Evolution equations: input functions & stability

• Applicants: Professorin Dr. Birgit Jacob; Professor Dr. Felix Schwenninger
• Subject Area: Mathematics
• Term: from 2020 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 445241640

Evolution equations describe the temporal development of a dynamical system driven by an initial state and a given input. In this project, we investigate evolution equations on an infinite-dimensional state space with the following two features: We allow for a considerable unboundedness of the input operator and for non- standard spaces of input functions. A natural example for such a space, where standard methods are limited, is given by the essentially bounded functions. Our setting encompasses many of the common partial differential equations with boundary input and control. The latter being the main motivation for the proposed research. The state- of-the art for related control problems requires relatively nice boundary operators and spaces of input functions, such as square integrable functions, that may be hard to check. Moreover, the natural choice of norm for the inputs may not fall into these classes. When trying to lift the known results to these more general spaces, one is soon faced with severe theoretical difficulties, e.g., the fact that the translation operator might not be strongly continuous anymore. Surprisingly, this even reveals several open problems in the (abstract) linear theory, as for instance the question whether mild solutions of an linear boundary control problem with essentially bounded inputs are always continuous. These open problems are explicitly part of the project. These remaining difficulties also indicate that the approach to the control of infinite-dimensional evolution equations with respect to non-standard function spaces has only emerged recently. In contrast, the underlying finite-dimensional theory is well-known in the case of essentially bounded inputs, going back to Eduardo Sontag, is well- known. The aim of this project is to develop further the well- posedness as well as the internal/external stability, both with respect to non-standard classes of input functions. The main focus lies on the space of bounded functions, which are of high relevance in application, but mathematically quite intricate. The project consists of two subprojects: parabolic equations and hyperbolic equations. In both subprojects, as most of questions are even open for the linear case, we first of all and mainly focus on this case. However, our long term goal are nonlinearities, in particular, we treat bilinear hyperbolic problems and semilinear parabolic evolution equations. A major novelty of this project is that we combine tools from operator theory, infinite-dimensional systems theory, functional calculus for bounded analytic functions and function spaces in order to investigate well- posedness and input-to-state stability of infinite-dimensional evolution equations with unbounded input operators.

• DFG Programme: Research Grants