Nonlinear Goursat-Darboux control problem: Conditions for the preservation of global solvability

We consider the nonlinear vector Goursat-Darboux problem of general form with a control on the right-hand side in the differential equation. We consider the case in which its solution necessarily belongs to the class of functions with p-integrable (for some p > 1) mixed and first partial derivatives. In this connection, we use the fact that the first derivatives of such a solution are elements of spaces with a mixed norm; these spaces are much narrower than L _p . Sufficient conditions for the preservation of the global solvability of the problem under control perturbations are proved.