Optimal bounds for the cost of fast controls of a KdV system

TL;DR

This paper investigates the minimal control effort required for rapid boundary control of linearized and nonlinear KdV systems, overcoming spectral analysis challenges by relating to a modified system.

Contribution

It introduces a novel approach to derive optimal control bounds for KdV systems despite non-self-adjoint operator complications.

Findings

Derived explicit bounds for control costs in KdV systems.

Extended results to both linearized and nonlinear cases.

Provided new insights into boundary control of non-critical length KdV systems.

Abstract

We study the cost of fast controls for a linearized KdV system and a nonlinear KdV system locally, using right Neumann boundary control for non-critical lengths. Since the operator associated with the linearized system is neither self-adjoint nor skew-adjoint, its (known) spectral properties are not directly amenable to the moment method, leaving optimal cost bounds an open problem. We address this difficulty by shifting attention to a related KdV system and deriving the optimal bounds from the new one.