On BIBO stability of infinite-dimensional systems

A control system is bounded-input bounded-output stable (or short BIBO stable) if uniformly bounded inputs in time give rise to outputs bounded uniformly in time and furthermore there exists a uniform relative bound between the respective supremum norms.

For linear systems, this notion translates to boundedness of a linear operator with respect to the supremum norms. The problem of checking boundedness of such operators is omnipresent in functional analysis, but also has significance in fields such as harmonic analysis or control theory.

BIBO stability of linear systems described by finite-dimensional state-space representations is a well-studied notion with applications in signal processing and controller design. Turning to infinite-dimensional state-space representations – e.g. arising from modelling heat flows or vibrations in mechanical systems – the situation becomes more complex and even the question of a proper definition of BIBO stability itself is non-trivial.

The aim of this thesis is to fill this gap and provide a thorough study of BIBO stability for such infinite-dimensional state-space systems. Employing tools from operator and semigroup theory, functional analysis, and systems and control theory, we establish characterisations of BIBO stability and study for instance its behaviour under perturbations of the underlying systems.

Furthermore, we derive sufficient conditions implying BIBO stability for special subclasses of systems, such as for instance port-Hamiltonian systems. These play an important role in the mathematical modelling of energy-conserving physical systems in particular those featuring effects such as vibrations, flows or transports. In addition, we study the close interrelation between the L^∞ to L^∞ boundedness in BIBO stability and the corresponding one from L¹ to L¹ that arises from duality transformations.

Finally, we consider control-theoretic applications such as the role BIBO stability plays in the application of funnel control to relative-degree systems and its potential use within the setting of the Predictive Avatar Control and Feedback project aimed at mitigating effects of time-delays in telerobotics.