Stability analysis in continuous and discrete time

Abstract

The relation between continuous time systems and discrete time systems is the main topic of this research. A continuous time system can be transformed into a discrete time system using the Cayley transform. In this process the generator of the semigroup is mapped to a difference operator, the cogenerator. Stability analysis plays a central role in the study of the relation between continuous and discrete time systems. Do stable continuous time systems correspond to stable discrete time systems? This is the main question of this dissertation. For many stable continuous time systems the corresponding discrete time systems are stable as well. In Banach spaces however, several examples are known of stable semigroups where the corresponding cogenerators have unsta- ble power sequences. In Hilbert spaces no such examples are known. It remains an open problem whether for every stable continuous time system the corre- sponding discrete time system is stable as well. This dissertation addresses the main question in three ways. First, a growth bound for the cogenerator is provided for exponentially stable semigroups in Hilbert spaces. Using Lyapunov equations it is shown that for such semigroups the power sequence of the corresponding cogenerator cannot grow faster than ln(n). Second, we extend the class of stable continuous time systems for which the corresponding discrete time systems are stable as well. For this, the notion of Bergman distance is introduced. The Bergman distance defines a metric for semigroups and a metric for power sequences. If the Bergman distance is finite, the two semigroups have the same stability behaviour. This holds for two power sequences as well. Furthermore, the Bergman distance is preserved by the Cayley transform. This enables us to extend this class. Third, the inverse of the generator is taken into account in the stability analysis. For exponentially stable semigroups on Banach spaces similarity is shown between the growth of the semigroup of the inverse and the growth of the cogenerator. For bounded semigroups on Hilbert spaces it is shown that if the semigroup generated by the inverse is bounded, the growth of the cogenerator is bounded as well.