Nontraditional conditions for optimal control existence for Gursat-Darboux system

In this paper we consider a Gursat-Darboux control system and the following classes of cost functionals: 1) the cost functionals that depend directly on the state and both its lower derivatives; 2) the cost functionals that depend directly on the state and only one of its lower derivatives; and 3) the cost functionals that depend on the state alone. For each of these classes we extract for particular consideration the subclass of functionals that are convex in state and its lower derivatives. For any of these classes and subclasses we derive necessary and sufficient conditions for existence of an optimal control. More precisely, they are the conditions which guarantee that any functional from the class under consideration reaches the optimum over all trajectories of the system.