The fixed point method for the problems of nonlinear systems optimization on the managing functions and parameters

A new approach to the class of nonlinear optimal control problems containing both managing functions and parameters, is proposed on basis of the solution of special fixed point problems for operators constructed in the space of controls. The fixed point problems make it possible to build improving controls and obtain new conditions for optimal control in the class of optimization problems. The problem of control improvment as a problem of a fixed point is formed on the basis of a control increment formula without the residual terms of expansions. This formula is constructed using the differential-algebraic modification of the standard adjoint system. The method of sequential solving of improvement problems in the form of fixed point problems is characterized by nonlocal improvement of management, lack of search procedure improving control in a sufficiently small neighborhood of current control and the ability to improve suboptimal controls, satisfying the maximum principle. To search for controls, satisfying the maximum principle, instead of the boundary value problem in the space of states are invited to consider the fixed point problem in space of controls. Examples illustrating the basic properties of the method are given.