A note on stability conditions for planar switched systems

Balde, Moussa; Boscain, Ugo; Mason, Paolo

A note on stability conditions for planar switched systems

Authors

Balde, Moussa

Boscain, Ugo

Mason, Paolo

Description

This paper is concerned with the stability problem for the planar linear switched system $\dot x(t)=u(t)A_1x(t)+(1-u(t))A_2x(t)$, where the real matrices $A_1,A_2\in \R^{2\times 2}$ are Hurwitz and $u(\cdot) [0,\infty[\to\{0,1\}$ is a measurable function. We give coordinate-invariant necessary and sufficient conditions on $A_1$ and $A_2$ under which the system is asymptotically stable for arbitrary switching functions $u(\cdot)$. The new conditions unify those given in previous papers and are simpler to be verified since we are reduced to study 4 cases instead of 20. Most of the cases are analyzed in terms of the function $Γ(A_1,A_2)={1/2}(\tr(A_1) \tr(A_2)- \tr(A_1A_2))$.
9 pages, 3 figures