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Averaged Controllability of Time-Fractional Schrödinger Equations with Random Quantum Diffusivity

Jon Asier Bárcena-Petisco, Salah-Eddine Chorfi, Fouad Et-tahri, Lahcen Maniar

TL;DR

This work analyzes averaged controllability for the time-fractional Schrödinger equation with random diffusivity $\xi$ (order $\alpha\in(0,1)$). It introduces a two-parameter fractional characteristic framework based on Mittag–Leffler functions to study averaging and proves that simultaneous null controllability holds only for a countable set of $\xi$ realizations, ruling out absolutely continuous laws. It also shows lack of exact averaged controllability for absolutely continuous $\xi$, and identifies a class $\mathcal{C}_{\alpha}$ for which null averaged controllability holds at any time from any sensor set of positive measure, via a parameter-free open-loop control. As a corollary, null controllability for the fractional biharmonic diffusion equation follows, and the results point to further numerical and analytical investigations of averaged controllability in fractional PDEs.

Abstract

This paper addresses the problem of averaged controllability for the time-fractional Schrodinger equation, where the quantum diffusivity parameter is a random variable with a general probability distribution. First, by exploiting the analyticity of the Mittag-Leffler function and Muntz's theorem, we show that the simultaneous null controllability of the system can occur only for a countable set of realizations of the random diffusivity. In particular, this implies the impossibility of simultaneous null controllability for absolutely continuous random diffusivity. Next, we prove the lack of exact averaged controllability for absolutely continuous random variables, irrespective of the control time. Furthermore, we introduce a new two-parameter fractional characteristic function, which allows us to construct a class of random variables satisfying null averaged controllability at any time from any arbitrary sensor set of positive Lebesgue measure. This is achieved using an open-loop control belonging to L^\infty and independent of the random parameter. In particular, we obtain the null controllability of the fractional biharmonic diffusion equation. Finally, we conclude with several remarks and open problems that merit future investigation.

Averaged Controllability of Time-Fractional Schrödinger Equations with Random Quantum Diffusivity

TL;DR

This work analyzes averaged controllability for the time-fractional Schrödinger equation with random diffusivity (order ). It introduces a two-parameter fractional characteristic framework based on Mittag–Leffler functions to study averaging and proves that simultaneous null controllability holds only for a countable set of realizations, ruling out absolutely continuous laws. It also shows lack of exact averaged controllability for absolutely continuous , and identifies a class for which null averaged controllability holds at any time from any sensor set of positive measure, via a parameter-free open-loop control. As a corollary, null controllability for the fractional biharmonic diffusion equation follows, and the results point to further numerical and analytical investigations of averaged controllability in fractional PDEs.

Abstract

This paper addresses the problem of averaged controllability for the time-fractional Schrodinger equation, where the quantum diffusivity parameter is a random variable with a general probability distribution. First, by exploiting the analyticity of the Mittag-Leffler function and Muntz's theorem, we show that the simultaneous null controllability of the system can occur only for a countable set of realizations of the random diffusivity. In particular, this implies the impossibility of simultaneous null controllability for absolutely continuous random diffusivity. Next, we prove the lack of exact averaged controllability for absolutely continuous random variables, irrespective of the control time. Furthermore, we introduce a new two-parameter fractional characteristic function, which allows us to construct a class of random variables satisfying null averaged controllability at any time from any arbitrary sensor set of positive Lebesgue measure. This is achieved using an open-loop control belonging to L^\infty and independent of the random parameter. In particular, we obtain the null controllability of the fractional biharmonic diffusion equation. Finally, we conclude with several remarks and open problems that merit future investigation.
Paper Structure (20 sections, 20 theorems, 118 equations)

This paper contains 20 sections, 20 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Miscellanea on fractional calculus and averaged control
  3. Fractional calculus
  4. Proof:
  5. Proof:
  6. Proof:
  7. Averaged controllability
  8. Proof:
  9. Proof:
  10. Lack of simultaneous null controllability
  11. Proof:
  12. Proof:
  13. Proof:
  14. Proof:
  15. Lack of exact averaged controllability
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let $\alpha>0$ and $1\le p\le \infty$. Then $I^{\alpha}_{0,t}$ acts continuously on $L^{p}(0,T;H)$, and we have

Theorems & Definitions (36)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Remark 1
  • Remark 2
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • Remark 3
  • ...and 26 more