Local controllability of the Cahn-Hilliard-Burgers' equation around certain steady states

TL;DR

This paper establishes the local controllability of the one-dimensional Cahn-Hilliard-Burgers' equation around certain steady states using linearization, Carleman inequalities, and fixed point arguments.

Contribution

It introduces a novel controllability analysis for the coupled second and fourth order parabolic system with localized control, including new Carleman inequalities.

Findings

Null controllability of the linearized system is proven.

Local controllability of the nonlinear system is achieved.

A new Carleman inequality for coupled parabolic systems is derived.

Abstract

In this article we study the local controllability of the one-dimensional Cahn-Hilliard-Navier-Stokes equation, that is Cahn-Hilliard-Burgers' equation, around a certain steady state using a localized interior control acting only in the concentration equation. To do it, we first linearize the nonlinear equation around the steady state. The linearized system turns out to be a system coupled between second order and fourth order parabolic equations and the control acts in the fourth order parabolic equation. The null controllability of the linearized system is obtained by a duality argument proving an observability inequality. To prove the observability inequality, a new Carleman inequality for the coupled system is derived. Next, using the source term method, it is shown that the null controllability of the linearized system with non-homogeneous terms persists provided the non-homogeneous terms satisfy certain estimates in a suitable weighted space. Finally, using a Banach fixed point theorem in a suitable weighted space, the local controllability of the nonlinear system is obtained.