Boundary observation and control for fractional heat and wave equations

TL;DR

The paper addresses boundary observability and null controllability for nonlocal diffusion modeled by the fractional Laplacian $(-\Delta)^s$ on a bounded domain, focusing on the physically relevant range $s\in(1/2,1)$. The authors synthesize a fractional Pohozaev identity for the adjoint wave equation, transmutation techniques to carry observability from the wave to the heat equation, and a Lebeau–Robbiano–style frequency-by-frequency iteration to achieve full null controllability of finite-energy solutions for all $T>0$, with explicit dependence on spectral data via $T_0(J)=C\lambda_J^{\gamma(s)}$. This framework enables a novel multidimensional boundary observability result and establishes boundary controls supported on the active part of the boundary $\partial\Omega^+$, advancing the theory beyond known one-dimensional or spectral settings. The work highlights a sharp threshold at $s=1/2$ for controllability of the fractional heat equation and suggests directions for Carleman estimates and micro-local analysis to further extend these results. Overall, the study provides a robust toolkit for nonlocal diffusion control with potential applications in models featuring long-range interactions and memory effects.

Abstract

We establish boundary observability and control for the fractional heat equation over arbitrary time horizons $T > 0$, within the optimal range of fractional exponents $s \in (1/2, 1)$. Our approach introduces a novel synthesis of techniques from fractional partial differential equations and control theory, combining several key ingredients in an original and effective manner: 1. Boundary observability for low-frequency solutions of the fractional wave equation. We begin by analyzing the associated fractional wave equation. Using a fractional analogue of Pohozaev's identity, we establish a partial boundary observability result for the low-frequency solutions. The corresponding observability time horizon increases with the eigenmode frequency, reflecting the inherently slower propagation speed of the fractional waves. 2. Transmutation to the parabolic setting. Using transmutation techniques, we transfer the observability results from the wave setting to the parabolic one. This yields a frequency-dependent observability inequality for the fractional heat equation, which - via duality - enables control of its low-frequency components. 3. Frequency-wise iteration. Leveraging the dissipative nature of the fractional heat equation, we develop an iterative procedure to successively control the entire frequency spectrum of solutions. The condition $s \in (1/2, 1)$ is crucial in this analysis, as it guarantees sufficient decay of high-frequency components, enabling the convergence of the iteration. 4. Duality. By a duality argument, we derive boundary observability from the boundary controllability of the fractional heat equation. Remarkably, this type of boundary observability result is entirely new in the multi-dimensional setting and appears to be out of reach for existing methods. \end{itemize}

Theorems & Definitions (43)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Proposition 2.1
• Remark 2.2
• Remark 2.3: Hidden regularity of the fractional normal derivative
• Proposition 3.1
• Remark 3.2
• Lemma 4.1
• Proposition 4.2