  This paper considers the optimal control problem of connecting two periodic
trajectories with maximal persistence. A maximally persistent trajectory is
close to the periodic type in the sense that the norm of the image of this
trajectory under the operator defining the periodic type is minimal among all
trajectories. A solution is obtained in this paper for the case when the two
trajectories have the same period but it turns out to be only piecewise
continuous and so an alternate norm is employed to obtain a continuous
connection. The case when the two trajectories have different but rational
periods is also solved. The problem of connecting periodic trajectories is of
interest because of the observation that the operating points of many
biological and artificial systems are limit cycles and so there is a need for a
unified optimal framework of connections between different operating points.
This paper is a first step towards that goal.
