Stability of Planar Switched Systems: the Nondiagonalizable Case
Abstract
Consider the planar linear switched system
x
˙
(t)=u(t)Ax(t)+(1−u(t))Bx(t),
where
A
and
B
are two
2×2
real matrices, $x \in \R^2$, and
u(.):[0,∞[→{0,1}
is a measurable function. In this paper we consider the problem of finding a (coordinate-invariant) necessary and sufficient condition on
A
and
B
under which the system is asymptotically stable for arbitrary switching functions
u(.)
. This problem was solved in previous works under the assumption that both
A
and
B
are diagonalizable. In this paper we conclude this study, by providing a necessary and sufficient condition for asymptotic stability in the case in which
A
and/or
B
are not diagonalizable. To this purpose we build suitable normal forms for
A
and
B
containing coordinate invariant parameters. A necessary and sufficient condition is then found without looking for a common Lyapunov function but using "worst-trajectory'' type arguments.