Insensitizing control problems for the stabilized Kuramoto-Sivashinsky system

TL;DR

This paper investigates insensitizing controls for a coupled nonlinear stabilized Kuramoto-Sivashinsky system, using Carleman estimates and local inversion to ensure certain system functionals are insensitive to initial data perturbations.

Contribution

It introduces a novel approach to insensitizing control for coupled parabolic systems by reducing the problem to null-controllability and employing Carleman estimates tailored to the extended system.

Findings

Established null-controllability for the linearized cascade system

Proved local insensitizing control for the nonlinear system

Developed control strategies depending on the sentinel choice

Abstract

In this work, we address the existence of insensitizing controls for a nonlinear coupled system of fourth- and second-order parabolic equations known as the stabilized Kuramoto-Sivashinsky model. The main idea is to look for controls such that some functional of the state (the so-called sentinel) is locally insensitive to the perturbations of the initial data. Since the underlying model is coupled, we shall consider a sentinel in which we may observe one or two components of the system in a localized observation set. By some classical arguments, the insensitizing problem can be reduced to a null-controllability one for a cascade system where the number of equations is doubled. Upon linearization, the null-controllability for this new system is studied by means of Carleman estimates but unlike other insensitizing problems for scalar models, the election of the Carleman tools and the overall control strategy depends on the initial choice of the sentinel due to the (lack of) couplings arising in the extended system. Finally, the local null-controllability of the extended (nonlinear) system (and thus the insensitizing property) is obtained by applying a local inversion theorem.