As far as numerical methods for optimization problems are concerned, in practice it is a matter of generating sequences of solutions and associated multipliers which are expected to obey, asymptotically, (at least) the first-order necessary optimality conditions. In mathematical programming, this procedure is theoretically validated by means of the so-called asymptotic KKT conditions. In optimal control theory, there is a lack of necessary optimality conditions of this kind. In this paper, we propose a new set of necessary optimality conditions for mixed-constrained optimal control problems in the form of an asymptotic (weak) maximum principle. We also propose, inspired by the augmented Lagrangian method for nonlinear programming, a method of multipliers for numerically solving mixed-constrained optimal control problems in which the generated sequences of solutions and multipliers satisfy the conditions of the asymptotic weak maximum principle. We discuss its convergence properties in terms of optimality and feasibility. To demonstrate the practical viability of the proposed theory, we provide some numerical results.