Nonlocal descent method at a set of admissible controls in optimal control problems with constraints on the state

The article considers the method of constructing a relaxation sequence in the class of admissible controls in optimal control problems with constraints in the state. Relaxation is carried out according to the functional of auxiliary extension problem and based on the construction of nonlocal conditions for improving control in the extension problem in the form of a fixed point problem. This form makes it possible to apply and modify the well-known apparatus of fixed-points theorem and methods for the search of improving admissible control. The construction of improving controls in the class of admissible controls allows us to apply the theory of extension for justification of sufficient conditions for constructing a minimizing sequence. The comparative effectiveness of the proposed descent method is illustrated by calculation of the well-known model problem.