About singular controls of pointwise maximum principle for optimization problem connected with Goursat-Darboux system

In this paper, we consider the problem of maximizing a sufficiently general functional defined on solutions of the controlled nonlinear Goursat-Darboux system. The right-hand side of the differential equation is a Caratheodory function. We study singular controls in the sense of the pointwise maximum principle, i.e. the controls for which this principle degenerates. We consider strong degeneration of the pointwise maximum principle (this principle is the necessary first-order optimality conditions by using of needle-shaped variation of a control) when the maximum principle degenerates together with second-order optimality conditions. Sufficient conditions for the strong degeneration of maximum principle and necessary conditions for the optimality of corresponding singular controls are given. These conditions generalize the conditions that are known for the case of the terminal quality functional and the case of the smoother right side of the equation.