Analysis and control of evolutionary plates and elastic structures

在这个项目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在自然和人造世界中看到。目前考虑的弹性演化PDE模型也可能受到系统外部的影响；例如，弹性体在某些边界界面上受到阻尼力的作用。因此，我们将分析的控制PDE模型可以想象地构成特征完全不同的PDE动力学的耦合；例如，冯·卡门动力板PDE与热过程耦合将产生耦合PDE系统，该系统具有双曲型和抛物线型PDE的现象学特征，但不能说是严格的双曲型或抛物线型。对于这样的线性和非线性演化方程，我们打算解决以下问题：(1)描述弹性结构与周围流体介质之间相互作用的耦合PDE模型的精确可控性和一般可达性。根据物理应用，弹性体的运动是通过流体边界控制的间接手段来控制的。(2)结构声系统的均匀稳定特性。在这种情况下，声波，室内的几何形状，耦合到弹性方程，该方程模拟了部分腔壁上的弯曲振动；这里的弹性分量将表现出一些可量化的阻尼，从弱粘性到超强开尔文-沃伊特阻尼。对于这些系统，在适当的几何假设下，我们打算研究弹性阻尼在整个复合系统中传播的可能性，在某种程度上，每个分量——表面上无阻尼的波以及结构上阻尼的弹性壁分量——以某种可识别的速率衰减。(3)关于一类非线性演化板PDE系统解或流的渐近行为的结果。特别地，我们将集中于那些“非梯度”的系统；也就是说，在相关的有限能量空间上没有可用的Lyapunov函数可以用来跟踪给定的，可能是非耗散的PDE的长时间行为。希望我们在这方面的工作最终将使这种非线性过程稳定为一个全局紧致吸引子。我们相信，这个项目的结果可以带来的好处远远超过它们作为数学学科贡献的内在价值。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发挥作用的特定腔室几何形状。因此，我们相信我们的研究努力可以给出那些结构声学几何形状的精确特征，这些几何形状将在很长一段时间内产生相对静止的内部声场。这样的几何情况可以避免，或者至少减少，主动工程控制噪声的需要。此外，我们在分析非线性演化板块动力学的长时间行为方面的计划工作可能具有直接的控制工程意义：如果我们发现，对于给定的非线性PDE系统，相应的流收敛于有限分形维数的全局紧致吸引子，那么可以想象，系统可以通过构造的有限维反馈进行主动数值控制。此外，本项目所产生的研究成果将作为我们为对数值偏微分方程和一般数学感兴趣的本科生制定和实施研究计划的基础。特别值得一提的是，pi将于2013年夏季运营内布拉斯加大学林肯分校“本科生研究经验”网站。在规范的一维环境下，通过商业计算机代数软件包的部分代理，我们将向本科生参与者教授在我们项目工作过程中发展起来的非线性理论的各个方面。此外，我们将积极地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.