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

Manika Bag, Sheetal Dharmatti, Subrata Majumdar, Debanjana Mitra

TL;DR

This work studies local null controllability for the one-dimensional Cahn-Hilliard-Burgers' equation around a steady state, using a localized interior control acting in the concentration equation. The authors linearize about a steady state to obtain a coupled system of a second-order heat equation and a fourth-order Cahn-Hilliard equation, and prove null controllability of the linearized system via a novel Carleman-based observability inequality for the coupled system. They then extend the result to the nonlinear system by employing the source-term method and a Banach fixed-point argument in weighted spaces, yielding local null controllability of the nonlinear CHB around the steady state. The analysis hinges on a new joint Carleman inequality for the coupled second- and fourth-order parabolic equations with the given boundary conditions, enabling explicit control costs and a robust framework that could inform higher-dimensional CHNS controllability studies.

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.

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

TL;DR

This work studies local null controllability for the one-dimensional Cahn-Hilliard-Burgers' equation around a steady state, using a localized interior control acting in the concentration equation. The authors linearize about a steady state to obtain a coupled system of a second-order heat equation and a fourth-order Cahn-Hilliard equation, and prove null controllability of the linearized system via a novel Carleman-based observability inequality for the coupled system. They then extend the result to the nonlinear system by employing the source-term method and a Banach fixed-point argument in weighted spaces, yielding local null controllability of the nonlinear CHB around the steady state. The analysis hinges on a new joint Carleman inequality for the coupled second- and fourth-order parabolic equations with the given boundary conditions, enabling explicit control costs and a robust framework that could inform higher-dimensional CHNS controllability studies.

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.
Paper Structure (15 sections, 15 theorems, 170 equations)

This paper contains 15 sections, 15 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Well-posedness of the linear system
  3. Carleman inequality
  4. Weight function
  5. Some useful notations
  6. Some standard Carleman estimates
  7. Proof of \ref{['thm-Fur-Ima']}
  8. Controllability of the linear system
  9. Observability inequality
  10. Proof of \ref{['Thm-linear']}
  11. Null controllability of the linearized model with nonhomogeneous source term
  12. Local null controllability of the nonlinear system
  13. Further remarks and open problems
  14. Acknowledgment
  15. Some auxiliary result

Key Result

Theorem 1.3

Let $(\overline{u}, \overline{\phi})$ be as mentioned in rem-assump-steadystate and let $\mathcal{O}$ be any open subset of $(0, 1)$. For any time $T>0$ and any initial data $(w_0,\psi_0)\in (L^2(0,1))^2,$ there exists a control $h\in L^2(0, T; L^2(\mathcal{O}))$ such that the system CH--in admits Moreover, the control function $h$ has the following estimate: where the constant $M>0$ is indepen

Theorems & Definitions (27)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • Theorem 2.4
  • ...and 17 more