STUDY LEAVE SEMINAR Speaker: Dr Björn Rüffer, CARMA, The University of Newcastle Title: Nonlinear Stability Analysis to Douglas-Rachford Location: Room SR192, SR Building (Callaghan Campus) The University of Newcastle Time and Date: 10:00 am, Thu, 24th Jan 2019 Abstract: During my study leave in 2018 I have applied nonlinear stability analysis techniques to the Douglas-Rachford Algorithm, with the aim of shedding light on the interesting non-convex case, where convergence is often observed but seldom proven. The Douglas-Rachford Algorithm can solve optimisation and feasibility problems, provably converges weakly to solutions in the convex case, and constitutes a practical heuristic in non-convex cases. Lyapunov functions are stability certificates for difference inclusions in nonlinear stability analysis. Some other recent nonlinear stability results are showcased as well. [Permanent link] CARMA SEMINAR Speaker: Dr Björn Rüffer, CARMA, The University of Newcastle Title: Separable Lyapunov functions for monotone systems Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 4:00 pm, Thu, 24th Sep 2015 Abstract: We consider monotone systems defined by ODEs on the positive orthant in $\mathbb{R}^n$. These systems appear in various areas of application, and we will discuss in some level of detail one of these applications related to large-scale systems stability analysis. Lyapunov functions are frequently used in stability analysis of dynamical systems. For monotone systems so called sum- and max-separable Lyapunov functions have proven very successful. One can be written as a sum, the other as a maximum of functions of scalar arguments. We will discuss several constructive existence results for both types of Lyapunov function. To some degree, these functions can be associated with left- and right eigenvectors of an appropriate mapping. However, and perhaps surprisingly, examples will demonstrate that stable systems may admit only one or even neither type of separable Lyapunov function. [Permanent link]