A new algorithm of constructing asymptotic solution of singularly perturbed optimal control problems with intersecting trajectories of degenerate state equation

The paper deals with a new method of constructing asymptotic approximations of any order to a solution of a two-point boundary value problem following from control optimality conditions for singularly perturbed optimal control problems with a weak control in a critical case. Namely, differential state equations contain a small parameter before a derivative for fast variables. If this parameter is equal to zero, then the degenerate state equation for the fast variable has two different solutions with respect to this fast variable and some corresponding trajectories for slow variables intersect each other at one internal point. The asymptotics contains regular functions, depending on the original argument, and boundary functions of four types, two of them are essential in a vicinity of the intersection point. The suggested new method of asymptotics construction is based on solving equations with a fixed value in the initial point or in the end point of the considered interval for the independent variable. Solutions of boundary value problems are not used. Acknowledgements The work of Galina Kurina was supported by the Russian Science Foundation, Project No. 21-11-00202.