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Optimal bounds for the boundary control cost of one-dimensional fractional Schrödinger and heat equations

Hoai-Minh Nguyen

TL;DR

This work determines sharp bounds on the boundary-control cost for one-dimensional fractional Schrödinger and heat equations with $s\in(\tfrac12,1)$. It combines two complementary approaches: a singular-control/complex-analytic route to obtain tight lower bounds, and a moment-method construction with carefully designed entire functions to achieve matching upper bounds. The results yield exponential-in-$T^{\!-\tau}$ cost growth, with $\tau=\tfrac{1}{2s-1}$ for the Schrödinger case and corresponding constants $\nu_s,\mu_s$ governing the rate, under general spectral assumptions $\lambda_n=\gamma_n^{\alpha/2}$ ($\alpha=2s$). The analysis also provides generality with respect to eigenvalue sequences satisfying quantified gaps and growth, and extends classical 1D results to the fractional setting. These bounds illuminate the cost of rapid boundary control and underpin strategies for stabilization via Gramian-based or related approaches in fractional diffusion contexts.

Abstract

We derive sharp bounds for the boundary control cost of the one-dimensional fractional Schrödinger and heat equations. The analysis of the lower bound is based on the study of the control cost of a related singular boundary control problem in finite time, using tools from complex analysis. The analysis of the upper bound relies on the moment method, involving estimates of the Fourier transform of a class of compactly supported functions.

Optimal bounds for the boundary control cost of one-dimensional fractional Schrödinger and heat equations

TL;DR

This work determines sharp bounds on the boundary-control cost for one-dimensional fractional Schrödinger and heat equations with . It combines two complementary approaches: a singular-control/complex-analytic route to obtain tight lower bounds, and a moment-method construction with carefully designed entire functions to achieve matching upper bounds. The results yield exponential-in- cost growth, with for the Schrödinger case and corresponding constants governing the rate, under general spectral assumptions (). The analysis also provides generality with respect to eigenvalue sequences satisfying quantified gaps and growth, and extends classical 1D results to the fractional setting. These bounds illuminate the cost of rapid boundary control and underpin strategies for stabilization via Gramian-based or related approaches in fractional diffusion contexts.

Abstract

We derive sharp bounds for the boundary control cost of the one-dimensional fractional Schrödinger and heat equations. The analysis of the lower bound is based on the study of the control cost of a related singular boundary control problem in finite time, using tools from complex analysis. The analysis of the upper bound relies on the moment method, involving estimates of the Fourier transform of a class of compactly supported functions.
Paper Structure (13 sections, 11 theorems, 199 equations)

This paper contains 13 sections, 11 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Lower bound on the cost of fast controls for the Schrödinger equation
  3. Preliminaries
  4. Lower bound on the cost of fast controls for a singular Schrödinger control system
  5. Proof of part $a)$ of \ref{['thm-S']}
  6. Lower bound on the cost of fast controls for the heat equation
  7. Preliminaries
  8. Lower bound on the cost of fast controls for a singular heat control system
  9. Proof of part $a)$ of \ref{['thm-H']}
  10. Technical lemmas
  11. Upper bound of the cost of the fast controls
  12. Proof of part $b)$ of \ref{['thm-S']}
  13. Proof of part $b)$ of \ref{['thm-H']}

Key Result

Theorem 1.1

Let $1/2 < s < 1$ and $0< T < 1/2$, and let $y_0 \in L^2(0, 1)$. Set We have Here $c_1, c_2, c_3$ are positive constants depending only on $s$, and the constant $\nu_s$ is defined by

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof
  • Remark 2.1
  • Lemma 3.1
  • proof
  • ...and 16 more