Operator equations and maximum principle algorithms in optimal control problems

The article deals with a new developing approach to the numerical solution of nonlinear optimal control problems, based on the construction of operator equations in the form of fixed point problems characterizing optimal control conditions. This form makes it possible to apply and modify the well-known apparatus of the theory and methods of fixed points for searching of extremal controls. The proposed iterative algorithms of fixed points of the maximum principle have the nonlocality property of successive control approximations and the absence of a parametric search procedure for improving approximations at each iteration, which is characteristic of the well-known standard methods of the gradient maximum principle. We have considered the conditions for convergence of the constructed iterative processes based on the principle of contracting mappings.