Analysis and control of evolutionary plates and elastic structures

演化板块和弹性结构的分析与控制

• 批准号: 1211232
• 负责人: George Avalos
• 金额: $ 29.28万
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211232&HistoricalAwards=false
• 关键词: Analysis; control; evolutionary; plates; elastic

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.

该项目中的努力将集中在偏微分方程（PDE）模型的数学分析和控制上，这些模型描述了某些在自然和人造世界中可以看到的弹性动力学。目前考虑的弹性进化PDE模型也可能受到系统外部的影响。例如，弹性物体在某些边界界面上受到阻尼力。因此，我们将分析的管理PDE模型可以想象构成PDE动力学的耦合，而PDE动力学的性质却大不相同。例如，与热过程耦合的von Karman动力板PDE PDE会产生一个耦合的PDE系统，该系统表达了双曲线和抛物线PDE的现象学特征，但不能说是严格的屈服或抛物线。对于这种演化方程，线性和非线性，我们打算解决以下问题：（1）那些耦合的PDE模型的精确可控性和一般可及性能，这些模型描述了弹性结构与周围流体介质之间的相互作用。 与物理应用一致，弹性体的运动应由流体边界控制的间接手段控制。 （2）结构声系统的统一稳定性。在这种情况下，与腔室几何形状内部的声波耦合到弹性方程，该方程模拟了腔室壁的一部分上的弯曲振动。这里的弹性成分将显示一些可量化的阻尼度量，从弱粘稠到超强的开尔文 - 宣传阻尼。 对于这些系统以及在适当的几何假设下，我们打算调查弹性阻尼在整个复合系统中传播的可能性，以至于每个组件（稳定的未阻尼波以及结构上阻尼的弹性壁成分）的可能性是以某种可见的速度以某种可见的速度进行的。 （3）关于某些非线性进化板PDE系统的溶液或流动渐近行为的结果。特别是，我们将专注于那些“非毕业者”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，可以使用它来跟踪给定（可能是非疾病的PDE）的长时间行为。希望我们在这方面的工作将最终使这种非线性过程稳定为全球紧凑型吸引子。我们相信，该项目的结果可以带来超出其内在价值的好处，因为对数学学科的贡献。例如，在我们上述和预期的结构声学均匀衰减调查中，我们预计稳定性结果将急剧取决于正在发挥的特定室内几何形状。结果，我们认为我们的研究工作可以给出那些结构性声学几何形状的精确表征，这些几何形状将在长期以来引起相对静止的内部声学场。然后，这种几何情况可以想到，或者至少可以减少对声音噪声的主动工程控制的需求。此外，我们在分析非线性进化板动力学的长时间行为方面的预期项目可能具有直接的控制工程意义：对于给定的非线性PDE系统而言，是否应该发现相应的流量会融合到有限的分形尺寸的全球紧凑型吸引子，然后可以想象地通过构建的有限限制了数值的数值。此外，该项目产生的研究将作为我们将为对数值PDE感兴趣的大学生以及数学一般的大学生开发和实施研究计划的基础。 特别是，PIS将在2013年夏季运营内布拉斯加州林肯大学的“本科生研究经验”。在规范的一维环境中，并通过商业计算机代数软件包的部分代理，我们将教给我们的本科参与者，非线性理论在我们项目项目中开发的非线性理论的方面。此外，我们将积极参与有关解决方案或流量的数值近似的研究项目，这些研究与我们项目工作的伞下的那些一维非线性过程相对应。