It is proved in this paper that the existence of a delay-dependent suitable Lyapunov function is a necessary and sufficient condition for a discrete-time fully nonlinear time-delay system, with given delays digraph, to be globally asymptotically stable. The same result is provided for the input-to-state stability. The less is the number of edges in the delays digraph, the less is the number of inequalities that are involved in the provided necessary and sufficient Lyapunov conditions. The case of arbitrary time-varying time delays, with no constraints as long as bounded, is covered as a special case.

中文翻译：

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具有约束时滞和时滞相关李雅普诺夫函数的离散时间系统

本文证明了时滞相关的合适李雅普诺夫函数的存在是具有给定时滞有向图的离散时间全非线性时滞系统全局渐近稳定的充分必要条件。为输入到状态的稳定性提供了相同的结果。延迟有向图中的边数越少，所提供的充分必要和充分的李雅普诺夫条件中所涉及的不等式的数量就越少。任意时变时间延迟的情况，只要有界就没有约束，作为一种特殊情况被涵盖。