Stability of nonlinear discrete time-varying retarded systems (Q1573529)

In the past, the stability problem of discrete time-varying retarded systems was mainly treated via the use of Lyapunov functions, which in many cases is difficult to find. This paper establishes some verifiable sufficient conditions for asymptotic stability of discrete time-varying systems, without using the Lyapunov functions approach. The results of the paper are applicable to a general class of nonlinear systems and to systems with time delays, namely \[ x(k+1)= f(k,x(k), x(k-r)),\quad f(k,0,0)= 0 \] and \[ x(k+1)= f(k,x(k), x(k-1),\dots, x(k-r)),\quad f(k,0,0,\dots, 0)=0. \] For the second system the result is that if it satisfies the condition \[ \|f(k,x_1,x_2,\dots, x_{r+1})\|\leq \sum^m_{i=1} \alpha_i \prod^{r+1}_{j=1} \|x_j\|^{p_{ij}} \] with \(\alpha_i> 0\), \(p_{ij}> 0\) \((i= 1,2,\dots, m; j= 1,2,\dots, r+1)\) and \(p_{11}+\cdots+ p_{1r+1}< p_{21}+\cdots+ p_{2r+1}<\cdots< p_{m1}+\cdots+ p_{mr+1}\), then it is asymptotically stable if one of the two following conditions holds (a) \(p_{11}+ p_{12}+\cdots+ p_{1r+1}> 1\) and \(\alpha_1> 0\), (b) \(p_{11}+ p_{12}+\cdots+ p_{1r+1}\geq 1\) and \(\alpha_1< 1\).