  This work studies the problem of searching for homogeneous polynomial
Lyapunov functions for stable switched linear systems. Specifically, we show an
equivalence between polynomial Lyapunov functions for systems of this class and
quadratic Lyapunov functions for a related hierarchy of Lyapunov differential
equations. This creates an intuitive procedure for checking the stability
properties of switched linear systems and a computationally competitive
algorithm is presented for generating high-order homogeneous polynomial
Lyapunov functions in this manner. Additionally, we provide a comparison
between polynomial Lyapunov functions generated with our proposed approach and
polynomial Lyapunov functions generated with a more traditional sum-of-squares
based approach.
