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Optimal control is a special type of the optimization problem and has tremendous applications. It deals with the study of systems. The early developments of the theories were given by engineers whose systems were machines and their interactions were controls. However, nowadays the system covers a much wider range, such as the human body, a particular industry or even the whole economy of a country. In this book, the existence and uniqueness of the state of these systems have been derived, then the set of adjoint state equations and inequalities that have been described the optimality conditions are given.