Nonlinear Optimal Control Theory

118 Nonlinear Optimal Control Theory

Definition 5.3.2. A set F of absolutely c ontinuous functions f defined on

[a, b] is said to b e equi-absolutely continuous if given an ε > 0 there is a

δ > 0 such that for any finite collectio n of non-overlapping intervals (α

i

, β

i

)

contained in [a, b], with

P

i

|β

i

−α

i

| < δ, the inequality

P

i

|f(β

i

) −f(α

i

)| < ε

holds for all f in F .

We leave it to the reader to verify that a set of absolutely continuous

functions is equi-absolutely continuous if and only if the derivatives f

′

have

equi-absolutely continuous integrals.

Lemma 5.3.3. Let {f

n

} be a sequence of equi-absolutely continuous func-

tions defined on an interval [a,