QUADRATIC FORMS AS LYAPUNOV FUNCTIONS IN THE STUDY OF STABILITY OF SOLUTIONS TO DIFFERENCE EQUATIONS

A system of linear autonomous difference equations x(n + 1) = Ax(n) is considered, where x 2 R-k, A is a real nonsingular k x 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 V-m = V(A(m)x) -V(x) = W(x) holds if and only if mu(i)mu(j) not equal 1 (i = 1, 2 ... k; j = 1, 2 ... k) where mu 1, mu 2, ... , 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.