Is exponential stability achievable in singular perturbed delayed systems with time-varying parameters? A comprehensive analysis.

The present article conducts an investigation into the phenomenon of exponential stability within singular perturbed delayed systems, incorporating time-varying parameters. Singularly perturbed systems serve as essential tools in modeling intricate systems characterized by multiple time scales, wherein one subsystem exhibits significantly faster evolution than the others. The presence of small delays introduces complexities, influencing both state derivatives and delays, further accentuating the intricacies of the system. Drawing upon the principles of singular perturbation theory, the article introduces a novel approach to analyzing the stability of these complex systems, eschewing the conventional assumption of exponential stability in the fast subsystem. Within the scope of this study, we propose a rigorous stability analysis, utilizing Linear Matrix Inequality (LMI) methods, while considering time-varying parameters that exert substantial influence on the system's dynamics. The proposed methodology enables the exploration of system stability beyond conventional assumptions, imparting valuable insights into the behavior of singular perturbed delayed systems amidst varying conditions. Through extensive numerical simulations, the effectiveness and robustness of the approach are validated, illuminating the stability properties of these intricate systems. Comparative studies with existing techniques, which assume exponential stability in the fast subsystem, demonstrate the distinct advantages and uniqueness of the presented approach. The findings underscore the significance of accounting for time-varying parameters in achieving a comprehensive understanding of the exponential stability inherent in singular perturbed delayed systems. This research makes substantial contributions to the field of system stability analysis, particularly in the context of singular perturbed delayed systems featuring time-varying parameters. The originality of our approach lies in introducing a comprehensive analysis framework that overcomes the limitations of existing methodologies. By integrating a novel stability analysis method based on Linear Matrix Inequalities (LMIs), we offer a fresh perspective on achieving exponential stability in such complex systems. Significantly, our work addresses a critical gap in current literature by challenging the assumption of exponential stability in the fast subsystem, a key feature of singularly perturbed systems. Through a meticulous examination of time-varying parameters, we unveil their profound impact on system dynamics, thus enriching the understanding of stability behaviors. The potential real-world applications of our findings span diverse fields, ranging from engineering to mathematical modeling. Performance metrics are a key focal point of our research. Numerical simulations employing our proposed LMIs serve as a robust benchmark, demonstrating the superior stability achieved in comparison to existing methods. This performance-driven evaluation ensures the practical applicability and reliability of our analysis approach across various scenarios.