Asymptotics of a solution to a terminal control problem with two small parameters

An optimal control problem is considered in the class of piecewise continuous controls with smooth geometric constraints on a fixed interval of time, for a linear autonomous system with two small positive independent parameters, one of which, ε, multiplies some derivatives in the equations of the system, while the other, μ, is involved in the initial conditions. The quality functional is convex and terminal, and depends only on the values of the slow variables at the terminal time. A limit relation as the small parameters tend independently to zero is verified for the vector describing the optimal control. Two cases are considered: the regular case, when the optimal control in the limiting problem is continuous, and the singular case, when this control has a singularity. In the regular case the solution is shown to expand in a power series in ε and μ, while in the singular case the solution is asymptotically represented by an Erdélyi series - in either case the asymptotics is with respect to the standard gauge sequence εᵏ+μᵏ, as ε +μ →0. Bibliography: 23 titles.