Nonlinear Optimal Control Theory

Existence Theorems; Non-Compact Constraints 133

Since the functions {

e

ψ

k

} are equi-absolutely continuous and converge uni-

formly to

e

ψ on I, the function

e

ψ is a bsolutely co ntinuous. Hence it is differ-

entiable almost everywhere and

e

ψ(t) =

e

ψ(t

0

) +

Z

t

t

0

e

ψ

′

(s)ds.

Let I = [a, b]. Since

e

ψ is constant on [a, t

0

] and on [t

1

, b], we have

e

ψ

′

(t) = 0

on (a, t

0

) and (t

1

, b). Hence we can write the preceding r elation as

e

ψ(t) =

e

ψ(a) +

Z

t

a

e

ψ

′

ds

for all t in [a, b]. We also have

e

ψ

k

(t) =

e

ψ

k

(a) +

Z

t

a

e

ψ

′

k

ds.

It then follows from (5.4.27) that for all t in I,

Z

t

a

e

ψ

′

k

ds →

Z

t

a

e

ψ

′

k

ds.

Since the functions {

e

ψ

k

} are equi-absolutely continuous, so are the integ rals

of the functions {

e

ψ

′

k

}. Hence by Theo rem 5.3.5

e

ψ

′

k

→

e

ψ

′

weakly in L

1

[a, b].

We a lso note that

e

ψ is in R

0

and (