Asymptotic expansion of solution of one singularly perturbed optimal control problem with convex integral performance index and cheap control

We consider the problem of optimal control for a linear system with constant coefficients with convex integral performance index contains small parameter in integral part in the class of piecewise continuous controls with a smooth control constraints. The article is based on asymptotic of the initial vector of the adjoint state, which determines the type of optimal control. In the time-optimal control problem limit problem has a solution with discontinuous control but the perturbed problem has continuous control. It is proved that in this case the solution is decomposed in an series with a complex structure. But optimal control is decomposed in a power series of expansion in small parameter in the cheap control problem.