# Basic Concepts

• Henning Tolle
Part of the Universitext book series (UTX)

## Abstract

The simplest optimization problem is concerned with the determination of the values of independent variables which maximize or minimize a given function of these variables. That is, in the simplest case we have to determine the values of t for which
$$i = l(t) = \left\{ {\begin{array}{*{20}{l}} {\max .} \\ {or\min .} \end{array}} \right.$$
(1)
holds. The problem next in complexity is to find the derivatives of functions of variables which allow the integral of a function of functions of variables and their derivatives to assume an extremal value. The simplest requirement here is: Find the values of y′(t) for which
$$J = \int\limits_{{t_A}}^{{t_E}} L [t,y(t),{y^1}(t)]dt = \left\{ {\begin{array}{*{20}{c}} {\max .} \\ {or\min .} \end{array}} \right.withy({t_A}) = A;y({t_E}) = E$$
(2)
is satisfied.

## Keywords

Euler Equation Gradient Method Solution Curve Pontryagin Maximum Principle Lagrange Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.