Boundary observation and control for fractional heat and wave equations

Abstract

We establish boundary null controllability of the heat equation driven by the integral fractional Laplacian $(-\Delta {)}^s$ on a bounded smooth domain for every $T>0$ and every fractional exponent $s\in (1/2,1)$. The control acts through the singular boundary trace naturally associated with the fractional Dirichlet problem. The main ingredient is a frequency-dependent boundary observability inequality for the associated fractional wave equation, obtained by combining multiplier arguments with the fractional Pohozaev identity. In contrast with the classical wave equation, the observability time deteriorates with the spectral cutoff, reflecting the slow propagation of high-frequency fractional waves. We transfer this estimate to the parabolic problem by transmutation and combine the resulting low-frequency controllability estimates with the high-frequency dissipation of the fractional heat semigroup through a Lebeau-Robbiano iteration. The balance between these two mechanisms yields null controllability precisely in the range $s>1/2$.