Mathematics

We study the stability of the origin of the dynamical system: (1) \dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t), where A and B are two 2 x 2 real matrices with eigenvalues having strictly negative real part, x \in R^2 and u(.) : [0,\infty[ \to [0,1] is an arbitrary measurable control function. In particular we find a necessary and sufficient condition on A and B under which the origin of the system (1) is asymptotically stable for each function u(.). The result is obtained studying the locus in which the two vector fields Ax and Bx are collinear. There are only three relevant parameter: the first depends on the eigenvalues of A, the second on the eigenvalues of B, the third contains the interrelation between the two systems and it is the cross ratio of the four points in the projective line CP^1 that correspond to the four eigenvectors of A and B. In the space of this parameters we study the shape and the convexity of the stability domain. This bidimensional problem assumes particular interest since systems of higher dimension can be reduced to our situation.