On the nonlocal stabilization by starting control of the normal equation generated from Helmholtz system

TL;DR

This paper proves that a simplified control function can stabilize the 3D Helmholtz system without the previously necessary term, improving the understanding of control strategies for such PDEs.

Contribution

It demonstrates that the control term previously thought essential for stabilization can be omitted, simplifying the control approach for the Helmholtz system.

Findings

The control function can be modified to exclude the problematic term.

Stabilization to zero is achievable with the revised control.

The results extend previous stabilization methods for the Helmholtz system.

Abstract

We consider the problem of stabilization to zero of semilinear normal parabolic equations connected with the 3D Helmholtz system with periodic boundary conditions and arbitrary initial datum. This problem was previously studied in \cite{FSh16}. As it was recently revealed, the control function suggested in that work contains a term impeding transference the stabilization construction on the 3D Helmholtz system. The main concern of this article is to prove that this term is not necessary for the stabilization result, and therefore the control function can be changed by a proper way.