  We consider a vibrating string that is fixed at one end with Neumann control
action at the other end. We investigate the optimal control problem of steering
this system from given initial data to rest, in time T , by minimizing an
objective functional that is the convex sum of the L 2-norm of the control and
of a boundary Neumann tracking term. We provide an explicit solution of this
optimal control problem, showing that if the weight of the tracking term is
positive, then the optimal control action is concentrated at the beginning and
at the end of the time interval, and in-between it decays exponentially. We
show that the optimal control can actually be written in that case as the sum
of an exponentially decaying term and of an exponentially increasing term. This
implies that, if the time T is large the optimal trajectory approximately
consists of three arcs, where the first and the third short-time arcs are
transient arcs, and in the middle arc the optimal control and the corresponding
state are exponentially close to 0. This is an example for a turnpike
phenomenon for a problem of optimal boundary control. If T = +$\infty$
(infinite horizon time problem), then only the exponentially decaying component
of the control remains, and the norms of the optimal control action and of the
optimal state decay exponentially in time. In contrast to this situation if the
weight of the tracking term is zero and only the control cost is minimized,
then the optimal control is distributed uniformly along the whole interval [0,
T ] and coincides with the control given by the Hilbert Uniqueness Method.
