Operator methods for searching degenerate extremal controls in optimal control problems linear in control

New methods of searching for extremal controls in the class of optimal control problems linear in control with degenerate conditions of the maximum principle are considered. The proposed approach is based on special forms of the maximum principle, which have the form of operator problems on a fixed point in the space of controls, which are equivalent to the well-known condition of the maximum principle in the considered class of optimal control problems. The considered operator forms of the maximum principle make it possible to obtain new equivalent formulations of the known conditions for the degeneracy of the maximum principle and to construct new algorithms for searching for extremal controls in degenerate problems of the class under consideration. A comparative analysis of the effectiveness of new algorithms for finding extremal controls is carried out on the example of a well-known model optimization problem for a quantum system, which is characterized by the degeneracy of the maximum principle.