A Hurwitz Like Characterization of GUAS Planar Switched Systems

where the real matrices \(A_1 , A_2 \in \mathbb {R}^2\) are Hurwitz and u(.) : [0, +∞)→{0, 1} is a measurable function. We give a Hurwitz like characterization of globally uniformly asymptotically stable planar switched systems. Another contribution of this paper is a new version of the main result in Shorten and Narendra (Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for two stable second order linear time-invariant systems, in Proceedings of the 1999 American Control Conference (1999), pp. 1410–1414) using real algebraic geometry tools. This new approach gives a Hurwitz like characterization of switched systems which share a same strict or large common quadratic Lyapunov function and improves the main result in Balde et al. (Int J Control 82(10):1882–1888, 2009).