Existence and approximate controllability for random functional differential equations with finite delay

Abstract

This study examines second-order equations with delays, which frequently arise in various scientific and engineering applications. Within Banach spaces, these equations introduce unique challenges and opportunities for analysis and control. By exploring the existence and approximate controllability of solutions, the research enhances the understanding of dynamical systems with delayed feedback. Using mathematical tools such as cosine family theory and the Leray-Schauder theorem, it establishes rigorous conditions for solution existence, contributing to both theoretical and practical advancements. Additionally, the study incorporates empirical validation through a practical example, offering insights into the real-world behavior of these equations. This empirical analysis bridges the gap between theory and application, supporting the development of effective control strategies and engineering solutions. Ultimately, this research deepens the understanding of complex dynamical systems with delays and provides valuable
contributions to both theoretical progress and practical implementation.