Mathematical research of dynamic sysytems and control theory

This thesis is review-based study on the place and importance in mathematical control theory of the dynamic systems and control systems The system is a structure that is consisting of inputs and outputs. This structure can be mathematical or it can be used to represent real-world systems in different disciplines. The system receives inputs from its environment, processes it, changes its state or behavior, and interacts with its environment by transmitting outputs to it. This input-output interaction is important for system analysis and control. Control, or "controllability" in a mathematical sense, refers to the ability of a system or mathematical model to be effectively controlled towards a specific goal or situation.By examining the input-output relations, the behavior of the system can be understood, controllability can be evaluated and necessary arrangements can be made for the system to produce the desired outputs. In this thesis, especially linear time systems, continuous and discrete time systems and time varying systems have been studied. Examples are given and calculations are provided for both controllable and non-controllable systems. Considering observability, and similarly, examples are given for both observable and unobservable systems. In addition, duality theorem, which provides a relationship between controllability and observability, is given and canonical forms and Kalman decomposition, which are important for control theory, are mentioned. The aim of this thesis is to examine the controllability and observability problems of control systems mathematically, starting with the historical development and change of dynamic systems, which is a multidisciplinary field of study in this respect. Keywords: Control Theory, Controllability, Dynamic Systems, Observability