Liapunov’s characterization of stable matrices. A Liapunov function for x’ = Ax

Abstract

Although Liapunov did not consider difference equations, what we do here is the exact analog of what Liapunov did for linear differential equations. In the context of differential equations a matrix is said to be stable if ${e^{At}} \to 0$ as $t \to \infty$, and for difference equations Aⁿ is the analog of e^At.