Uniqueness of a cycle with discounting that is optimal with respect to the average time profit

For cyclic processes modeled by periodic motions of a continuous control system on a circle, we prove the uniqueness of a cycle maximizing the average one-period time profit in the case of discounting provided that the minimum and maximum velocities of the system coincide at some points only and the profit density is a differentiable function with a finite number of critical points. The uniqueness theorem is an analog of Arnolds theorem on the uniqueness of such a cycle in the case when the profit gathered along the cycle is not discounted.