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On the controllability of some equations of Sobolev-Galpern type

F. W. Chaves-Silva, Diego A. Souza

Abstract

In this paper we deal with the controllability problem for some Sobolev type equations. We show that the equations cannot be driven to zero if the control region is strictly supported within the domain. Nevertheless, we also prove that it is possible to control the equations using controls which have a moving support, under some assumptions on its movement.

On the controllability of some equations of Sobolev-Galpern type

Abstract

In this paper we deal with the controllability problem for some Sobolev type equations. We show that the equations cannot be driven to zero if the control region is strictly supported within the domain. Nevertheless, we also prove that it is possible to control the equations using controls which have a moving support, under some assumptions on its movement.

Paper Structure

This paper contains 9 sections, 13 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Negative Controllability Results
  3. Barenblatt-Zheltov-Kochina with fixed controls
  4. Benjamin-Bona-Mahony with fixed controls
  5. Positive Controllability Results
  6. Barenblatt-Zheltov-Kochina with moving controls
  7. Benjamin-Bona-Mahony with moving controls
  8. Carleman inequality for the Laplace operator
  9. Carleman inequality for the Laplace operator in $H^{-1}(\Omega)$

Key Result

Theorem 1.1

Let $T>0$ and $\omega \subsetneqq \Omega$ be a fixed open set. If $\mathcal{O}=\omega\times (0,T)$ then system eq:BZK is not null controllable at time $T$, i.e., there exists $y_0\in H^2(\Omega) \times H^1_0(\Omega)$ such that the null controllability of system eq:BZK fails.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.4: Moving control region
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • proof : Proof of Claim \ref{['C:C1']}
  • proof : Proof of Claim \ref{['C:C2']}
  • proof
  • ...and 14 more