Complete characterization of quadratic Lyapunov functions for planar systems in LMI regions

This paper discusses the problem of finding a common quadratic Lyapunov function (CQLF) for planar systems in linear matrix inequality (LMI) regions. The main contribution of this paper is to give a complete characterization of quadratic Lyapunov functions (QLFs) for a given planar system, which is stable in an LMI region, and obtain a necessary and sufficient condition for such a class of planar systems to share a CQLF in the LMI region. The results are not only verifiable but also constructive, which can be realized by computer easily.