Nonlinear Optimal Control Theory

50 Nonlinear Optimal Control Theory

Then, |T | = |I| and for each i the mapping τ → L

τ

(g

i

) is differentiable at all

points of T .

We now show that for each g in C(I × Z) the mapping τ → L

τ

(g) is

differentiable at all points of T . Let τ

0

be a point in T and for τ 6= τ

0

define

∆

τ,τ

0

(g) =

L

τ

(g) − L

τ

0

(g)

τ − τ

0

.

Then to show that τ → L

τ

(g) is differentiable at τ

0

it suffices to show that

for τ

1

, τ

2

→ τ

0

that

∆

τ

1

,τ

0

(g) − ∆

τ

2

,τ

0

(g) → 0. (3.3.6)

For g

i

as in the preceding paragraph,

|∆

τ

1

,τ

0

(g) − ∆

τ

2

,τ

0

(g)| ≤ |∆

τ

1

,τ

0

(g − g

i

)| + |∆

τ

2

,τ

0

(g − g

i

)|

+ |∆

τ

1

,τ

0

(g

i

) − ∆

τ

2

,τ

0

(g

i

)|.

From the definition of ∆

τ,τ

0

(g) and (3.3.5) we get that each of the first two

terms on the right do not