Abstract: To address the phenomenon whereby the discriminating theorem of the discriminating domain of differential games is difficult to implement, we study the discriminating domain for pursuit-evasion differential games in a region defined by inequality constraints as being either smooth or unsmooth. Using the equivalence theorem of the Gordan Lemma, we obtain the discriminating domain in a manner similar to the Karush-Kuhn-Tucker (KKT) condition in the optimization. The discriminating domain is easy to implement. Through an example, we illustrate the effectiveness of the method and generalize the results to nonlinear control systems.