On singular controls of a maximum principle for the problem of the Goursat–Darboux system optimization

The paper deals with the terminal optimization problem connected with the Goursat–Darboux control system. The right-hand side of the differential equation is a full nonlinear Caratheodory function. We consider the case in which solutions of the Goursat–Darboux system necessarily belong to a class of functions with $p$-integrable (for some $p>1$) mixed derivatives. In our case a choice of this class is defined by boundary functions. We study singular controls in the sense of the pointwise maximum principle that are controls for which this principle is strong degenerate, i.e., degenerate together with second-order optimality conditions. It is shown that for strong degeneration of the pointwise maximum principle it is sufficient that right-hand side with respect to state derivatives is affine and these derivatives and control are separated additively. Necessary optimality conditions of the singular controls are given for this case. These conditions generalize similar necessary optimality conditions which were obtained for more smooth right-hand sides in the case of solutions with bounded mixed derivatives.