On the Construction of Diagonal Lyapunov Functions for Linear Systems

The paper generalizes the concept of diagonal-type Lyapunov functions for arbitrary Holder vector p-norms, 1lesplesinfin. For p=2 this is equivalent with the usual quadratic form V(x)=xTDeltax, 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 majored the matrices defining the dynamics.