On a construction of approximations for the optimal control in quasi-linear partial differential equations of the first order

We propose an approximate method of studying the optimal control problem for a quasilinear partial differential equations of first order. We consider the control bounded by a constant and quadratic criterion type. For each set of given coordinates and controls the Cauchy problem is reduced to an integral equation. It is considered the case when all the variables are integer values. The integral equation is replaced by the discrete analog. The existence and uniqueness of solution of this equation is proven. We use the method of successive approximations combined with the method of compressing maps.