，我们相信我们的研究工作可以给出这些结构声学几何形状的精确表征，这些结构声学几何形状将在很长一段时间内产生相对静止的内部声场。这样的几何形状的情况下，可以想象，减少，或至少减少，需要主动工程控制的声学噪声。此外，我们在分析非线性演化板动力学的长期行为的预期项目工作可能会有直接的控制工程的影响：如果我们发现，对于一个给定的非线性PDE系统，相应的流收敛到一个有限分形维数的全球紧凑吸引子，那么可以想象的系统可以通过构建有限维反馈积极控制数字。此外，该项目产生的研究将作为基础，我们将开发和实施对数值偏微分方程和数学感兴趣的本科生的研究计划。 特别是，PI将在2013年夏季运行内布拉斯加大学林肯分校的“本科生研究经验”网站。在一个规范的一维设置的背景下，并通过商业计算机代数软件包的部分代理，我们将教，我们的本科生参与者，在我们的项目工作过程中开发的非线性理论的各个方面。此外，我们将积极让他们参与有关解决方案或流动的数值近似的研究项目，这些解决方案或流动对应于属于我们项目工作范围内的一维非线性过程。

The effort in this project will be focused on the mathematical analysis and control of partial differential equation (PDE) models which describe certain elastic dynamics which are seen in the natural and man-made world. The elastic evolutionary PDE models under present consideration might also be subjected to influences external to the system; e.g., elastic bodies subjected to damping forces across some boundary interface. In consequence, the governing PDE models we will analyze could conceivably constitute a coupling of PDE dynamics which are quite different in character; e.g., a von Karman dynamical plate PDE coupled to a thermal process would give rise to a coupled PDE system which evinces phenomenological traits of both hyperbolic and parabolic PDE, yet could not be said to be either strictly hyperbolic or parabolic. For such evolution equations, linear and nonlinear, we intend to address the following problems: (1) Exact controllability and general reachability properties of those coupled PDE models which describe the interaction between an elastic structure and a surrounding fluid medium. In line with the physical application, the movements of the elastic body are to be controlled by the indirect means of fluid boundary control. (2) Uniform stability properties of structural acoustic systems. In this situation, acoustic waves, interior to a chamber geometry, are coupled to elastic equations which model the flexural vibrations on a portion of the chamber wall; the elastic component here will manifest some quantifiable measure of damping, from weak viscous to super-strong Kelvin-Voight damping. For these systems, and under appropriate geometrical assumptions, we intend to investigate the possibility that the elastic damping is propagated throughout the entire composite system, to the extent that each component--ostensibly undamped wave as well as structurally damped elastic wall component--decays at some discernible rate. (3) Results concerning the asymptotic behavior of solutions, or flows, of certain nonlinear evolutionary plate PDE systems. In particular, we shall concentrate on those systems which are "non-gradient"; that is, there is no available Lyapunov function on the associated finite energy space which can employed to track the long time behavior of the given, possibly non-dissipative, PDE. It is hoped that our work in this connection will culminate in the stabilization of such nonlinear processes to a global compact attractor.We believe that the results of this project could give benefit much beyond their intrinsic worth as contributions to the discipline of mathematics. For example, in our aforesaid and intended structural acoustics uniform decay investigation, we anticipate that the stability results will depend critically upon the particular chamber geometry which is in play. As a consequence, we believe our research efforts could give a precise characterization of those structural acoustic geometries which will give rise, in long time, to relatively quiescent interior acoustic fields. Such geometrical situations could then conceivably obviate, or at least lessen, the need for the active engineering control of acoustic noise. Moreover, our intended project work in analyzing the long time behavior of nonlinear evolutionary plate dynamics could have immediate Control Engineering implications: Should we find, for a given nonlinear PDE system, that the corresponding flows converge to a global compact attractor of finite fractal dimension, then conceivably the system could be actively controlled numerically by means of a constructed finite-dimensional feedback. In addition, the research generated by this project will serve as the basis from which we will develop and implement a research program for undergraduates interested in numerical PDEs, and in mathematics generally. In particular, the PIs will run the University of Nebraska-Lincoln "Research Experience for Undergraduates" site in Summer 2013. Within the context of a canonical one dimensional setting, and through the partial agency of a commercial computer algebra software package, we will teach, to our undergraduate participants, aspects of the nonlinear theory developed in the course of our project work. Moreover we will actively involve them in research projects concerning the numerical approximation of the solutions, or flows, which correspond to those one dimensional nonlinear processes which fall under the umbrella of our project work.