Solution of the problem of the extremum of the system final state norm based on finite-dimensional approximations of control

The problem of optimizing of the final state norm of a nonlinear system in the class of piecewise constant controls is considered. In the class of discrete-continuous controlled systems under consideration, a constructive condition for control optimality is constructed in the form of a fixed point problem in the space of admissible control parameters. The proposed approach allows one to apply the well-known theory and methods of fixed points to search for extremal controls. Illustrative examples of searching for extremal controls by the proposed fixed point method are given in the well-known problem of finding the extremum of the final state norm of a linear controlled system. The obtained extremal controls are compared with known solutions obtained within the framework of applying an alternative approach of control parameterization to the considered examples.