Heat equation on the Heisenberg group: observability and applications

TL;DR

This paper studies the observability and stability of the heat equation on the Heisenberg group, showing that observability requires a minimal time depending on the observation region’s distance from the boundary, with implications for control theory.

Contribution

It establishes the minimal time needed for observability and Lipschitz stability of the Heisenberg heat equation, combining Fourier analysis and Carleman inequalities, and extends results to unbounded domains.

Findings

Observability fails for any positive time without a minimal time threshold.

Both observability and Lipschitz stability hold after the minimal time depending on the observation region.

The minimal time depends on the distance between the observation region and the boundary.

Abstract

We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain $$\Omega = (-1,1)\times\mathbb{T}\times\mathbb{T}$$ taking as observation regions slices of the form $\omega=(a,b) \times \mathbb{T} \times \mathbb{T}$ or tubes $\omega = (a,b) \times \omega_y \times \mathbb{T}$, with $-1<a<b<1$. We prove that observability fails for an arbitrary time $T>0$ but both observability and Lipschitz stability hold true after a positive minimal time, which depends on the distance between $\omega$ and the boundary of $\Omega$: $$T_{\min} \geqslant \frac{1}{8} \min\{(1+a)^2,(1-b)^2\}.$$ Our proof follows a mixed strategy which combines the approach by Lebeau and Robbiano, which relies on Fourier decomposition, with Carleman inequalities for the heat equations that are solved by the Fourier modes. We extend the analysis to the unbounded domain $(-1,1)\times\mathbb{T}\times\mathbb{R}$.