Quadratic Lyapunov functions for systems with state-dependent switching

In this paper, we consider the existence of quadratic Lyapunov functions for certain types of switched linear systems. Given a partition of the state-space, a set of matrices (linear dynamics), and a matrix-valued function A (x) constructed by associating these matrices with regions of the state-space in a manner governed by the partition, we ask whether there exists a positive definite symmetric matrix P such that A (x)^T P + PA (x) is negative definite for all x (t). For planar systems, necessary and sufficient conditions are given. Extensions for higher order systems are also presented.