Null Controllability of the Kuramoto-Sivashinsky Equation on star-shaped trees

TL;DR

This paper investigates the null controllability of the linear Kuramoto-Sivashinsky equation on star-shaped networks, demonstrating controllability under specific boundary conditions and parameters, with implications for control theory on complex structures.

Contribution

It establishes null controllability results for the Kuramoto-Sivashinsky equation on star-shaped trees with boundary controls, highlighting the role of the anti-diffusion parameter and boundary conditions.

Findings

Controllability holds if the anti-diffusion parameter is outside a critical countable set.

Different boundary conditions lead to different critical sets for controllability.

Boundary controls on external vertices can achieve null controllability under specified conditions.

Abstract

In this paper we treat controllability properties for the linear Kuramoto-Sivashinsky equation on a network with two types of boundary conditions. More precisely, the equation is considered on a star-shaped tree with Dirichlet and Neumann boundary conditions. By using the moment theory we can derive null-controllability properties with boundary controls acting on the external vertices of the tree. In particular, the controllability holds if the anti-diffusion parameter of the involved equation does not belong to a critical countable set of real numbers. We point out that the critical set for which the null-controllability fails differs from the first model to the second one.