## ABSTRACT

In this chapter, we engage with different types of optimal control problems and derive the necessary conditions for the minimum. The constraints in the problems are handled using the Lagrange Multiplier Rule and the John Multiplier Theorem. Since derivatives are available in most of the engineering problems, we assume the functions involved are sufficiently differentiable and the constraint qualifications explained in Chapter 4 are satisfied. Simply put, we take for granted the existence of a set of Lagrange multipliers in the augmented functional incorporating all constraints of an optimal control problem.