We prove weak duality between two recent convex relaxation methods for
bounding the optimal value of a constrained variational problem in which the
objective is an integral functional. The first approach, proposed by Valmorbida
et al. (IEEE Trans. Automat. Control 61(6):1649--1654, 2016), replaces the
variational problem with a convex program over sufficiently smooth functions,
subject to pointwise non-negativity constraints. The second approach, discussed
by Korda et al. (arXiv:1804.07565v1 [math.OC]), relaxes the variational problem
into a convex program over scaled probability measures. We also prove that the
duality between these infinite-dimensional convex programs is strong, meaning
that their optimal values coincide, when the range and gradients of admissible
functions in the variational problem are constrained to bounded sets. For
variational problems with polynomial data, the optimal values of each convex
relaxation can be approximated by solving weakly dual hierarchies of
finite-dimensional semidefinite programs (SDPs). These are strongly dual under
standard constraint qualification conditions irrespective of whether strong
duality holds at the infinite-dimensional level. Thus, the two relaxation
approaches are equivalent for the purposes of computations.