A Geometrical Treatment for Obtaining Necessary and Sufficient Conditions for Joint Quadratic Lyapunov Function Existence for State-Dependent, Switched Systems: A Two-Dimensional Case

The question of existence of joint quadratic Lyapunov functions (QLFs) for state-dependent, switched dynamical systems is given a preliminary geometrical treatment in this paper. The joint QLF problem for a switched system and a collection of regions defined by state vectors that determine when switching occurs consists of finding nonempty intersections of convex sets of QLFs. The existence of a joint QLF guarantees switched system stability. Necessary and sufficient conditions for the existence of a joint QLF are obtained for a two-dimensional problem