A theorem of existence of an optimal control for the Goursat–Darboux problem without convexity assumptions

We prove a theorem on the existence of solutions of the minimization problem for the integral functional on solutions of the controlled system described by the Goursat–Darboux equation, which is linear with respect to the phase variables and their derivatives, with constraints on the control, the phase variables, and their first partial derivatives.
The system is controlled by means of boundary and distributed controls. We do not assume that the functional to be minimized is convex with respect to the controls. The sets of admissible controls and the constraints on the phase variables and their first partial derivatives also are non-convex.