Pontryagin’s maximum principle analog and linearized necessary conditions of optimality in one nonlinear control problem of a Gurs-Darboux system with a variable structure

One two-stage optimal control problem is considered, which is described by systems of second-order hyperbolic equations with Goursat boundary conditions. The formula of the first order increment of the functional is constructed which allows is to prove the necessary optimality condition of the type maximum principle L.S. Pontryagin. In the case of convexity of control domains a linearized integral maximum condition is proved. An analogue of the differential maximum condition is given. Assuming the openness of the control domains an analogue of the Euler equations is established.