The Calculus of Variations deals with optimisation of quantities which can be expressed as integrals of some unknown function and its derivatives. The goal is to find the unknown function that minimises or maximises an integral. We find that this unknown function is a solution of a differential equation.
Control theory deals with the control of differential equations. The objective is to apply inputs to drive the system to a desired state, while minimising any delay, or cost. Control theory is a generalisation of the calculus of variations and it has a vast range of applications in industry and the sciences.
The finite element method (FEM) is a method of approximating solutions to differential equations and it is based on weak forms of the differential equation, which involve integrals. The mathematics is very closely related to the mathematics of the calculus of variations. FEM is the industry standard approach computing solutions to partial differential equations. We make use of the method for solving differential equations, calculus of variations problems, and control problems.