Geodesic
Generic profit singularities in time-averaged optimization for cyclic processes in polydynamical systems
Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Mathematical Control Theory and Mechanics (Suzdal, July 3–7, 2009), Tome 42 (2011), pp. 95-117.

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We consider the optimization problem of maximizing the time-averaged profit for the motion of a smooth polydynamical system on the circle in the presence of a smooth profit density. If the problem depends on a $k$-dimensional parameter, then the optimal averaged profit is a function of the parameter. It is known from [4] that an optimal motion can always be selected among stationary strategies and a special type of periodic motions called $level cycles$. We present a classification of all generic singularities of the optimal averaged profit if $k\le2$ and the maximum is provided by level cycles. 
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A. A. Davydov; H. Mena-Matos; C. S. Moreira. Generic profit singularities in time-averaged optimization for cyclic processes in polydynamical systems. Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Mathematical Control Theory and Mechanics (Suzdal, July 3–7, 2009), Tome 42 (2011), pp. 95-117. http://geodesic.mathdoc.fr/item/CMFD_2011_42_a9/

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