 
Summary: To appear in SIAM J. Control Optim. 1
Liealgebraic stability criteria for switched systems
Andrei A. Agrachev
Steklov Math. Inst., Moscow, Russia
and S.I.S.S.A.{I.S.A.S., Trieste, Italy
agrachev@sissa.it
Daniel Liberzon
Coordinated Science Laboratory
Univ. of Illinois at UrbanaChampaign, U.S.A.
liberzon@uiuc.edu
January 19, 2001
Abstract
It was recently shown that a family of exponentially stable linear systems whose matrices gener
ate a solvable Lie algebra possesses a quadratic common Lyapunov function, which implies that the
corresponding switched linear system is exponentially stable for arbitrary switching. In this paper we
prove that the same properties hold under the weaker condition that the Lie algebra generated by given
matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact Lie group.
The corresponding local stability result for nonlinear switched systems is also established. Moreover,
we demonstrate that if a Lie algebra fails to satisfy the above condition, then it can be generated by
a family of stable matrices such that the corresponding switched linear system is not stable. Relevant
