Functional analytic methods for evolution equations

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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.