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

The problem of global asymptotic stability (GAS) of a time-variant m-th order difference equation y(n)=aT(n)y(n-1)=a1(n)y(n-1)+···+am(n)y(n-m) for ||a(n)||1<1 was addressed, whereas the case ||a(n)||1=1 has been left as an open question. Here, we impose the condition of convexity on the set C0 of the initial values y(n)=[y(n-1),...,y(n-m)]T εRm and on the set AεRm of all allowable values of a(n)=[a1(n),...,am(n)]T, and derive the results from [1] for ai≥0, i=1,...,n, as a pure consequence of convexity of the sets C0 and A. Based upon convexity and the fixed-point iteration (FPI) technique, further GAS results for both ||a(n)||i<1, and ||a(n)||1=1 are derived. The issues of convergence in norm, and geometric convergence are tackled.