Problems in the optimal control of linear systems with convex costs are recast in a primal-dual (minimax) framework. An approximation scheme which leads to primal and dual optimal control problems in discrete time having similar structure to the original primal-dual pair is introduced. The discretization is shown to be variationally consistent in the sense oi epi/hypoconvergence, so that any limit point of solutions for the approximate minimax problems will solve the original primal and dual problems.