Contraction Analysis of Monotone Systems via Separable Functions

In this paper, we study incremental stability of monotone nonlinear systems through contraction analysis. We provide sufficient conditions for incremental asymptotic stability in terms of the Lie derivatives of differential one-forms or Lie brackets of vector fields. These conditions can be viewed as sum- or max-separable conditions, respectively. For incremental exponential stability, we show that the existence of such separable functions is both necessary and sufficient under standard assumptions for the converse Lyapunov theorem of exponential stability. As a by-product, we also provide necessary and sufficient conditions for exponential stability of positive linear time-varying systems. The results are illustrated through examples

Topological Dynamics via Structured Koopman Subsystems

This thesis deals with the interplay of quotient systems of a topological dynamical system and subsystems of its corresponding Koopman system. It begins with a historical „prelude“ (in German) where biographical aspects of the involved mathematicians are highlighted. In Chapter 1 topological dynamical systems and their corresponding Koopman systems are introduced and the correspondence of quotient systems and subsystems is explained. Chapter 2 is devoted to the simplest subsystem of a Koopman system, the fixed space. A dynamical description of the corresponding quotient system of the dynamical system is derived via a hierarchy of transfinite orbits. In particular, this leads to the characterization of a one-dimensional fixed space. In Chapter 3 the Lyapunov algebra is defined which is generated by so-called Lyapunov functions. Its properties, special cases and its connection to the generalized recurrent set are discussed. Also algebras which are generated by a single Lyapunov function are considered and extended Lyapunov functions are introduced. Finally, decompositions of the state space obtained by the Lyapunov algebra are studied