Nonlinear Optimal Control Theory

Existence Theorems; Compact Constraints 109

Define a linear mapping T from L

2

[t

0

, t] to R

n

by the formula

T ρ =

Z

t

t

0

Ψ

−1

(s)ρ(s)ds.

The mapping T is a continuous map from L

2

[t

0

, t] to R

n

.

The point a is in the set T (Θ). The set T

−1

(a) is non-empty, is closed, and

is convex in L

2

[t

0

, t]. Let Σ denote the intersection of T

−1

(a) a nd Θ. Then

Σ is weakly closed and convex. Since Θ is bounded, so is Σ. Since Σ is also

strongly closed and convex, by Mazur’s Theorem, Σ is also weakly closed. Since

Θ is weakly compact, Σ is weakly compact. By the Krein-Milman Theorem

(Lemma 4.7.5), Σ has an extreme point θ

0

. Since θ

0

∈ Θ it follows that θ

0

has a representation given ...