Nonlinear Optimal Control Theory

94 Nonlinear Optimal Control Theory

The set T is Lebesgue measura ble and thus is a measure space. The set Z is

clearly Hausdorff. Since Ω is u.s .c.i., it follows from Lemma 3.3.11 that ∆ is

compact. From this and the continuity of f

0

we get that D is closed. If D is

bounded, then D is compact. Otherwise D is the countable union of compact

sets D

i

, where each D

i

is the intersection of D with the c losed ball centered

at the origin with radius i.

Let Γ denote the mapping from T to Z defined by Γ(t) =

(t, ψ(t), ψ

′

(t), ψ

0

′

(t)). Since ea ch of the functions

b

ψ and

b

ψ

′

is measurable,

so is Γ. Let ϕ denote the mapping from D to Z defined by

ϕ(t, x, z, η) = (t, x, ...