One approach to improving control in systems with constraints based on a boundary value problem

State-nonlinear optimal control problems are considered in the presence of an additional constraint on the phase trajectory of terminal type. For the considered problems, a new approach to improving admissible controls is proposed based on the exact formula for the increment of the Lagrange functional by solving a special boundary value problem, which is significantly simpler than the boundary value problem of the maximum principle. To solve it, a special iterative process is proposed based on the sequential solution of Cauchy problems and the determination of an auxiliary factor from the condition of fulfilling the functional constraint. Unlike most standard numerical methods for optimal control problems (needle linearization, conditional gradient), the proposed approach does not use the operation of changing a small parameter in the vicinity of the current approximation. In addition, it has the ability to strictly improve non-optimal extremal controls.