  In the paper we study the following problem: given a Hamilton-Jacobi equation
where the Hamiltonian is convex with respect to the last variable, are there
any optimal control problems representing it? In other words, we search for an
appropriately regular dynamics and a Lagrangian that represents the Hamiltonian
with given properties. This problem was lately researched by
Frankowska-Sedrakyan (2014) and Rampazzo (2005). We introduce a new method to
construct a representation of a wide class of Hamiltonians, wider than it was
achieved before. Actually, we get two types of representations: with compact
and noncompact control set, depending on regularity of the Hamiltonian. We
conclude the paper by proving the stability of representations.
