Abstract. The author proposes simplified schemes for the construction of the lower Pontryagin’s alternating integral in pursuit games described by the differential inclusion z(t) ∈F(t, u), where F is a continuous compact-valued mapping. Based on this schemes, the author proves that for the initial states to which the lower alternating integral can be applied, there exists a pursuer’s strategy that guarantees exact completion of the pursuit and has piecewise constant realizations.

Keywords: lower alternating integral, partial sum, differential inclusion, strategy, control.