Optimal control theory has widespread applications across various fields such as aerospace, robotics, economics, and process engineering. It enables the design of control strategies that can achieve superior performance by systematically considering the trade-offs and constraints inherent in dynamic systems.The purpose of this book is to present the basic elements of optimal control. The first chapter will introduce the optimization criteria encountered. Concepts based on the calculus of variations and Pontryagin's minimum principle, for minimizing cost functions, will be covered in the second chapter. The third chapter will extend Pontryagin's minimum principle to linear systems with quadratic criteria (LQ theory). At the end of this document (Chapter 4), examples of implementation will be provided to deepen the understanding of the introduced concepts. The prerequisites for this course on optimal control include fundamental notions of continuous and sampled linear control systems as well as state representation.