Spherical Logvinenko-Sereda-Kovrijkine type inequality and null-controllability of the heat equation on the sphere

TL;DR

This paper establishes a new inequality for polynomials on spheres, leading to sharp estimates for controlling the heat equation on spherical domains, with implications for spectral analysis and control theory.

Contribution

It introduces a spherical Logvinenko-Sereda-Kovrijkine inequality and derives explicit control cost estimates for the spherical heat equation.

Findings

Proves a polynomial restriction inequality on spheres.

Derives spectral inequalities for the Laplace-Beltrami operator.

Provides sharp control cost estimates for the spherical heat equation.

Abstract

It is shown that the restriction of a polynomial to a sphere satisfies a Logvinenko-Sereda-Kovrijkine type inequality (a specific type of uncertainty relation). This implies a spectral inequality for the Laplace-Beltrami operator, which, in turn, yields observability and null-controllability with explicit estimates on the control costs for the spherical heat equation that are sharp in the large and in the small time regime.