Nonlinear Optimal Control Theory

164 Nonlinear Optimal Control Theory

pair, then by Theorem 3.2.11 there exists a discrete measure control eµ such

that

b

ψ

′

(t) =

b

f(t, ψ(t), µ

t

) =

b

f(t, ψ(t), eµ

t

),

where

b

f = (f

0

, f

1

, . . . , f

n

). Thus, for every admissible pair (

b

ψ, µ), there exists

a discrete measure control eµ such that (ψ, eµ) is admissible and J(ψ, µ) =

J(ψ, eµ). Therefore, in consider ing necessary conditions if the co nstraint sets

Ω(t) are compact we need only consider discrete measure controls.

If µ

t

is a Dirac measure concentrated at u(t), then H

r

(t, x, µ

t

, q

0

, q) =

H(t, x, u(t), q

0

, q). In general, if µ

t

is a discrete measure control then

H

r

(t, x, µ

t

, q

0

, q) = hbq,

b

f(t, x, µ

t

)i (6.3.11)

=

n+1

X

j=0

n+2

X

k=1

p

k

(t)f

j

(t, x, u

k

(t))q

j

!

=

n+2

X

k=1

p

k

(t)





n+1

X

j=0

q

j

f

j

(t, x, u

k

(t))





=

n+2

X

k=1

p

k