Generalized Lyapunow functions and semialgebraic geometry

The development of novel powerful computational tools plays an increasingly important role in science and engineering. Recent developments in computational semialgebraic geometry, in particular in semidefinite programming and sum of squares techniques (SDP-SOS techniques), have led to new and very promising algorithms for the analysis of dynamical systems in control. However, there is still a significant number of unsolved problems when using SDP-SOS techniques for the analysis of dynamical systems. Firstly, SDP-SOS techniques are currently not applicable to real world applications, because these methods only allow to address small size problems. Secondly, many analysis methods deal mainly with stability-type analysis problems. For other problems, not so many methods based on SDP-SOS techniques have been so far developed. For example, a very important but unaddressed problem with many applications in science and technology is the analysis of complex dynamical systems driven by oscillatory external signals. The main goal of the proposed research project is to address the above mentioned problems and to develop novel analysis methods which are applicable to real world problems. The proposed research project is twofold: In a first step, novel analysis methods based on generalized Lyapunov functions and SDP-SOS techniques are developed. These methods allow in particular the analysis of complex dynamical systems which are driven by oscillatory external signals. In a second step, the efficiency of the analysis methods is increased such that they are applicable to real world problems. Therefore, the proposed research project is focused on the interface between control and computation with the goal to provide scientists and control engineers with sophisticated computational tools to enable a better understanding of complex dynamical systems in areas like internet congestion control, quantum control, or gene regulatory networks.