Nonlinear Optimal Control Theory

Existence Theorems; Non-Compact Constraints 123

end condition

(t

0

, ψ(t

0

), t

1

, ψ(t

1

)) ∈ B (5.4.4 )

and control constraints on v(t) = (p(t), v(t))

p(t) ∈ Π

n+2

u

i

(t) ∈ Ω(t, ψ(t)) i = 1, . . . , n + 2. (5.4.5)

If we set

e

Ω(t, x) = Ω(t, x) ×. . . ×

n+2 t imes

Ω(t, x)

then we may w rite (5.4.5) as

v(t) ∈

e

Ω(t, ψ(t)). (5.4.6)

We emphasiz e the observation made in Remark 3.5.8 that the relaxed

problem just formulated can be viewed as an ordinary problem with controls

v = (p, u

1

, . . . , u

n+2

).

We a ssume that the

b

f, g, B, and Ω satisfy the following:

Assumption 5.4.1. (i) The function

b

f = (f

0

, f) = (f

0

, f

1

, . . . , f

n

) is de-

fined on a set G = I × X × U, whe re I is ...