  We develop an asymptotical control theory for one of the simplest distributed
(infinite dimensional) oscillating systems, namely, for a closed string under a
bounded load applied to a single distinguished point. We find exact classes of
string states that admit complete damping and an asymptotically exact value of
the required time. By using approximate reachable sets instead of exact ones,
we design a feedback control, which turns out to be asymptotically optimal. The
main results are an exact algebraic formula for asymptotic shape of the
reachable sets, asymptotically optimal time of motion, and an asymptotically
optimal control thus constructed.
