We investigate the existence of solutions of the Dirichlet problem for the differential inclusion $0\in \Delta x\left(y\right)+{\partial }_{x}G(y,x\left(y\right))$ for a.e. y ∈ Ω, which is a generalized Euler-Lagrange equation for the functional $J\left(x\right)={\int }_{\Omega }1/2\left|\nabla x\right(y\left)\right|²-G(y,x(y\left)\right)dy$. We develop a duality theory and formulate the variational principle for this problem. As a consequence of duality, we derive the variational principle for minimizing sequences of J. We consider the case when G is subquadratic at infinity.

Given a deterministic optimal control problem (OCP) with value function, say $J^{*}$, we introduce a linear program $\left(P\right)$ and its dual $\left(P^{*}\right)$ whose values satisfy $sup\left(P^{*}\right)\le inf\left(P\right)\le J^{*}(t,x)$. Then we give conditions under which (i) there is no duality gap

For a multidimensional control problem ${\left(P\right)}_K$ involving controls $u\in L_{\infty }$, we construct a dual problem ${\left(D\right)}_K$ in which the variables ν to be paired with u are taken from the measure space rca (Ω,) instead of $\left(L_{\infty }\right)*$. For this purpose, we add to ${\left(P\right)}_K$ a Baire class restriction for the representatives of the controls u. As main results, we prove a strong duality theorem and saddle-point conditions.