WG1 - C0 - semigroups and beyond
Overview
The theory of C0-semigroups, time-continuous linear dynamical systems, is a useful tool to study evolution equations. There is a one-to-one correspondence between these objects and well-posed abstract Cauchy problems on Banach spaces. In many situations, however, the notion of C0-semigroups is not satisfactory, and depending on the framework there are a couple of competing approaches. This Working Group will generalize the theory of semigroups by relaxing the classical theory and studying some novel continuity concepts. These are, for example, important to obtain the well-posedness on Lor continuous function spaces on infinite networks.
Especially important for applications will be the study of systems driven by two competing processes: (Additive) perturbation theory asks for conditions on a linear operator B such that the sum A + B (defined in an appropriate sense) generates a C0-semigroup, whenever A does. In this case, a formula (or an approximation) relating the perturbed and the unperturbed semigroups is also desired.
Tasks
- Generalize the theory of C0-semigroups by modifying analytic or algebraic features.
- Study examples which are appropriate for the theory (in collaboration with W2 and W3).
- Obtain stability results for the semigroup in terms of spectral properties of the generating operator.
- Develop perturbation results for the semigroups discussed above.
- In collaboration with WG5 develop appropriate numerical methods for applications.
- Use perturbation results for questions regarding control of the system.
