A common Lyapunov function for a differential inclusion with matrices having small commutators

Abstract

Let A1 and A2 be asymptotically stable matrices. We consider switched continuous time linear systems described by the differential inclusion ẋ(t)={y(t):y(t)=Ax(t),A∈{A1,A2}}. The paper demonstrates that a common quadratic Lyapunov function exists for such systems, provided the commutator [A1,A2]=A1A2−A2A1 has a sufficiently small norm. Our results generalize the ones of Narendra and Balakrishnan (1994).