CONTROLLABILITY AND COMPUTATION OF CONTROL FOR INFINITE DIMENSIONAL FRACTIONAL DIFFERENTIAL SYSTEMS

Abstract

Controllability of dynamical systems governed by fractional evolution equations has emerged as a comprehensively researched area in the scientific community. Controllability is an intrinsic and qualitative property of a dynamical system that enables the dynamical system to behave in a desired way between two arbitrary initial and final states within the stipulated time corresponding to adequate control functions. Control theory, an area of mathematics and engineering that crosses disciplinary boundaries, is appreciably used in various fields, including robotics, automotive, aerospace, electrical and computer engineering, image processing, biomathematical modeling, and fluid dynamics. This thesis focuses on studying the existence of solution and controllability results of various fractional evolution equations in infinite dimensional spaces. In addition to analyse the fractional differential systems for controllability results, it also thrives to compute the control for approximately controllable fractional differential models. The investigation is carried out both analytically and numerically with graphical demonstrations. Throughout the thesis, the studied fractional differential systems are governed by the two classical fractional derivatives, Caputo and Riemann-Liouville. The motivation behind studying the impulsive and fractional differential systems, and the impact and importance of the results obtained are discussed further. The research work documented in this thesis is an interesting blend of the existence of solutions, approximate controllability, and control computation for linear and nonlinear deterministic fractional differential systems with and without the presence of non-instantaneous impulses. To establish the existence of solutions and controllability results, the use of C0-semigroup, fractional semigroup, fractional resolvent, and fixed point theory has been taken into account, involving several other xi xii techniques. To compute the control for approximately controllable fractional order systems, the theory of linear operators, ill-posed problems, and the method of Tikhonov regularization have been employed. Some examples are included to illustrate the methodology and analysis performed. Several real-life phenomena, suitably modeled in terms of impulsive fractional evolution equations, can be analyzed using the results obtained. The organization of the thesis is as follows: Chapter 1 briefly introduces mathematical control theory, ill-posed problems, regularization theory, and their related literature survey stimulating the study presented in this thesis. Chapter 2 provides a quick referral to some fundamental concepts and definitions from control theory, semigroup theory, ill-posed problems, regularization theory, fractional calculus, and nonlinear functional analysis, which are beneficial for the smooth study of the thesis. Chapter 3 analyzes the infinite dimensional abstract nonlinear fractional control systems governed by Riemann-Liouville fractional derivatives. The existence of mild solutions is deduced under the Lipschitz continuity of the involved nonlinear term (depending on the state and the control parameter) using semigroup theory and a fixed point approach. The system is proved approximately controllable using Cauchy convergence through iterative and approximate technique. Lastly, an application is provided as an example to support the proposed methodology.