The aim of this book is to comprehensively present the mathematical aspects of control systems for graduate-level courses in control theory and optimal control theory. Since the subject requires foundational knowledge in linear algebra, functional analysis, and ordinary differential equations, three chapters are devoted to concisely covering the necessary topics from these areas in order to make the book as self-contained as possible.

In Chapter 1, simple mathematical models of various dynamical and control systems are presented to illustrate key concepts of control systems theory, including controllability, observability, stability, and optimal control.

In Chapter 2, essential results from matrix theory that are fundamental to understanding control theory are presented. Specifically, sections on eigenvalues and eigenvectors, canonical forms, matrix functions, and special matrices establish a strong foundation for finite dimensional control systems theory.

In Chapter 3, definitions, theorems, and techniques from linear algebra and functional analysis are briefly reviewed through examples. The primary purpose of this chapter is to introduce the notations and concepts that will be used in subsequent chapters. The main result is the Banach fixed point theorem, presented in Section 3.5 and applied in Chapter 4 to establish the existence and uniqueness theorem. Section 3.6 on compact operators is included primarily to support Chapter 10 on infinite dimensional systems.

Systems of first-order ordinary differential equations play a vital role in control systems theory. In Chapter 4, key results from theory of differential equations, essential for developing control theory, are presented. Theorem 4.5.1 on the existence and uniqueness of solutions to initial value problems, Section 4.6 on systems of linear ODEs, Section 4.8 on fundamental matrices and the state transition matrix, and Theorem 4.10.1 on Gronwall's inequality are key focal points for better understanding the material in Chapters 5–8. Controllability and observability are desirable properties of a control system. Controllability implies that a control input can steer the system to any arbitrary state, while observability means that the system's state over a period of time can be uniquely determined from given observation data.