Nonlinear Optimal Control Theory

220 Nonlinear Optimal Control Theory

and the Cauchy-Schwarz inequality we get that for ψ in D

ε

|ψ(t) − ψ

∗

(t)| ≤ |ψ(0) −ψ

∗

(0)| + kψ

′

− ψ

′

∗

k. (7.5.3)

Since the graph of ψ

∗

is compact and is a t distance ε

′

1

from the boundary of

I

0

×X

0

, it follows from (7.5.3) and the first two inequalities in (7.5.2) that all

points ψ(t), 0 ≤ t ≤ 1 are in a fixed compact set X that is independent of ψ

and is contained in X

0

. Thus, the graph of ψ is contained in I

0

× X

0

. Hence

D

ε

is contained in D and F is defined on D

ε

× (0, ε).

For each 0 < ε < ε

1

, we define the ε-problem to be: Minimize F (ψ, µ, K, ε)

over the set D

ε

.

Step II consists of establishing the fo llowing lemma.

Lemma