Research on Controllability of Partial Differential Equations

偏微分方程的可控性研究

• 批准号: 0808130
• 负责人: Oleg Emanouilov
• 金额: $ 10.86万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0808130&HistoricalAwards=false
• 关键词: Research; Controllability; Partial; Differential; Equations

The focus of the proposed research is to analyze global controllabilityand stabilizability properties of several different types of physical processes described by nonlinear partial differential equations (PDEs). The control acts through actuators located on the boundary or concentrated in some subdomain. The primary goals are to establish the theoretical possibility of global controllability, and to obtain an explicit construction of the control strategy. The latter will be in the form of asymptotic formulae or by reduction of original nonlinear problem to a much simpler controllability problem for linear PDEs or integral equations capable of numerical solution. For example, if a physical process is described by anintegrable nonlinear PDE , we reduce the question of controllability to the analysis of a controllability problem for the Gelfand-Levitan-Marchenko integral equation and then try to solve this problem asymptotically. For a physical process described by a nonintegrable PDE we plan to adapt Coron's return method, which allows construction of asymptotic solutions to a large class of controllability problems with control distributed over the wholeboundary or just a part of the boundary. The main analytical tools will be the inverse scattering technique, microlocal analysis, the Carleman typeestimates and soliton theory.Understanding the control and stabilization of physical processes described by nonlinear partial differential equations is becoming increasingly important due to the rapid advances in material science, physics, aircraft design, fiber-optic communication systems. The nonlinear partial differential equations used as models are mathematically challenging and widely used in physics and engineering. The global controllability problems for these equations are motivated by practical real-world applications which include suppression of turbulence, control of waves in channels, control of plasma flow, design of a new generation of amplifiers which will allow the preservation of the shape of optical pulse in fiber-optical communication systems.

研究的重点是分析几种不同类型的非线性偏微分方程（PDE）描述的物理过程的全局可解性和可稳性。控制通过位于边界上或集中在某个子域中的致动器起作用。 主要目标是建立全局可控性的理论可能性，并获得一个明确的结构的控制策略。 后者将采用渐进公式的形式，或者通过将原始非线性问题简化为能够数值求解的线性偏微分方程或积分方程的更简单的可控性问题。例如，如果一个物理过程是由一个可积的非线性偏微分方程描述，我们减少了可控性问题的分析的可控性问题的Gelfand-Levitan-Marchenko积分方程，然后试图解决这个问题的渐近。 对于一个物理过程所描述的不可积偏微分方程，我们计划适应科龙的回报方法，它允许建设的渐近解的一大类可控性问题的控制分布在整个边界或只是一部分的边界。主要的分析工具是逆散射技术、微局域分析、Carleman型估计和孤子理论。由于材料科学、物理学、飞行器设计、光纤通信系统的快速发展，理解由非线性偏微分方程描述的物理过程的控制和稳定变得越来越重要。作为模型的非线性偏微分方程在数学上具有挑战性，在物理和工程中有着广泛的应用。这些方程的全局可控性问题的动机是实际的现实世界中的应用，其中包括抑制湍流，在通道中的波的控制，等离子体流的控制，设计的新一代的放大器，这将允许在光纤通信系统中的光脉冲的形状的保存。