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Local exact controllability to constant trajectories for Navier-Stokes-Korteweg model

Adrien Tendani Soler

Abstract

In this article, we study the local exact controllability to a constant trajectory for a compressible Navier-Stokes-Korteweg system on the torus in dimension $ d\in\{1,2,3\}$ when the control acts on an open subset. To be more precise, we obtain the local exact controllability to the constant state $(ρ_{\star},0)$ for arbitrary small positive times and without any geometric condition on the control region. In order to do so, we analyze the control properties of the linearized equation, and present a detailed study of the observability of the adjoint equations. In particular, we shall exhibit the parabolic (possibly also dispersive) structure of these adjoint equations. Based on that, we will be able to recover observability of the adjoint system through Carleman estimates.

Local exact controllability to constant trajectories for Navier-Stokes-Korteweg model

Abstract

In this article, we study the local exact controllability to a constant trajectory for a compressible Navier-Stokes-Korteweg system on the torus in dimension when the control acts on an open subset. To be more precise, we obtain the local exact controllability to the constant state for arbitrary small positive times and without any geometric condition on the control region. In order to do so, we analyze the control properties of the linearized equation, and present a detailed study of the observability of the adjoint equations. In particular, we shall exhibit the parabolic (possibly also dispersive) structure of these adjoint equations. Based on that, we will be able to recover observability of the adjoint system through Carleman estimates.
Paper Structure (12 sections, 16 theorems, 234 equations)

This paper contains 12 sections, 16 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Controllability of the heat equation
  3. Construction of the weight function
  4. Controllability results for the complex coefficients heat equation
  5. Strategy
  6. Diagonalizable case
  7. Non-diagonalizable case
  8. Controllability of (\ref{['NSK quantique Slcc**']}): The diagonalizable case.
  9. Controllability of (\ref{['NSK quantique Slcc** Jordan']}): The non-diagonalizable case
  10. Proof of Theorem \ref{['theorem 4.1 EGG']}
  11. Proof of Theorem \ref{['Controle intern de NSK quantique sur le tore']}
  12. Proof of the Carleman estimate

Key Result

Theorem 1.1

Let $d\in\{1,2,3\}$, $L>0$, and $\omega$ be a non-empty open subset of $\mathbb{T}_L$. Let $\rho_{\star}>0$, and let us assume that Then there exists $\delta>0$ such that, for all $(\rho_0,u_0)\in H^2(\mathbb{T}_L)\times H^1(\mathbb{T}_L)$ satisfying there exist a control $(v_{\rho},v_u)\in L^2(0,T;H^2(\mathbb{T}_L))\times L^2(0,T;H^1(\mathbb{T}_L))$ and a corresponding controlled trajectory $(\

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Theorem 3.1
  • Theorem 3.2
  • ...and 15 more