Nonlinear Optimal Control Theory

Relaxed Controls 55

Theorem 3.3.12. Let I be a compact interval in R

1

. Let Ω: t → Ω(t) be a

mapping from I to subsets of R

k

that is upper semi-continuous with respect

to inclusion on I and such that for each t in I the set Ω(t) is compact. Let

{µ

n

} be a sequence of relaxed controls s uch that for each n the m easure µ

nt

is for almost every t concentrated on Ω(t). Then there exists a subsequence of

the sequence {µ

n

} that converges weakly to a relaxed control µ such that for

almost all t the measure µ

t

is concentrated on Ω(t).

Proof. It follows from the upper semi-continuity with respect to inclusio n of

Ω and Lemma 3.3.11 that there exists a compact set ...