Nonlinear Optimal Control Theory

230 Nonlinear Optimal Control Theory

Hence there exists a constant B > 0 such that for all 0 ≤ t ≤ 1 and 0 < ε ≤ ε

0









Z

t

0

P (ε, s)N (ε, s)ds









≤ B. (7.7.12)

By (7.7.2) the sequence {λ(ε

n

, 0)} is bounded. From this and from (7.7.7),

(7.7.11), and (7.7.12) the inequality (7.7.8) now follows.

We now show that the integrals

R

E

|λ

′

(ε, t)|dt are equi-absolutely continu-

ous. From (7.7.3), (7.7.4), (7.7.8), (7.7.9) and the inequality 0 ≤ M(ε)

−1

≤ 1,

we get the existence of a function M in L

2

[0, 1] such that for any measura ble

set E ⊂ [0, 1]

Z

E

|λ

′

(ε, t)|dt ≤ (C + 1)

Z

E

M(t)dt + 2

Z

E

M(t)|ψ

′

ε

(t) − ψ

′

∗

(t)|dt.

From the Cauchy-Schwarz inequality and ψ

ε

∈ D

ε

, we get that the second

integral on the right is less than or equal to



Z

E

M

2

(t)dt



1/2

kψ

′

ε

− ψ

′

∗

k < (ε/2)



Z

E

M

2

(t