In this thesis we present some results about analytic machines regarding computability over Q and R, solutions of differential equations, and the stability problem of dynamical systems. We first explain the machine model, which is a kind of Blum-Shub-Smale machine enhanced by infinite convergent computations. Next, we compare the computational power of such machines over the fields Q and R showing e.g. that finite computations with real numbers can be simulated by infinite converging computations on rational numbers, but the precision of the approximation is not known during the process. Analytic computations over R are strictly more powerful than over Q. Our attention is then shifted to ordinary differential equations (ODEs) where we establish sufficient criteria for the computability of their solutions within our model. We investigate dynamical systems described by ODEs and show the undecidability of a class of stability problems for dynamical systems.

We consider the problem of diagnosis in complex systems with the example of tank ballast systems. It is shown that known approaches for diagnosis either restricting to quantitative or qualitative methods are not suitable. Therefore, we introduce a new approach combining quantitative and qualitative aspects. It does not renounce to complete information available in quantitative measurements as it is inevitably the case for qualitative methods that only use a classification of the measurements. Decisions for the diagnosis process however are made with the help of qualitative comparisons. Therefore, the approach is robust against noise. Finally, the performance of this new approach is demonstrated by many random tests.

Die vorliegende Arbeit schildert die wesentlichen Aspekte eines Projektes, dem die Aufgabe zugrundelag, ein Gesamtsystem von der Art einer `Werkbank' für eine repräsentative Anwendung, das symbolische Arbeiten mit mathematischen Formeln, prototypisch zu realisieren.