Controllability of control systems and mathematical models

Some Systems in Control Theory and their controllability characterizations are studied, and some existed mathematical models are given. This work consists of three chapters. In the first chapter, manifold, tangent vector, tangent space and vector fields notions which are fundamentals in Control Theory are contained. Moreover, Lie group, Lie algebra, their connection via exponential map and adjoint representation, and examples are contained. In the second chapter, definition of some classes of control systems and their existed controllability characterizations are given. In the third chapter, controllabilities of control systems are studied on some mathematical models. Keywords: Control systems, controllability, mathematical models, Lie group, Lie algebra