New Algorithm for Constructing Asymptotic Solutions of Singularly Perturbed Optimal Control Problems with Intersecting Trajectories of the Degenerate State Equation

The paper deals with a new method for 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, the differential state equations contain a small parameter multiplying the derivative for the fast variables. If this parameter is zero, then the degenerate state equation for the fast variable has two distinct solutions for this fast variable and some of the corresponding trajectories for the slow variables meet at one interior point. The asymptotics contains regular functions depending on the original argument and boundary functions of four types, two of which are significant in a neighborhood of the point of intersection. The new method suggested for constructing the asymptotics is based on solving equations with a given value at the initial point or at the endpoint of the considered interval of the independent variable. Solutions of boundary value problems are not used.