Asymptotic expansion of a solution for the singularly perturbed optimal control problem with a convex integral quality index and smooth control constraints

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 smooth control constraints. In a general case, to solve such a problem, the Pontryagin maximum principle is applied as the necessary and sufficient optimum condition. The main difference from the preceding article [5] is that the terminal part of the convex integral quality index depends not only on slow, but also on fast variables. In a particular case, we derive an equation that is satisfied by an initial vector of the conjugate system. Then this equation is extended to the optimal control problem with the convex integral quality index for a linear system with the fast and slow variables. It is shown that the solution of the corresponding equation as $\varepsilon\to0$ tends to the solution of an equation corresponding to the limit problem. The results obtained are applied to study a problem which describes the motion of a material point in $\mathbb{R}^n$ for a fixed interval 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.