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Non null controllability of Stokes equations with memory

Enrique Fernández-Cara, José Lucas F. Machado, Diego A. Souza

Abstract

In this paper, we study the null controllability of the three-dimensional Stokes equations with a memory term. For any positive final time $T>0$, we construct initial conditions such that the null controllability does not hold even if the controls act on the whole boundary. Moreover, we also prove that this negative result holds for distributed controls.

Non null controllability of Stokes equations with memory

Abstract

In this paper, we study the null controllability of the three-dimensional Stokes equations with a memory term. For any positive final time , we construct initial conditions such that the null controllability does not hold even if the controls act on the whole boundary. Moreover, we also prove that this negative result holds for distributed controls.

Paper Structure

This paper contains 13 sections, 2 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. The radially symmetric eigenfunctions of the Stokes operator
  3. Lack of null controllability
  4. Construction of $\varphi^M$
  5. Estimate from below
  6. Estimate from above
  7. Construction of non-controllable initial data
  8. Additional comments and questions
  9. Lack of null controllability for the two-dimensional Stokes equations with a memory term
  10. Heat equation with memory
  11. Controls with less components
  12. Hyperbolic equations with memory
  13. Nonlinear systems with memory

Key Result

Theorem 1

Let $T>0$ be given. There exists initial data $y^0 \in H(\Omega)$ such that, for any control $v \in L^2(\gamma\times(0,T))$, the associated solution to eq_stokes+memory is not identically zero at time $T$.

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 1
  • Remark 1.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 4.1
  • Remark 4.2