Evolution equations: input functions & stability
Final Report Abstract
Evolution equations describe the temporal development of a dynamical system driven by an initial state and a given input. In this project, we investigated evolution equations on an infinite-dimensional state space with the following two features: We allowed 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 were 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 were a main motivation for the project. Former results for related control problems required 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, we were faced with severe theoretical difficulties, e.g., the fact that the translation operator might not be strongly continuous anymore. Surprisingly, this even left 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. 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 was 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 lay on the space of bounded functions, which are of high relevance in application, but mathematically quite challenging. The project consisted of two subprojects: parabolic equations and hyperbolic equations. In both subprojects, as most of questions were even open for the linear case, we first of all and mainly focussed on this case. However, we also treated bilinear hyperbolic problems. A major novelty of this project is that we combined 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.
Publications
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Integral input-to-state stability of unbounded bilinear control systems. Mathematics of Control, Signals, and Systems, 34(2), 273-295.
Hosfeld, René; Jacob, Birgit & Schwenninger, Felix L.
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Characterization of Orlicz admissibility. Semigroup Forum, 106(3), 633-661.
Hosfeld, René; Jacob, Birgit & Schwenninger, Felix
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BIBO Stability for Funnel Control: Semilinear Internal Dynamics with Unbounded Input and Output Operators. Trends in Mathematics, 189-217. Springer Nature Switzerland.
Hastir, Anthony; Hosfeld, René; Schwenninger, Felix L. & Wierzba, Alexander A.
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Input-to-State Stability for Bilinear Feedback Systems. SIAM Journal on Control and Optimization, 62(3), 1369-1389.
Hosfeld, René; Jacob, Birgit; Schwenninger, Felix L. & Tucsnak, Marius
