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Exact controllability to eigensolutions of the fractional heat equation via bilinear controls on N-dimensional domains

Rémi Buffe, Alessandro Duca

TL;DR

This work proves local exact controllability to eigensolutions for a bilinear fractional heat equation on $N$-dimensional domains under $s>\max(\frac{4N}{5},N-1)$, by linking local controllability to the null-controllability of linearized systems. The authors develop an abstract framework combining finite-dimensional moment problem solvability with the Lebeau--Robbiano--Miller lifting to extend controllability from a subspace to the full $L^2$ space, under Weyl asymptotics and spectral-gap conditions. They establish spectral tools, including an auxiliary sequence and block/weak gap conditions, and verify these in rectangle domains (Assumptions II), yielding explicit local controllability results (Main Theorem A and B) and corollaries. The results generalize bilinear controllability to higher dimensions and nonlocal diffusion, with concrete implications for rectangle geometries where the spectral gaps can be ensured via number-theoretic properties.)

Abstract

The exact controllability of heat-type equations in the presence of bilinear controls has been successfully studied in recent works, motivated by numerous applications to engineering, neurobiology, chemistry, and life sciences. Nevertheless, the result has only been achieved for $1$-dimensional domains due to the limitations of the existing techniques. In this work, we consider a fractional heat-type equation as $\partial_tψ+(-Δ)^sψ+\langle v(t), Q\rangle ψ(t)=0$ with $s>0$ and on a domain $Ω\subset \mathbb R^N$ for $N\in\mathbb N^*.$ We study the so-called exact controllability to the eigensolutions of the equations when $s>\max(\frac{4N}{5},N-1)$. The result is implied by the null controllability of a suitable linearized equation, and the main novelty of the work is the strategy of its proof. First, the null controllability in a finite-dimensional subspace has to be ensured via the solvability of a suitable moment problem. Explicit bounds on the control cost with respect to the dimension of the controlled space are also required. Second, the controllability can be extended to the whole Hilbert space, thanks to the Lebeau-Robbiano-Miller method, when the control cost does not grow too fast with respect to the dimension of the finite-dimensional subspace. We firstly develop our techniques in the general case when suitable hypotheses on the problem are verified. Secondly, we apply our procedure to the bilinear heat equation on rectangular domains, and we ensure its exact controllability to the eigensolutions.

Exact controllability to eigensolutions of the fractional heat equation via bilinear controls on N-dimensional domains

TL;DR

This work proves local exact controllability to eigensolutions for a bilinear fractional heat equation on -dimensional domains under , by linking local controllability to the null-controllability of linearized systems. The authors develop an abstract framework combining finite-dimensional moment problem solvability with the Lebeau--Robbiano--Miller lifting to extend controllability from a subspace to the full space, under Weyl asymptotics and spectral-gap conditions. They establish spectral tools, including an auxiliary sequence and block/weak gap conditions, and verify these in rectangle domains (Assumptions II), yielding explicit local controllability results (Main Theorem A and B) and corollaries. The results generalize bilinear controllability to higher dimensions and nonlocal diffusion, with concrete implications for rectangle geometries where the spectral gaps can be ensured via number-theoretic properties.)

Abstract

The exact controllability of heat-type equations in the presence of bilinear controls has been successfully studied in recent works, motivated by numerous applications to engineering, neurobiology, chemistry, and life sciences. Nevertheless, the result has only been achieved for -dimensional domains due to the limitations of the existing techniques. In this work, we consider a fractional heat-type equation as with and on a domain for We study the so-called exact controllability to the eigensolutions of the equations when . The result is implied by the null controllability of a suitable linearized equation, and the main novelty of the work is the strategy of its proof. First, the null controllability in a finite-dimensional subspace has to be ensured via the solvability of a suitable moment problem. Explicit bounds on the control cost with respect to the dimension of the controlled space are also required. Second, the controllability can be extended to the whole Hilbert space, thanks to the Lebeau-Robbiano-Miller method, when the control cost does not grow too fast with respect to the dimension of the finite-dimensional subspace. We firstly develop our techniques in the general case when suitable hypotheses on the problem are verified. Secondly, we apply our procedure to the bilinear heat equation on rectangular domains, and we ensure its exact controllability to the eigensolutions.
Paper Structure (16 sections, 19 theorems, 166 equations)

This paper contains 16 sections, 19 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Local controllability to eigensolutions in the abstract setting
  3. Local controllability to eigensolutions and null-controllability of the linearized problem
  4. Null-controllability in filtered spaces using the moment method
  5. The Lebeau-Robbiano-Miller method
  6. Some spectral properties and well-posedness
  7. An auxiliary spectral sequence
  8. The block gap condition
  9. The weak gap condition
  10. Spectral behavior in the rectangle case
  11. Local exact controllability to the eigensolutions
  12. Proof of Main Theorem A and Corollary \ref{['main_coro_1_gen']}
  13. Proof of Main Theorem B and Corollary \ref{['main_coro_1_rec']}
  14. Proof of the local bilinear controllability
  15. A preliminary lemma
  16. ...and 1 more sections

Key Result

Corollary 1.2

Let Assumptions I be verified. For any time $T>0$ and $j\in\mathbb{N}^*$, there exists $\delta>0$ such that, for any $\alpha,\beta \in\mathbb{R}^*$ with the same sign and for any initial state $\psi_0\in L^2(\Omega,\mathbb{R})$ with there exist $v_1\in L^2((0,T),\mathbb{R})$ and $v_2\in\mathbb{R}$ such that $\psi(T;\psi_0, v)$, the solution to main with control $v=(v_1,v_2)$, verifies

Theorems & Definitions (51)

  • proof
  • Remark 1.1
  • Corollary 1.2
  • proof
  • Remark 1.3
  • proof
  • Corollary 1.4
  • proof
  • Definition 2.1
  • Definition 2.2
  • ...and 41 more