Cost for a controlled linear KdV equation

TL;DR

This paper introduces a constructive method to determine the quantitative observability constant for the controlled linear KdV equation with Neumann control, addressing a longstanding unknown in the mathematical control theory of this PDE.

Contribution

A new constructive approach is developed to explicitly compute the observability constant for the linearized KdV equation, improving upon previous contradiction-based proofs.

Findings

Explicit value of the observability constant obtained

Method applicable to similar PDE control problems

Enhances understanding of controllability in linear KdV equations

Abstract

The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [25]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathematical theories are built on this totally unknown constant. We introduce a constructive method that gives the quantitative value of this constant.