On quadratic stability of state-dependent planar switching systems

In this paper we consider the stability of a class of switched systems. Specifically, we ask the following question. Given a partition of the state space, and a set of linear dynamics, each of which are active in parts of the state space in a manner governed by the partition, does there exist a quadratic Lyapunov function for the resulting system? For planar systems with conic partitions and two dynamics, necessary and sufficient conditions are given for such a Lyapunov function. Examples are given to illustrate our results