Regularisation of maximum principle and improvement methods in quadratic problems of optimal control

Quadratic, nonconvex problem with bounded control is considered. A procedure of phase regularisation along the fixed admissible pair (control, state) is applied with the goal of improvement of the maximum principle and corresponding methods in given problem. The first result is a new optimality condition which includes the maximum principle as a corolary. The second result is a new method of successive approximations which has more effective characteristics of improement and convergence than known ones.