If the pair (x, u) is a minimizer for the OCP, then there exists an absolutely continuous function p and [mu] [member of] [C.

out]]) and that q is absolutely continuous function excepted at [t.

It is easy to see that for any locally absolutely continuous function f: [a, b] [right arrow] R, we have the identity

respectively, and to seek sharp upper bounds for these distances in terms of different measure that can be associated with f, where f is restricted to particular classes of functions including functions of bounded variation, Lipschitzian, convex and absolutely continuous functions.

2) is an absolutely continuous function, which satisfies that equation on (0, [infinity]) almost everywhere and condition (1.

4) is defined as a locally absolutely continuous function x(t), with [?

2 Let p [member of] [1, [infinity]) and let g : R [right arrow] C be a locally

absolutely continuous function with g and g' belonging to [L.

for x [member of] [a, b] where f : [a, b] [right arrow] R is an

absolutely continuous function on [a, b].

2]) [member of] (0,1] x (0,1], f : J x R [right arrow] R, is a given function and [phi] : [0,a] [right arrow] R, [psi] : [0,b] [right arrow] R are given

absolutely continuous functions with [phi](0) = [psi](0).

The rate is best possible amongst

absolutely continuous functions f on [-1, 1] whose derivative is bounded.

We denote the class of

absolutely continuous functions in Caratheodory's sense by [C.

loc] refers to the class of locally

absolutely continuous functions.