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, $\varepsilon$, multiplies some derivatives in the equations of the system, while the other, $\mu$, 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 instant. 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 if 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 $\varepsilon$ and $]\mu$, 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 $\varepsilon^k+\mu^k$, as $\varepsilon+\mu\to0$.