A finite-difference method of approximate optimal-control synthesis

A finite-difference method is proposed for design of optimal control where the function $\varphi(t,y)$ is a solution of the partial differential equation over the set under condideration with the use of the finite-difference method. This definition of the function $\varphi(t,y)$) reduces the problem of approximate design of the optimal control function $\tilde{u}(t,y)$ to a Cauchy problem for a system of common differential equations.