Methods of fixed points in one class of discrete-continuous problems of optimizing controlled systems

In the considered class of discrete-continuous control systems, formulas for the increment of the objective function of the standard form with remainder terms of the expansions and non-standard formulas that do not contain the remainder terms of the expansions are constructed. Based on the formulas obtained, conditions for nonlocal improvement and control optimality are constructed in the form of fixed point problems in the control space. Such a representation of conditions makes it possible to apply and modify the well-known theory and methods of fixed points for constructing iterative algorithms for searching for extremal controls and constructing relaxation sequences of controls in the considered discrete-continuous optimal control problems. The proposed iterative algorithms have the property of nonlocality of successive control approximations and the absence of a parametric search procedure for an improving approximation at each iteration, which is characteristic of gradient-type methods.