Nonlinear Optimal Control Theory

210 Nonlinear Optimal Control Theory

Since (λ

0

, λ, µ) is a unit vector, it is different from zero.

From Eq s. (7.2.7), (7.2.8), and (7.2.9) and the continuity of the partials of

f, g, and h, we ge t

λ

0

∂f

∂x

j

(0) +

r

X

i=1

λ

i

∂g

i

(0)

∂x

j

+

k

X

i=1

µ

i

∂h

i

(0)

∂x

j

= 0. (7.2.10)

From (7.2.6) and (7.2.9) we see that λ

i

≥ 0, i = 0, 1, . . . , r and that λ

i

= 0 for

i = r + 1, . . . , m. Thus, λ

0

≥ 0 and λ ≥ 0. Since g

i

(0) = 0 for i = 1, . . . , r and

λ

i

= 0 for i = r + 1 , . . . , m, we have that λ

i

g

i

(0) = 0 for i = 1 , . . . , m. Also,

we can take the upper limit in the se cond term in (7.2.10) to be m and write

λ

0

∂f

∂x

j

(0) +

m

X

i=1

λ

i

∂g

i

(0)

∂x

j

+

k

X

i=1

µ

i

∂h

i

(0)

∂x

j

= 0.

This completes the ...