Nonlinear Optimal Control Theory

240 Nonlinear Optimal Control Theory

if t ∈ E. Then

Z

1

0

{H

r

(t, ψ(t), µ

t

,

b

λ(t)) − H

r

(t, ψ(t), ν

t

,

b

λ(t))}dt

=

Z

E

p

k

(t){H(t, ψ(t), u

k

(t),

b

λ(t)) − H(t, ψ(t), z,

b

λ(t))}dt < 0.

This contradicts (6.3.7), which establishes (ii).

Remark 7.9.3. If we assume that

b

f is continuous on I

0

×X

0

×U

0

, the proof

of (ii) is somewhat simpler. We again note that if (6.3.15) wer e not true, then

there would exist a set E

k

⊂ P

k

of positive measure such that (7.9.1) holds.

The function h defined by h(t) = H(t, ψ(t), u

k

(t),

b

λ(t)) is measurable. Hence

by Lusin’s theorem there exists a closed set E

0

of positive measure such that

E

0

⊂ E

k

and h is continuous on E

0

. Let τ be a p oint of density ...