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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.