First asymptotic approximations to a solution of singularly perturbed optimal control problem with intersecting solutions of degenerate problem

Abstract

Using the direct scheme method for constructing asymptotic solution of optimal control problems with small parameters, consisting of immediate substituting a postulated asymptotic expansion of a solution into the problem condition and receiving problems for finding asymptotic terms, we construct formally the zero order asymptotic approximation to an optimal control and the first order approximation to an optimal trajectory of a singularly perturbed optimal control problem with a weakly controllable state equation, a cheap control, a not uniquely solvable degenerate state equation, fixed endpoints and with intersecting trajectories of the degenerate state equation corresponding to slow trajectories of the perturbed problem. Together with boundary-layer functions in the vicinities of both ends of the considered interval, the constructed asymptotics contains inner boundary-layer functions. An illustrative example is given.