Chapter 1
Introduction
Optimal Control is a generalization of the Calculus of Variations and the theory of di�erential
equations. It was in the late 1950s when problems arising in engineering and economics
were recognized and reformulated as optimal control problems and mathematicians started
developing the tools to analyze them. At �rst, di�erent approaches were used to solve
individual problems. Later on, it was discovered that all those problems have a common
mathematical structure and Optimal Control emerged then as a new branch of Applied
Mathematics.
The scienti�c formulation of an optimal problem must be based on three kind of infor-
mation. At �rst, we have evolution of the behavior of the system that mathematically can
be described in terms of di�erential equations or state equations called dynamics. The dy-
namics provide a mathematical model re�ecting the interactions between the state variables
and the input parameters called control. A second important aspect is given by constraints
on the system. Thirdly, we have the purpose of the control which must be mathematically
speci�ed in form of so-called objective functions which should be minimized or maximized.
The range of applicability of optimal control theory is very wide and diverse, ranging from
engineering to physics, to economic and more recently biomedical problems. This paper will
deal with applications of optimal control theory to biomedicine, namely to the analysis of a
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