Nonlinear Optimal Control Theory

Proof of t he Maximum Principle 225

and let

λ(ε, t) =

¯

λ(ε, t)/M(ε). (7.6.9)

We now divide through by M (ε) in (7.6 .7) and use (7.6.9) to get

λ

′

(ε, t) = M(ε)

−1

f

0

x

(ε, t) −f

x

(ε, t)

T

λ(ε, t) + 2M (ε)

−1

f

x

(ε, t)

T

(ψ

′

ε

(t) −ψ

′

x

(t)).

(7.6.10)

In summary, we have shown that since ψ

ε

minimizes (7.6.1 ) over T

ε/2

(ψ

∗

),

there exists an absolutely continuous function λ(ε, ·) such that (7.6.10) holds

a.e. on [0, 1].

Since ψ

ε

minimizes (7.6.1) over T

ε/2

(ψ

∗

), the transversality condition

(7.4.9) of Lemma 7.4 .8 holds. We suppose the e(ψ

ε

) is an interior point of

B. From the first equation in (7.6.2) we get tha t

γ

x

0

(e(ψ

ε

)) = g

x

0

(e(ψ

ε

)) + 2(ψ

ε

(0) − ψ

∗

(0))

γ

x

1

(e(ψ

ε

)) = g

x

1

(e(