Nonlinear Optimal Control Theory

Existence Theorems; Non-Compact Constraints 143

Proof of Theorem 5.5.3. We first note that conclusion (ii) follows from (i) by

virtue of Corollary 4.4.3 . Hence it suffices to prove (i).

If

b

f satisfies (5.5 .1), then

|

b

f

r

(t, x, π, z) −

b

f

r

(t, x

′

, π, z)| ≤

n+2

X

i=1

π

i

|

b

f(t, x, z

i

) −

b

f(t, x

′

, z

i

)|

≤

n+2

X

i=1

π

i

L(t)|x − x

′

| = L(t)|x − x

′

|.

Hence

b

f

r

satisfies (5.5.1) as well as the other hypothese s in Assumption 5.5.1.

Let

e

Ω be as in the proof of Theorem 5.4.4. Then

e

Ω is upper semi-continuous.

Hence we may proceed as in the proof of Theorem 5.4.4 and take the

relaxed problem to be an ordinary problem with control v = (p, v) =

(p

1

, . . . , p

n+2

, u

1

, . . . , u

n+2

).

The ...