Controllability of two-point boundary value problem for wave equations in $L^1$ and $L^2$ spaces: One dimensional case

TL;DR

This paper investigates the controllability of one-dimensional wave equations with two-point boundary conditions in $L^1$ and $L^2$ spaces, introducing new concepts and methods for minimum-input solutions.

Contribution

It introduces new concepts like TBVP input control, MS, and PMS, and studies their existence, uniqueness, and properties in $L^1$ and $L^2$ spaces.

Findings

Minimum inputs form a strip in $L^1$ space.

PMS always exists in $L^1$ and $L^2$ spaces.

An approximation method for constructing PMS is proposed.

Abstract

In this paper we discuss the controllability of two-point boundary value problem (TBVP) for one-dimensional wave equation. Some new concepts are introduced: TBVP input control problem, minimum-input solution (MS) and pre-minimum-input solution (PMS). We set the metric in $L^1$ and $L^2$ spaces on a closed set, and control the input to reach its minimum. And we mainly discuss the property of input, the existence and uniqueness of MS and PMS for $L^1$ and $L^2$ metric respectively. The minimum inputs lie on a strip in $L^1$ and PMS for $L^1$ and $L^2$ always exists. Furthermore, to construct PMS, we also introduce an approximation method which meets certain conditions.