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The authors consider one optimal control problem   with a variable structure, described in various domains by a Goursat-Darboux   system and a two-dimensional Volterra integral equation. Using one version of   the method of increments, a necessary and sufficient condition for optimality   is proved in the form of the Pontryagin maximum principle.

The Pontryagin maximum principle as a necessary and sufficient optimality condition in a variable structure optimal control problem