Nonlinear Optimal Control Theory

Bounded State Problems 325

φ

j

k

(t))dt remain bounded as k → ∞. If we next replace h by ζ(t) +

[∇G(t,

e

φ

ǫ

(t)) · ζ]ξ

ǫ

, ζ(0) = ζ(t

1

) = 0 we obtain

J(

e

φ

ǫ

, ζ) + J(

e

φ

ǫ

, [∇G(·,

e

φ

ǫ

(·)) · ζ]ζ

ǫ

) = 0

Now, using (11.6.9) and (11.6.10) we obtain

ψ(ǫ; t) + λ(ǫ; t)∇G(t,

e

φ

ǫ

(t)) (11.6.11)

=

Z

t

0

n

−[ψ(ǫ; s) + λ(ǫ; s)∇G(s,

e

φ

ǫ

(s))] · f

1

(ǫ; s) + f

0

1

(ǫ; s)

+ λ(ǫ; s)∇G(s,

e

φ

ǫ

(s)) · f

1

(ǫ; s)

+ λ(ǫ; s)

"

d∇G(s,

e

φ

ǫ

(s))

ds

#

+ 2(

e

φ

′

ǫ

(s) − φ

′

0

(s)) · f

1

(ǫ; s)

)

ds + c

ǫ

We next obtain an ǫ-minimum principle. From Remark 11.6.6 we know

that F

K(ǫ)

(

e

φ

ǫ

, ν

ǫ

) minimizes F

K(ǫ)

(

e

φ, ν) over B(ǫ). For 0 < ǫ < ǫ

′

< ǫ and

0 ≤ θ ≤ 1, let

ν(θ) = ν

ǫ

+ θ(ν − ν

ǫ

)

Since kν

ǫ

−ν

0

k < ǫ ∃ θ

0

> 0 such that 0 ≤ θ ≤ θ

0

implies ...