Nonlinear Optimal Control Theory

104 Nonlinear Optimal Control Theory

(iv) For each t in I the set Ω(t) is a co mpact subset of U and the mapping

Ω is u.s .c.i.

Lemma 4.7.2. Let the linear in the state system (4.7.1) satisfy (i), (ii), and

(iv) of Assumption 4.7.1. Then there exists a control u defined on I such that

u(t) ∈ Ω (t) for all t.

Proof. For each t in I let

d(t) = min{|z|z ∈ Ω (t)}.

Since the absolute value is a continuous function and Ω(t) is compact, we may

write min and the minimum is achieved at some point z

0

(t) in Ω(t). We assert

that the function d is lower semico ntinuous on I. To show this we shall show

that for each real α the set

E

α

= {t : d(t) ≤ α}

is closed. Let {t