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

Ignatyev, Alexander O. ; Ignatyev, Oleksiy

Electronic journal of differential equations, Tome 2011 (2011)

Résumé

A system of linear autonomous difference equations $x(n+1)=Ax(n)$ is considered, where $x\in \mathbb{R}^k, A$ is a real nonsingular $k\times k$ 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 $\mu_i\mu_j\neq1 (i=1, 2 \dots k; j=1, 2 \dots k)$ where $\mu_1,\mu_2,\dots,\mu_k$ are the roots of the equation $\det(A^m-\mu I)=0$. A number of theorems on the stability of difference systems have also been proved. Applying these theorems, the stability problem of the zero solution of the nonlinear system $x(n+1)=Ax(n)+X(x(n))$ has been solved in the critical case when one eigenvalue of a matrix $A$ is equal to minus one, and others lie inside the unit disk of the complex plane.

Détail

Classification : 39A11, 34K20
Keywords: difference equations, Lyapunov function