Nonlinear Optimal Control Theory

Existence Theorems; Compact Constraints 89

For each n, the admissible pair (ψ

n

, µ

n

) in the subsequence satisfies

ψ

n

(t) = ψ

n

(t

0n

) +

Z

t

t

0n

f(s, ψ

n

(s), µ

ns

)ds.

If we let n → ∞ and use (4.3.12), (4.3.13), and Lemma 4.3.3, we get that

ψ

∗

(t) = ψ

∗

(t

0

) +

Z

t

t

0

f(s, ψ

∗

(s), µ

∗

s

)ds (4.3.14)

for t ∈ I. A similar argument gives

lim

n→∞

Z

t

1n

t

0n

f

0

(s, ψ

n

(s), µ

n,s

)ds =

Z

t

1

t

0

f

0

(s, ψ

∗

(s), µ

∗

s

)ds. (4.3.15)

From (t

0

, x

0

, t

1

, x

1

) ∈ B, (4.3.13), (4.3.14), and the fact that µ

∗

t

is concentrated

on Ω(t) we get that (ψ

∗

, µ

∗

) is an admissible pair.

Since g is lower semicontinuous

lim inf

n→∞

g(e(ψ

n

)) ≥ g(e(ψ

∗

)).

From this and (4.3.11) a nd (4.3.15) we get that

m = lim

n→∞

J(ψ

n

, µ

n

) ≥ J(ψ

∗

, µ

∗