# Asymptotic control theory for a closed string

## TL;DR

This paper develops an asymptotic control theory for a closed string system, providing explicit solutions and feedback control strategies that are proven to be asymptotically optimal for damping and controlling the string's oscillations.

## Contribution

It introduces a novel asymptotic control approach for a distributed oscillating system, with explicit algebraic solutions and a proof of asymptotic optimality.

## Key findings

- Identified classes of string states allowing complete damping.
- Designed a feedback control that is asymptotically optimal.
- Proved the existence of motion under the control using explicit solutions.

## 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.