Asymptotics of Solution to Indirect Optimal Control Problem with Integral Convex Performance Index and Cheap Control

We consider the optimal control problem for a linear system with constant coefficients and a rapidly stabilizing indirect control with integral convex performance index and cheap control in the class of piecewise continuous controls with smooth geometric constraints on the control. We study the case where the optimal control of the limit problem remains unchanged, whereas for the original problem there is a unique point at which the control form is changed. Using the auxiliary parameter method, we show that the solution is expanded in a power series in a small parameter.