Application of machine learning frameworks in the controllability study of infinite-delay neutral integro-differential equations

Abstract The article delves into the controllability of second-order neutral functional integro-differential systems with infinite delay, which is a combination of neutrality, memory effects, and integral terms causing significant analytical challenges. We focus on the stochastic approach and the random fixed point theorem for set-valued operators to show that mild random solutions exist and also how randomml:mness can make the problem easier to solve. We then present rigorous controllability results, which are complicated by the presence of unbounded delay and impulsive effects. Theoretical contributions aside, the paper also demonstrates the use of such controllable systems to generate reliable simulated trajectories that are supportive of data-driven applications such as neural-network-based forecasting models. Thus, the integration confirms the practical relevance of the developed theory and its potential to solve real-world problems via advanced mathematical modeling. The primary objective of this study is to establish existence and controllability results for such systems within a unified stochastic framework. Using a random fixed-point theorem for set-valued operators, sufficient conditions for the existence of mild random solutions and controllability are derived. The novelty of the work lies in the combined treatment of infinite delay, neutrality, impulsive effects, and stochastic perturbations, together with its application to data-driven and machine-learning-based modeling.