Asymptotic control theory for a closed string II

TL;DR

This paper develops an asymptotic control theory for a closed string system, providing exact formulas for reachable states, optimal damping time, and feedback control design for infinite-dimensional oscillating systems.

Contribution

It introduces a novel asymptotic control framework for distributed oscillating systems, including explicit formulas and asymptotically optimal control strategies.

Findings

Exact algebraic formula for asymptotic reachable set shape

Asymptotically optimal damping time and control

Design of feedback control achieving asymptotic optimality

Abstract

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.