The problem of optimal control involving two opposing players is considered where optimality is understood in the minimax sense of achieving the best guaranteed outcome, and the control strategy is constructed with respect to the worst case admissible under the available measurements. The differential game problem is reduced to an optimal control synthesis problem by means of an abstract maximum principle using the method of Lagrange multipliers. A procedure is presented for applying the abstract maximum principle to the maximin formulation of the differential game problem within the Bellman framework in terms of dynamic programming. It is shown how the abstract maximum principle leads to the fundamental relations of Bellman’s optimization method for the differential game under consideration. The developed methodology for deriving optimality conditions in an antagonistic differential game using the abstract maximum principle can be applied to the analysis and design of nonlinear controlled dynamical systems with internally conflicting objectives.