August 2012
Intermediate to advanced
392 pages
10h 50m
English
Content preview from Nonlinear Optimal Control Theory
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Chapter 7
Proof of the Maximum Prin ciple
7.1 Introduction
In this chapter we prove Theorems 6.3.5 through 6.3.22 and their corol-
laries. Theorem 6.3.5 will be proved by a penalty function method, which
we outline here. For simplicity, let f be a real valued differentiable function
defined on an open set X in R
n
. Consider the unconstrained pr oblem of mini-
mizing f on X. If f attains a minimum at a point x
0
in X, then the necessary
condition df(x
0
) = 0 holds, where df is the differential of f. This condition is
obtained by making a perturbation x
0
+ εδx, where δx is arbitrary but fixed,
and ε is sufficiently small so that x
0
+εδx is in X. Then since f is differe ...
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