Nonlinear Optimal Control Theory

Relaxed Controls 65

Definition 3.5.5. An absolutely continuous function ψ = (ψ

1

, . . . , ψ

n

) de-

fined in interval [t

0

, t

1

] ⊆ I is a relaxed trajectory corresponding to a relaxed

control v if

(i) (t, ψ(t)) ∈ R for all t ∈ [t

0

, t

1

].

(ii) ψ is a solution of the differential equation

dx

dt

=

n+2

X

i=1

p

i

(t)f(t, x, u

i

(t)) =

Z

Ω(t)

f(t, x, z)dµ

t

, (3.5.2)

where µ

t

is as in (3.5.1).

The differential equation (3.5.2) is called the relaxed differential equa tion.

Definition 3.5.6. A re laxed trajectory ψ corresponding to a relaxed control

v is said to be admissible if

(i) (t

0

, ψ(t

0

), t

1

, ψ(t

1

)) ∈ B and the function

t →

n+2

X

i=1

p

i

(t)f

0

(t, ψ(t), u

i

(t)) =

Z

Ω(t)

f

0

(t, ψ(t), z)dµ

t

is integrable. ...