Mathematical types of control system and controllability of nilpotenet systems

The aim of this thesis is to present a result of control system theory from the differential geometric point of view. Especially, controllability property of invariant control systems on nilpotent simply connected Lie groups is studied. This work consists of five chapters. In the first chapter, literature review, objective of the thesis and a brief information about the hypothesis are introduced. In the second chapter, manifold, tangent vector, tangent space and vector fields notions which are fundamentals in Control Theory are contained. Lie group, Lie algebra, their connection via exponential map and adjoint representation and Lie derivative are studied. Moreover, examples and some basic theorems are mentioned. In the third chapter, after general control systems and fundamental problems of control are defined, some control systems are examined for illustrations. Difficulty of controllability problem is indicated with the help of a basic example. Besides, many mathematicians have been interested in controllability of control system in respect of differential geometry for 1970s. In the fourth, controllability of nilpotent systems which is the aim of the thesis is studied. First definition of invariant control system is given and then controllability is studied by analyzing the existence of symplectic vectors elaborately. Example is given for the last controllability result of the thesis. Finally, the fifth chapter is reserved for discussions and results.