Minimum calculation of an integral with differential constraints

The aim of this paper is to minimize a constrained integral in the form of differential equations with constraints. For this purpose, we have learned many methods of obtaining functions’ extremum in and functions’ extremal in calculus of variations. Then, we consider optimal control theory and look at it as a classical development of Euler–Lagrange theory. In such development, we are dealing with acceptable functions that are less well-behaved compared to normal functions. All of these cases reach us to Pontryagin's maximum principle (P.M.P).