Investigating Mild Solution and Optimal Control Results for Fractional-Order Semilinear Control System via Resolvent Operators

This paper investigates the existence of mild solutions and the derivation of optimal control results for a fractional integro-differential control system using resolvent operators and advanced operator theory. By employing mathematical tools such as the Banach Fixed Point Theorem, Gronwall's Inequality, and semigroup theory, the study addresses semilinear control systems governed by resolvent operators in the context of fractional-order dynamics. The paper establishes sufficient conditions for the existence and uniqueness of mild solutions under Lipschitz-type non-linearity and provides a framework for the analysis of optimal control strategies using minimizing sequences. Additionally, the work delves into the study of time-optimal control and time-dependent systems by defining appropriate transition times and controls within infinite-dimensional spaces. The contributions highlight the application of resolvent operators in complex dynamical systems, demonstrating the practical relevance of the derived results in engineering, biological models, and other scientific fields. Furthermore, the theoretical results are supplemented by examples that illustrate the applicability and significance of the findings in real-world control systems. This research not only extends the understanding of fractional-order systems but also provides a foundation for future studies on more complex non-linearities and control settings.