  In this paper we study the existence of sufficiently regular representations
of Hamilton-Jacobi equations in the optimal control theory with unbounded
control set. We use a new method to construct representations for a wide class
of Hamiltonians. This class is wider than any constructed before, because we do
not require Legendre-Fenchel conjugates of Hamiltonians to be bounded. However,
in this case we obtain representations with unbounded control set. We apply the
obtained results to study regularities of value functions and correlations
between variational and optimal control problems.
