Nonlinear Optimal Control Theory

60 Nonlinear Optimal Control Theory

will be called the open cube of radius a centered at the origin. If we denote

the closure of C(0, a) by C(0, a), then C(0 , a) = {z : kzk ≤ a} and C(0, a) =

{z : kzk < a}. Let

T

c

= {t : h(t, ·) is continuous on U},

and let A = T − T

c

. Then by hypothesis, meas A = 0.

For each positive integer a, let U

a

denote the closure of the intersection

of U and C(0, a). We first c onsider h o n the cartesian product T × U

a

and

assume that T is bounded.

Let ε > 0 be given. For each po sitive integer m define

E

εm

= {t: If z

1

, z

2

in U

a

, kz

1

− z

2

k

≤

√

n/m, then |h(t, z

1

) − h(t, z

2

)| ≤ ε/4}.

We assert that ea ch E

εm

is measurable. If E

εm

is empt