To solve the problem of pursuit in linear differential games, L. S. Pontryagin suggested two direct methods. Direct methods are of great importance in the development of the theory of differential games and in control theory under the conditions of uncertainty. It turned out to be useful also in solving the problem of control synthesis. Pontryagin direct methods have proved themselves as an effective means for solving problems of pursuit-evasion and control. These use integrals, having a number of significant differences from the classical integral. One of the differences consists in the use of multivalued mapping. Pontryagin's second direct method, based on concept of the alternating integral, which has no analogs in integration of real function. In definition of alternating integral participate of integration of setvalued mappings and geometric difference (Minkovski difference) of sets. These operations make difficulties for computation of alternating integral. From this point of view, the integral used by the first direct method has a simpler construction. Therefore, the question naturally arises of generalization the first direct method of pursuit. In this paper it will be studied a generalization of the first direct method for pursuit games, being described by differential inclusions <$E z dot~symbol <174>~-~F(t,~v)>, where F is a continuous multivalued mapping. This method will be called the modified first direct method of pursuit for differential inclusions. In particular, the class of stroboscopic strategies, the trajectory of the system are determined. For these classes games, it is proved that if the starting point belongs to the modified first integral (the integral from the multivalued mapping, which is present in the definition of the modified fist direct metod), then this is necessary and sufficient condition for completing the game in a fixed time instant in the class of stroboscobic strategies. The problem of computation this integral is important. In the present article it has also been proved that the union operations in the definition of the modified first integral can be narrowed down to the class of compact-valued mappings.

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