Nonlinear Optimal Control Theory

Proof of t he Maximum Principle 235

where cl denotes closure and B(0, k) is the ball in R

m

of radius k, centered at

the origin. All the sets Ω

k

(t) ar e compact and are contained in the compact

set cl B(0, k). By hypothesis, all the mappings Ω

k

: t → Ω

k

(t) are u.s.c.i. Also,

Ω

k

(t) ⊆ Ω

k+1

(t) and Ω(t) =

∞

[

k=1

Ω

k

(t).

For each positive integer k > k

0

we define Problem k to be:

Minimize: g(e(ψ)) +

Z

1

0

[X

E

k

(t)f

0

(t, x, µ

∗

t

) + X

G

k

(t)f

0

(t, x, µ

t

)] dt

Subject to:

dx

dt

= X

E

k

(t)f(t, x, µ

∗

t

) + X

G

k

(t)f(t, x, µ

t

)

µ

t

∈ Ω

k

(t) e(ψ) ∈ B,

where X

E

k

is the characteristic function of E

k

and X

G

k

is the characteristic

function of G

k

.

For each k > K define a relaxed control µ

k

as follows:

µ

kt