Conditions and methods for improving control in nonlinear systems with constraints

In nonlinear on a state optimal control problems with constraints, a new approach to improving admissible controls is proposed. Modification of the conjugate system as a system of differential-algebraic equations allows to construct exact formulas for the increment of the functional, which are the basis for constructing the corresponding procedures for improving control, which are of a nonlocal nature. The problem of admissible control improvement is presented in the form of a system of functional equations, for the solution of which the apparatus of fixed points known in mathematics is modified. The approach under consideration is free from the laborious operation of parametric variation to improve control, which is typical for gradient methods. In addition, the initial approximation of iterative processes may not be an admissible control, which leads to an increase in the efficiency of the developed improvement procedures. The results of numerical calculations of a model problem from spacecraft mechanics are presented, illustrating the effectiveness of the proposed methods for constructing relaxation control sequences.