A common Lyapunov function for a differential inclusion with matrices having small commutators. (30th December 2019) - Gil', Michael
Abstract: Let A 1 and A 2 be asymptotically stable matrices. We consider switched continuous time linear systems described by the differential inclusion x ̇ ( t ) = y ( t ) : y ( t ) = A x ( t ), A ∈ A 1, A 2 . The paper demonstrates that a common quadratic Lyapunov function exists for such systems, provided the commutator [ A 1, A 2 ] = A 1 A 2 − A 2 A 1 has a sufficiently small norm. Our results generalize the ones of Narendra and Balakrishnan (1994).