On one approach to searching for extreme controls in discrete-continuous systems

In the considered class of discrete-continuous controlled systems, an analogue of classical formulas for the increment of the objective function of the standard form with residual terms of expansions with the main part of the increment linear in the control increment is constructed. Based on the obtained formula, the optimality condition of control is constructed in the form of a problem about a fixed point in the space of admissible control parameters. The proposed approach allows applying the known theory and methods of fixed points to search for extremal controls. Illustrative examples of searching for extremal controls by the proposed method of fixed points in discrete-continuous problems on the extremum of the norm of the final state of a linear controlled system are given. The obtained extremal controls are compared with known solutions obtained within the framework of applying an alternative approach to control parameterization to the considered examples.