The asymptotics of the optimal value of the performance functional in a linear optimal control problem in the regular case

An optimal control problem is investigated for a linear system with fast and slow variables, a convex terminal performance functional depending on the slow variables, and smooth geometric constraints on the control. Sufficient regularity conditions are presented for the asymptotics of a solution of this problem, and a complete asymptotic expansion of the optimal value of the performance functional in powers of a small parameter is constructed.