Methods of bilinear approximations for solving optimal control problems

We attempt to solve a nonlinear for phase state optimal control problem basing on a quadratic approximation of the functional and on a procedure of weakly varying the controls. The auxiliary problem is bilinear for a pair "control variation–phase variation" and contains a parameter that characterizes the locality of the variation. The suggested iteration procedure improves the admissible controls that don't satisfy the maximum principle and also the singular controls that don't satisfy the second order optimality condition. A computational experiment for implementing the method to a number of applied problems was made.