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Control of the half-heat equation

Andreas Hartman, Armand Koenig

TL;DR

This work analyzes null-controllability for the half-heat equation on the unit circle with controls supported on a sub-arc {\omega}. It develops a complex-analytic framework, connecting the L^2 and H^2 settings via the Riesz projection and Bergman/Hardy spaces, and establishes a time-independent characterization of null-controllable states in the H^2 setting through an observability inequality on Bergman spaces over a geometry {\Omega_T}. A key achievement is expressing necessary and sufficient conditions in terms of boundary data supported on {\omega}, including a pseudo-Carleson-measure criterion, and linking these to a Poisson-kernel/Stokes-argument formulation. The L^2 problem is then related to the H^2 results by a frequency-separation argument, yielding density results for null-controllable data and a precise correspondence between NC_{L^2}(ω) and joint NC_{H^2}(ω) conditions. Overall, the paper blends complex analysis, Bergman/Hardy space theory, and control theory to characterize reachable/steerable states for the half-heat equation, with implications for fractional and Baouendi–Grushin-type controllability via similar analytic tools.

Abstract

In this paper we investigate null-controllable initial states of the half heat equation controlled from a sub-arc $ω$ of the unit circle. We also study the projection on positive frequencies of the half-heat equation. For this projected half-heat equation, we obtain necessary as well as sufficient conditions for an initial condition to be null-controllable. These conditions, which are almost sharp, are expressed in term of projections on positive frequencies of functions supported on $ω$. From these results, and with the help of classical results on sum of holomorphic and anti-holomorphic functions, we also treat the (unprojected) half-heat equation. Surprisingly, without using our conditions on null-controllable states, we are able to show that the space of null-controllable functions does not depend on time by using a result of separation of singularities for holomorphic functions.

Control of the half-heat equation

TL;DR

This work analyzes null-controllability for the half-heat equation on the unit circle with controls supported on a sub-arc {\omega}. It develops a complex-analytic framework, connecting the L^2 and H^2 settings via the Riesz projection and Bergman/Hardy spaces, and establishes a time-independent characterization of null-controllable states in the H^2 setting through an observability inequality on Bergman spaces over a geometry {\Omega_T}. A key achievement is expressing necessary and sufficient conditions in terms of boundary data supported on {\omega}, including a pseudo-Carleson-measure criterion, and linking these to a Poisson-kernel/Stokes-argument formulation. The L^2 problem is then related to the H^2 results by a frequency-separation argument, yielding density results for null-controllable data and a precise correspondence between NC_{L^2}(ω) and joint NC_{H^2}(ω) conditions. Overall, the paper blends complex analysis, Bergman/Hardy space theory, and control theory to characterize reachable/steerable states for the half-heat equation, with implications for fractional and Baouendi–Grushin-type controllability via similar analytic tools.

Abstract

In this paper we investigate null-controllable initial states of the half heat equation controlled from a sub-arc of the unit circle. We also study the projection on positive frequencies of the half-heat equation. For this projected half-heat equation, we obtain necessary as well as sufficient conditions for an initial condition to be null-controllable. These conditions, which are almost sharp, are expressed in term of projections on positive frequencies of functions supported on . From these results, and with the help of classical results on sum of holomorphic and anti-holomorphic functions, we also treat the (unprojected) half-heat equation. Surprisingly, without using our conditions on null-controllable states, we are able to show that the space of null-controllable functions does not depend on time by using a result of separation of singularities for holomorphic functions.

Paper Structure

This paper contains 23 sections, 36 theorems, 159 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Overview of the results
  4. Discussion and examples
  5. Bibliographical comments
  6. Control of the fractional heat equation
  7. Complex analysis tools for control theory
  8. Null-controllable states for other equations
  9. Notations
  10. The H² problem
  11. The observability inequality
  12. Independence of null-controllable states with respect to time
  13. Necessary conditions and sufficient conditions via complex analysis
  14. Characterization thanks to the Poisson kernel
  15. Formal argument
  16. ...and 8 more sections

Key Result

Proposition 1.2

Let $\omega$ be a strict open subset of $\mathbb T$ and $T>0$. Let $\Omega_T \mathrel{\mathord{:}\mathord=} \{ z\in \mathbb C,\ 1<|z|<\mathrm e^T,\ z/|z| \in \omega\}$ (see fig-Omega). Let $f_0 \in H^2$. The following assertions are equivalent: If these assertions are satisfied, we can choose the control $u$ that steers $f_0$ to $0$ such that $\|u\|_{L^2([0,T]\times\omega)} \leq \mathrm e^T C_{f

Figures (5)

  • Figure 1: Illustration of $\Omega_T^*$ (yellow partial ring inside of ${\mathbb{D}}$) and $\Omega_T$ (red partial ring outside ${\mathbb{D}}$).
  • Figure 2: A plot of $\theta\mapsto P_+g(\mathrm e^{\mathrm i\theta})$ for $\theta_0 = \pi/4$. The real part is in blue, the imaginary part in green.
  • Figure 3: Illustration of $\widehat{\Omega}_{T'}$ and $\Omega_T$ when $T'<T$. The partial ring $\Omega_T$ is in red (including the reddish-yellow part). $\widehat{\Omega}_{T'}$ is in yellow (also including the reddish-yellow part). The intersection of $\Omega_T$ and $\widehat{\Omega}_{T'}$ is $\Omega_{T'}$ colored in reddish-yellow.
  • Figure 4: Left figure: illustration of $\widetilde{\Omega}_T$ (in green), with $\Omega_T$ in red, the paths $\Gamma_i$ ($i=0,1,2$) and the points $\zeta_i'$ and $\mathrm e^{\eta_0}\zeta'_i$ ($i=1,2$). Right figure: illustration of $\widetilde{\Omega}_{T,\eta}$ (in green) and the path $\omega_\eta$.
  • Figure 5: Illustration of the paths $\gamma_\epsilon$ and $\gamma_+$. They are used in the proof of \ref{['th-lemma-ortho']} to prove with Morera's theorem that the function $H(z) = f(z)$ if $|z|<1$ and $f(z) = h(z)$ if $|z|\geq 1$ and $z\in\notin(-\infty,0]$ is holomorphic.

Theorems & Definitions (70)

  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 60 more