#TITLE_ALTERNATIVE# - KRISTINA RUSMALY (NIM 10104086), STEPHANI
Through the illustration of the Moon Lander, we arrived at the problem of the existence of optimal control. The optimal control must solve the governing equations, and at the same time must minimize the cost function. Using the machinery of functional analysis, it will be shown that such control exists. Furthermore, the optimal control is actually a bang-bang one, which is a simple control comprises of switches. The main result from functional analysis used here is the Krein-Milman Theorem. This theorem guarantees the existence of extreme points in a convex subset of a certain Banach space, which is compact in the weak topology. In this project we study the machinery of functional analysis which leads to the Krein-Milman theory. Subsequently we need Banach-Alaoglu theory, and the notion of weak topology of a Banach space, and Riesz Representation theory.