Asymptotically optimal control for a simplest distributed system

TL;DR

This paper develops an asymptotically optimal feedback control law for damping a distributed system modeled by a nonlinear wave equation, ensuring minimal time to reach a desired state with proven existence and uniqueness of solutions.

Contribution

It introduces a novel control law resembling dry friction that achieves asymptotic optimality for damping a string under bounded load.

Findings

Control law is asymptotically optimal for damping

Existence and uniqueness of solutions are established

System reaches a bounded neighborhood of the target state

Abstract

We study the problem of the minimum-time damping of a closed string under a bounded load, applied at a single fixed point. A constructive feedback control law is designed, which allows bringing the system to a bounded neighbourhood of the terminal manifold. The law has the form of the dry friction at the point, where the load is applied. The motion under the control is governed by a nonlinear wave equation. The existence and uniqueness of solution of the Cauchy problem for this equation are proved. The main result is the asymptotic optimality of the suggested control law.