Stability of Planar Switched Systems: The Linear Single Input Case

We study the stability of the origin for the dynamical system $\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where A and B are two 2 × 2 real matrices with eigenvalues having strictly negative real part, $x\in {\mbox{{\bf R}}}^2$, and $u(.):[0,\infty[\to[0,1]$ is a completely random measurable function. More precisely, we find a (coordinates invariant) necessary and sufficient condition on A and B for the origin to be asymptotically stable for each function u(.).

The result is obtained without looking for a common Lyapunov function but studying the locus in which the two vector fields Ax and Bx are collinear. There are only three relevant parameters: the first depends only on the eigenvalues of A, the second depends only on the eigenvalues of B, and the third contains the interrelation among the two systems, and it is the cross ratio of the four eigenvectors of A and B in the projective line CP1 . In the space of these parameters, the shape and the convexity of the region in which there is stability are studied.

This bidimensional problem assumes particular interest since linear systems of higher dimensions can be reduced to our situation.

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