Nonlinear Optimal Control Theory

Relaxed Controls 45

defines a continuous linear transformation with

|L

µ

(g)| ≤ kgk|I|,

where |I| denotes the length of I. If we take g to be the function identically

one on I × Z we get that

kL

µ

k = |I|. (3.3.2)

Henceforth, to simplify notation we take I to b e a generic compact interval

with the origin as left-hand end point.

We shall also use a theorem from real analysis, which was proved by

Urysohn in a more general context than the one we need. See [82].

Lemma 3. 3.1 (Ury sohn’s Lemma). Let K be a compact set in R

k

, let V be

an open set in R

k

, and let K ⊂ V . Then there exists a continuous nonnegative

function f with support contained in V , with 0 ≤