Research

Mathematical Sciences

Title :

Mathematical Analysis and Control Synthesis of Fractional-Order Differential Equations

Area of research :

Mathematical Sciences

Focus area :

Applied Mathematics, Control Theory

Principal Investigator :

Dr. Arthi Ganesan, PSGR Krishnammal College For Women, Coimbatore, Tamil Nadu

Timeline Start Year :

2023

Timeline End Year :

2026

Contact info :

Details

Executive Summary :

Fractional differential equations (FDEs) theory has emerged as a probable framework for modeling performance, due to its prominence on processes of coordination and control in movement systems. The theory describes the behaviors of complex and nonlinear phenomena in mathematics and physics, but it also has a long and rich tradition of classifications in engineering, biology and so forth. The research in this area is extremely important because of its wide range of applications in many areas of science such as aerospace, defense, robotics and other physical phenomena. On the other side, the existence and uniqueness of FDEs is a subject of both practical and theoretical importance, because it is essential to find the solution of differential equations to study the various qualitative behaviors. The existence and uniqueness of FDEs is longtime standing and has a rich history in the literature. However, there are some limitations on the existing works in the literature, for example, the qualitative behaviors like existence and uniqueness, controllability, and stability results for the fractional-order differential equations, especially for coupled FDEs, time-delay FDEs, FDEs with impulsive effects and stochastic disturbances by Levy noise are not yet fully developed. The focus of this project is to develop some new methodologies, mathematical techniques and numerical algorithms to deal the issues like time delay, perturbation, noise, impulses and other modeling problems like reliability, etc. Only few researchers have initialized the above issues, however, much work remains to be done in these cases. This is to be achieved partially through this project. So in this regard the project is an important contribution to the development of the field. In general, the well-posedness of all dynamical systems can be studied by different methods in both theoretically and numerically. The existence, uniqueness and controllability of linear and nonlinear FDEs are obtained by various types of fixed point method with different kinds of fractional derivatives like Caputo, Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu. The stability of FDEs is analyzed based on Gronwall inequality, Lyapunov method, Mittag-Leffler approach and linear matrix inequality technique, etc. The proposed research topic deals with the above mentioned analysis which could be useful in many practical systems especially in knowledge based systems. Since the non-integer differentiation has become a more and more suitable tool for modeling and controlling the behaviours of physical systems from diverse branches of physical systems. So, the proposed new control designs based on the above mentioned strategies will be significantly useful in practical situations. It broadens the applied research field of FDEs with various types by academic conferences, publishing research results and expanding the influence of fractional-order control all over the world.

Total Budget (INR):

18,42,016

Organizations involved