Geodesic
Non-traditional conditions for the existence of an optimal control for the Goursat--Darboux equations
Izvestiya. Mathematics , Tome 62 (1998) no. 5, pp. 929-951.

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We consider the optimal control problem for the non-linear Goursat–Darboux equations in the case when the quality functional depends only on the phase variable. We show that a natural analogue of the “traditional” criterion for the existence of an optimal control for this problem is not valid. Under certain assumptions on the equations we obtain necessary and sufficient conditions under which any functional of the type mentioned above attains a minimum. The traditional condition (the convexity of the set of contingencies) is supplemented here by the following essential condition: the right-hand side of the differential equation is affine with respect to each of the two lowest derivatives of the phase variable. 
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O. A. Danilova; A. S. Matveev. Non-traditional conditions for the existence of an optimal control for the Goursat--Darboux equations. Izvestiya. Mathematics , Tome 62 (1998) no. 5, pp. 929-951. http://geodesic.mathdoc.fr/item/IM2_1998_62_5_a2/

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