Bridging theoretical and applied gaps in the analysis and control of neutral fractional-order systems using advanced controllability and numerical analysis methodologies

The objective of this article is to create a comprehensive mathematical formula for the analysis and control of neutral fractional-order systems that are characterized by state-dependent delays and periodic coefficients. Some of the issues that this article will address involve a variety of important area, including, but not limited to, the existence, uniqueness, stability and controllability of neutral fractional-order systems and will develop efficient numerical methods. This study will also integrate recent developments in the fractional calculus with the application of develop sufficient conditions for asymptotic stability criteria for neutral fractional-order systems using the fractional Lyapunov-Krasovskii functional and an Algorithm of Linear Matrix Inequalities. The use of a predictor-corrector method with adaptive step size control and cubic spline interpolation with state-dependent delays will further aid in providing a theoretical basis for the development of adequate control of neutral fractional-order systems. Extensive numerical simulations and theoretical validation of the methods discussed in this paper have shown substantial improvements in control accuracy and computational efficiency when applied to biomedical engineering and smart grids.