On stability of relaxive systems described by polynomials with time-variant coefficients

摘要

The problem of global asymptotic stability (GAS) of a time-variant m-th order difference equation y(n)=a/sup T/(n)y(n-1)=a/sub 1/(n)y(n-1)+/spl middot//spl middot//spl middot/+a/sub m/(n)y(n-m) for /spl par/a(n)/spl par//sub 1/>1 was addressed, whereas the case /spl par/a(n)/spl par//sub 1/=1 has been left as an open question. Here, we impose the condition of convexity on the set C/sub 0/ of the initial values y(n)=[y(n-1),...,y(n-m)]/sup T/ /spl epsiv/R/sup m/ and on the set A/spl epsiv/R/sup m/ of all allowable values of a(n)=[a/sub 1/(n),...,a/sub m/(n)]/sup T/, and derive the results from [1] for a/sub i//spl ges/0, i=1,...,n, as a pure consequence of convexity of the sets C/sub 0/ and A. Based upon convexity and the fixed-point iteration (FPI) technique, further GAS results for both /spl par/a(n)/spl par//sub i/>1, and /spl par/a(n)/spl par//sub 1/=1 are derived. The issues of convergence in norm, and geometric convergence are tackled.