On quadratic Lyapunov functions

A topological structure, as a subset of [0,2(pi))/sup L/*R/sub +//sup n-1/, is proposed for the set of quadratic Lyapunov functions (QLFs) of a given stable linear system. A necessary and sufficient condition for the existence of a common QLF of a finite set of stable matrices is obtained as the positivity of a certain integral. The structure and the conditions are considerably simplified for planar systems. It is also proved that a set of block upper triangular matrices share a common QLF, if each set of diagonal blocks share a common QLF.