Asymptotic expansion of the solution to an optimal control problem for a linear autonomous system with a terminal convex quality index depending on slow and fast variables

In this paper, we investigate a problem of optimal control over a finite time interval for a linear autonomous system with slow and fast variables in the class of piecewise continuous controls with smooth geometric constraints in the form of a ball. We consider a terminal convex quality index that depends on slow and fast variables. We substantiate a limit relation for the vector determining the optimal control as the small parameter tends to zero. We refine the limit relation for the case of an indirect control problem with a terminal quality index, which is the sum of values of two strictly convex continuously differentiable functions, the first of which depends only on slow variables, and the second one depends only on fast variables and has a minimum at zero. In doing so, we show that the first component of the determining vector converges to the determining vector of the limit problem while the second component tends to zero. In the problem of indirect control of a system of material points in a medium with resistance, we obtain the complete asymptotics of the determining vector in powers of a small parameter.