Delay-dependent exponential stability criteria for nonlinear time-varying discrete systems with multiple time delays
The following nonlinear discrete time-delay system is considered: \[ x(n+1)= \sum^p_{i= 1}A_i(x,1\dots p)x(n- h_i), \] \[ x(n)= \phi(n),\quad -h_p\leq n\leq 0 \] \((0\leq h_1< h_2<\cdots< h_p)\). It is assumed that the nonlinearity \(A\) is bounded by \[ \| A_i(.,.)\|\leq b_i(n),\quad\forall i,n. \] Then if there exists \(J\) and \(0<r<1\) such that \[ \sum^p_{i=1} {1\over r^{1+ h_i}} b_i(n+ J+ h_i)< 1,\quad\forall n>0 \] it is shown that the equation is stable with decay rate at least \(r\). A simple example is given.
scientific article; zbMATH DE number 1028993