Quadratic forms as Lyapunov functions in the study of stability of solutions to difference equations

Abstract

A system of linear autonomous difference equations $x(n+1)=Ax(n)$ is considered, where $xin mathbb{R}^k$, $A$ is a real nonsingular $kimes k$ matrix. In this paper it has been proved that if $W(x)$ is any quadratic form and $m$ is any positive integer, then there exists a unique quadratic form $V(x)$ such that $Delta_m V=V(A^mx)-V(x)=W(x)$ holds if and only if $mu_imu_jeq1$ ($i=1, 2 dots k; j=1, 2 dots k$) where $mu_1,mu_2,dots,mu_k$ are the roots of the equation $det(A^m-mu I)=0$.

Keywords

• Difference equations
• Lyapunov function