On a method for finding extremal controls in systems with constraints

In the class of controlled systems with constraints, the conditions for improving and optimality of control are constructed and analyzed in the form of fixed point problems. This form allows one to obtain enhanced necessary optimality conditions in comparison with the known conditions and makes it possible to apply and modify the theory and methods of fixed points to search for extreme controls in optimization problems of the class under consideration. Fixed-point problems are constructed using the transition to auxiliary optimal control problems without restrictions with Lagrange functionals. An iterative algorithm is proposed for constructing a relaxation sequence of admissible controls based on the solution of constructed fixed point problems. The considered algorithm is characterized by the properties of nonlocal improvement of admissible control and the fundamental possibility of rigorous improvement of non-optimal controls satisfying the known necessary optimality conditions, in contrast to gradient and other local methods. The conditions of convergence of the control sequence for the residual of fulfilling the necessary optimality conditions are substantiated. A comparative analysis of the computational and qualitative efficiency of the proposed iterative algorithm for finding extreme controls in a model problem with phase constraints is carried out.