Abstract This note pays precise attention to derive the γ-exponential stability results of positive delayed systems via non-fragile control. By using Lyapunov–Krasovskii theory some necessary conditions are derived. Moreover, by using results from spectrum analysis the sufficient conditions for positive delayed systems with non-fragile control are analyzed. Our main intension is to locate the ideal application of this type of systems. In later part of this paper, the novel mathematical model which describes the life stages of Aedes Aegypti mosquitoes is proposed. In that model, the life shorting bacteria called Wolbachia is used as a biological control. By moderately thinking about the uncertain conditions in the biological control we implemented the non-fragile control. By applying the proposed results the γ-exponential stability of this population system is derived. Finally, some numerical examples are provided to expose the effectiveness of our predominant results.

中文翻译：

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具有不确定控制输入的延迟阳性系统的延迟依赖稳定性标准：在蚊媒疾病控制中的应用

摘要 本文着重通过非脆弱控制推导正延迟系统的γ-指数稳定性结果。利用李雅普诺夫-克拉索夫斯基理论推导出一些必要条件。此外，通过使用频谱分析的结果，分析了具有非脆弱控制的正延迟系统的充分条件。我们的主要意图是定位此类系统的理想应用。在本文的后半部分，提出了描述埃及伊蚊生命阶段的新数学模型。在那个模型中，称为沃尔巴克氏体的缩短寿命的细菌被用作生物控制。通过适度考虑生物防治中的不确定条件，我们实施了非脆弱性防治。通过应用所提出的结果，可以推导出该种群系统的 γ 指数稳定性。最后，提供了一些数值例子来展示我们主要结果的有效性。