Asymptotic expansion of a solution for one singularly perturbed optimal control problem in $\mathbb{R}^n$ with a convex integral quality index

The paper deals with the problem of optimal control with a convex integral quality index for a linear steady-state control system in the class of piecewise continuous controls with a smooth control constraints. In a general case, for solving such a problem, the Pontryagin maximum principle is applied as the necessary and sufficient optimum condition. In this work, we deduce an equation to which an initial vector of the conjugate system satisfies. Then, this equation is extended to the optimal control problem with the convex integral quality index for a linear system with a fast and slow variables. It is shown that the solution of the corresponding equation as $\varepsilon\to 0$ tends to the solution of an equation corresponding to the limit problem. The results received are applied to study of the problem which describes the motion of a material point in $\mathbb{R}^n$ for a fixed period of time. The asymptotics of the initial vector of the conjugate system that defines the type of optimal control is built. It is shown that the asymptotics is a power series of expansion.