Abstract
This study is dedicated to a comprehensive exploration aimed at advancing our understanding of stability within dynamic systems. The focus is particularly on the intricate domain of delayed systems characterized by gapped gamma distributions. The primary objective of this investigation revolves around evaluating the pragmatic application and efficacy of Jensen's integral inequality in combination with the powerful analytical tools provided by Linear Matrix Inequalities (LMIs). This evaluation is crucial for rigorously assessing exponential stability within these complex systems. Central to our investigative framework is the strategic deployment of augmented Lyapunov functions. These functions play a crucial role in unraveling the intricate stability properties of delayed systems featuring gapped gamma distributions, allowing for a nuanced examination of their inherent stability characteristics under various conditions. The mathematical formulation crafted in this exploration intricately captures the interplay between the distinctive attributes of the gapped gamma distribution and the complex dynamics of the loop traffic flow model within the overarching delayed system. This interconnection serves as the fundamental basis for the stability analysis, providing insights into the interdependence of these key elements. The noteworthy contribution of this study lies in the systematic construction of a robust analytical framework explicitly tailored for stability assessment. A comprehensive investigation is undertaken to elucidate critical aspects, including the convergence rate and the attainment of asymptotic stability within the considered delayed system. Additionally, a dedicated simulation section, focusing on Vehicle Active Suspension Control, has been incorporated to further validate and showcase the applicability of the proposed methodology.