Title: Asymptotic control theory for a closed string,
Abstract: We develop an asymptotical control theory for one of the simplest distributed
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
dry-friction like feedback control, which turns out to be asymptotically
optimal. We prove the existence of motion under the control using a rather
explicit solution of a nonlinear wave equation. Remarkably, the solution is
determined via purely algebraic operations. The main result is a proof of
asymptotic optimality of the control thus constructed.