Nonlinear Optimal Control Theory

40 Nonlinear Optimal Control Theory

Remark 3.2.4. The set of relaxed controls properly contains the set of ordi-

nary controls. To see this, let u be a control defined on [t

0

, t

1

] w ith u(t) ∈ Ω(t)

a.e. Let δ

u(t)

be the Dirac measure on Ω(t) that is equal to one at the point

u(t) and equal to zero o n any set that does no t contain u(t). Then δ

u(t)

is a

probability measure and for any g as in Definition 3.2.2 with I = [t

0

, t

1

], the

function h defined by

h(t) = g(t, u(t)) =

Z

Ω(t)

g(t, z)dδ

u(t)

is measurable. Thus, the mapping t → δ

u(t)

is a relaxed control. We can

therefore consider an ordinary control to be a special type of relaxed control.

We now exhibit relaxed controls that are not ordinary controls. Let

p

1

, . . . , p

k

be nonneg ative measurable functions ...