  Nonconvex optimal-control problems governed by evolution problems in
infinite-dimensional spaces (as e.g. parabolic boundary-value problems) needs a
continuous (and possibly also smooth) extension on some (preferably convex)
compactification, called relaxation, to guarantee existence of their solutions
and to facilitate analysis by relatively conventional tools. When the control
is valued in some subsets of Lebesgue spaces, the usual extensions are either
too coarse (allowing in fact only very restricted nonlinearities) or too fine
(being nonmetrizable). To overcome these drawbacks, a compromising convex
compactification is here devised, combining classical techniques for Young
measures with Choquet theory. This is applied to parabolic optimal control
problems as far as existence and optimality conditions concerns.
