Nonlinear Optimal Control Theory

128 Nonlinear Optimal Control Theory

where z

i

∈ Ω(t, x), α

i

> 0 and

P

α

i

= 1. But y

k

→ y, so

y =

s

X

i=1

α

i

f(t, x, z

i

). (5.4.20)

It follows from Eqs. (5.4.11), (5.4.14), (5.4.16), a nd (5.4.5) and the lower

semicontinuity of f

0

that

y

0

=

n+2

X

i=1

η

i

≥

s

X

i=1

η

i

=

s

X

i=1

lim

k→∞

(α

ki

y

0

ki

) (5.4.21)

≥

s

X

i=1

lim inf

k→∞

(α

ki

f

0

(t, x

ki

, z

ki

))

≥

s

X

i=1

α

i

f

0

(t, x, z

i

),

where z

i

∈ Ω(t, x) and α

i

≥ 0 for i = 1, . . . , s and

P

α

i

= 1. From (5.4.20) and

(5.4.21) we get that by = (y

0

, y) is in co Q

+

(t, x). But Q

+

r

(t, x) = co Q

+

(t, x),

so by ∈ Q

+

r

(t, x). Thus, Q

+

r

has the weak Cesari property at (t, x).

Remark 5.4.8. If we consider neighborhoods N

δ

(t, x) and in (5.4.12) con-

sider sequences of points