On the construction of diagonal Lyapunov functions for linear systems

The paper generalizes the concept of diagonal-type Lyapunov functions for arbitrary Wilder vector p-norms, 1 <= p <= alpha. For p =2 this is equivalent with the usual quadratic form vertical bar'(x) = x(T)Delta x, where Delta is a positive definite diagonal matrix, x is a real vector, and T denotes transposition. We provide concrete expressions for the Lyapunov function candidates that allow testing if a discrete- or continuous time system is asymptotically stable or not. These concrete expressions are constructed from the Perron or Perron-Frobenius eigenvectors of some matrices which either describe the system dynamics or majorize the matrices defining the dynamics.