Functional analytic methods for evolution equations

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。我们研究了这类方程解的定性性质，重点是稳定性、收敛性和控制理论性质。我们的方法是基于算子半群理论、谱理论、变换方法和不动点定理。在一个项目中，我们研究了周期轨道邻域的非线性抛物型边值问题，目标是不变流形及其吸引性。利用多项式稳定性的新理论结果，我们进一步研究了可能与抛物型问题耦合的线性双曲型问题的反馈稳定性。适定线性系统理论为这类问题和其他控制理论问题提供了一个强有力的框架。我们希望在第三个项目中为具有控制和观测的半线性系统发展相应的理论。

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.