Functional analytic methods for evolution equations

演化方程的泛函分析方法

• 批准号: 189767991
• 负责人: Professorin Dr. Birgit Jacob
• 金额: --
• 依托单位: Arbeitsgruppe Funktionalanalysis
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/189767991?language=en
• 关键词: Functional; analytic; methods; evolution; equations

Evolution equations describe the temporal development of a dynamical system with a given initial state and a given input. We study the qualitative behavior of the solutions of such equations focussing on stability, convergence and control theoretic properties. Our approach is based on the theory of operator semigroups, spectral theory, transform methods, and fixed point theorems. In one project we treat nonlinear parabolic boundary value problems in the neighborhood of a periodic orbit aiming at invariant manifolds and their attractivitity. Using new theoretical results on polynomial stability, we further investigate the feedback stabilizability of linear hyperbolic problems, possibly coupled withparabolic ones. The theory of well-posed linear systems gives a powerful framework for such and other control theoretic questions. We want to develop a corresponding theory for semilinear systems with control and observation in a third project.

演化方程描述了具有给定初始状态和给定输入的动力系统的时间发展。我们研究此类方程解的定性行为，重点关注稳定性、收敛性和控制理论特性。我们的方法基于算子半群理论、谱理论、变换方法和不动点定理。在一个项目中，我们处理周期性轨道邻域中的非线性抛物线边值问题，旨在解决不变流形及其吸引力。利用多项式稳定性的新理论结果，我们进一步研究了线性双曲问题（可能与抛物线问题相结合）的反馈稳定性。适定线性系统理论为此类和其他控制理论问题提供了强大的框架。我们希望在第三个项目中为具有控制和观察的半线性系统开发相应的理论。