Equivalent Subgradient Versions of Hamiltonian and Euler-lagrange Equations in Variational

Much effort in recent years has gone into generalizing the classical Hamiltonian and Euler-Lagrange equations of the calculus of variations so as to encompass problems in optimal control and a greater variety of integrands and constraints. These generalizations, in which nonsmoothness abounds and gradients are systematically replaced by subgradients, have succeeded in furnishing necessary conditions for optimality which reduce to the classical ones in the classical setting, but important issues have remained unsettled, especially concerning the exact relationship of the subgradient versions of the Hamiltonian equations versus those of the Euler-Lagrange equations. Here it is shown that new, tighter subgradient versions of these equations are actually equivalent to each other. The theory of epi-convergence of convex functions provides the technical basis for this development.