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In this paper we study the existence of sufficiently regular representationsof Hamilton-Jacobi equations in the optimal control theory with unboundedcontrol set. We use a new method to construct representations for a wide classof Hamiltonians. This class is wider than any constructed before, because we donot require Legendre-Fenchel conjugates of Hamiltonians to be bounded. However,in this case we obtain representations with unbounded control set. We apply theobtained results to study regularities of value functions and correlationsbetween variational and optimal control problems.

Representation of Hamilton-Jacobi equation in optimal control theory with unbounded control set