Nonlinear Optimal Control Theory

310 Nonlinear Optimal Control Theory

Proof. Le t t

′′

> t

′

≥ 0 be Lebesgue points of q. Taking

w(t) =



















(t − t

′

)/ǫ, t

′

≤ t ≤ t

′

+ ǫ

1, t

′

+ ǫ ≤ t ≤ t

′′

− ǫ

(t

′′

− t)/ǫ, t

′′

− ǫ ≤ t ≤ t

′′

0, 0 ≤ t ≤ t

′

, t

′′

≤ t ≤ t

1

gives the result.

By the Riesz representation theorem, there exists a function s 7→

e

Γ

i

(t,

¯

φ

ǫ

(·), ν

ǫ

t

, s) defined for s ∈ I

t

1

−r

, of bo unded variation and continuous

from the right such that, for each ζ ∈ C(I

t

1

−r

), we have

dh

i

(t,

¯

φ

ǫ

(·), ν

ǫ

t

)(ζ) =

Z

t

1

−r

ζ(s) ·d

s

e

Γ

i

(t,

¯

φ

ǫ

(·), ν

ǫ

t

, s) (11.3.11)

Here, dh

i

(t,

¯

φ

ǫ

(·), ν

ǫ

t

) denotes the Frechet derivative of h

i

(t, ·, ν

ǫ

t

) at

¯

φ

ǫ

. Since h

i

does not depend on

¯

φ

ǫ

(s) fo r s > t, s 7→

e

Γ

i

(t,

¯

φ

ǫ

(·), ν

ǫ

t

, s) is constant ...