By optimal control problem we mean the minimization of a functional over arcs that satisfy certain constraints (dynamics, control, endpoint and state constraints). 

After a brief introduction on the subject, we will discuss the notion of closure of trajectories associated with a controlled differential equation, so that to present an existence theorem for optimal control problems. Then, we will announce the Pontryagin's Maximum Principle, that is, the most known set of necessary conditions that has to be fulfilled by a minimizer. Afterwards, we will introduce the most common extensions for the optimal control problems which do not admit minimizers and we will analyze the properness of such extensions. In particular, we will deal with the issue of gap phenomena between an optimal control problem and an its extension and we will prove a link between this occurrence and a topological property of the trajectories which is usually called isolation. Finally, we will establish that isolated trajectories satisfy the Maximum Principle in abnormal form, i.e. there exists at least a set of multipliers with cost multiplier equal to zero. We will conclude with some examples that illustrate the outcomes.