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Approximate null-controllability of discrete heat equations with potentials on lattices

Yann Bourroux, Philippe Jaming, Yunlei Wang

Abstract

We investigate approximate null-controllability for semi-discrete heat equations on the lattice $h\mathbb{Z}^d$ with a potential. By establishing spectral inequalities for the discrete Schr{ö}dinger operator $P_h = -Δ_h + V$ on equidistributed sets, we derive observability estimates via the Lebeau-Robbiano method and the Hilbert Uniqueness Method. For bounded potentials, we obtain quantitative controllability results with explicit dependence on the potential and show near optimality of the geometric condition on the observation set. We also treat polynomial growth potentials, for which similar properties hold with weaker control cost estimates. These results extend discrete Carleman techniques to the full-space lattice setting and provide new spectral estimates for discrete Schr{ö}dinger operators.

Approximate null-controllability of discrete heat equations with potentials on lattices

Abstract

We investigate approximate null-controllability for semi-discrete heat equations on the lattice with a potential. By establishing spectral inequalities for the discrete Schr{ö}dinger operator on equidistributed sets, we derive observability estimates via the Lebeau-Robbiano method and the Hilbert Uniqueness Method. For bounded potentials, we obtain quantitative controllability results with explicit dependence on the potential and show near optimality of the geometric condition on the observation set. We also treat polynomial growth potentials, for which similar properties hold with weaker control cost estimates. These results extend discrete Carleman techniques to the full-space lattice setting and provide new spectral estimates for discrete Schr{ö}dinger operators.
Paper Structure (19 sections, 15 theorems, 218 equations)

This paper contains 19 sections, 15 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Main contributions.
  3. Spectral estimates.
  4. Approximate null-controllability.
  5. Observability inequalities.
  6. Organization of the paper.
  7. Setting and preliminaries on discrete Schrödinger operators
  8. Discrete calculus
  9. Potentials and control sets
  10. Schrödinger operators for bounded potentials
  11. Schrödinger operators for power growth potentials
  12. Proof of main theorems
  13. Carleman estimates
  14. Proof of the spectral inequalities for bounded potentials
  15. Proof of spectral inequalities for power growth potentials
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\omega$ be equidistributed and let $P_h=-\Delta_h+V$ with associated spectral projector $\Pi_{\mu,h}$. --- If $V\in \mathcal{C}_b(\mathbb R^d)$, there exist constants $C,\kappa>0$, $\varepsilon_0>0$, and $h_0>0$ depending only on $\omega$ such that for any $h\le h_0(1+\|V\|_{L^\infty}^{2/3})^{- --- If $V\in \mathcal{C}_b^1(\mathbb R^d)$, there exist constants $C>0$, $\varepsilon_0>0$, and $h_

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 20 more