Spectral inequalities for elliptic pseudo-differential operators on closed manifolds

TL;DR

This paper proves a spectral inequality for elliptic pseudo-differential operators on closed manifolds and applies it to establish null-controllability of associated heat equations, extending classical control theory results.

Contribution

It introduces a spectral inequality for positive elliptic pseudo-differential operators on closed manifolds and demonstrates null-controllability of related heat equations, solving an open problem.

Findings

Spectral inequality established for elliptic pseudo-differential operators.

Null-controllability of fractional heat equations on closed manifolds proven.

Introduces a periodization approach inspired by global pseudo-differential calculus.

Abstract

Let $(M,g)$ be a closed Riemannian manifold. The aim of this work is to prove the Lebeau-Robbiano spectral inequality for a positive elliptic pseudo-differential operator $E(x,D)$ on $M,$ of order $\nu>0,$ in the H\"ormander class $\Psi^\nu_{\rho,\delta}(M).$ In control theory this has been an open problem prior to this work. As an application of this fundamental result, we establish the null-controllability of the (fractional) heat equation associated with $E(x,D).$ The sensor $\omega\subset M$ in the observability inequality is an open subset of $M.$ The obtained results (that are, the corresponding spectral inequality for an elliptic operator and the null-controllability for its diffusion model) extend in the setting of closed manifolds, classical results of the control theory, as the spectral inequality due to Lebeau and Robbiano and their result on the null-controllability of the heat equation giving a complete picture of the subject in the setting of closed manifolds. For the proof of the spectral inequality we introduce a periodization approach in time inspired by the global pseudo-differential calculus due to Ruzhansky and Turunen.