Nonlinear Optimal Control Theory

Relaxed Controls 75

Integrating both sides of this inequality from t

0

to t and using (3.6.24) gives

log ρ(t) ≤ log



α +

Z

t

t

0

µρds



≤ log α +

Z

t

t

0

µds.

From this we get (3.6.25).

If α = 0, then (3.6.24) holds for all α

1

> 0. Hence (3.6.25) holds for all

α

1

> 0. Letting α

1

→ 0 now yields ρ(t) ≡ 0. Hence (3.6.25) is trivially true.

Remark 3.6.7. The proof shows that if α > 0 and strict inequality holds in

(3.6.24), then strict inequality holds in (3.6.25).

Theorem 3.6.8. Let I be a compact interval in R

1

, let X be a compact

interval in R

n

, and let R = I×X. Let G = R×U, where U is a region of R

m

,

and let f be a continuous mapping from G to R

n

. Let Ω be a mapping