Control of waves on Lorentzian manifolds with curvature bounds

TL;DR

This paper establishes boundary controllability for wave equations on Lorentzian manifolds with curvature bounds using a new Carleman estimate, enabling observability, unique continuation, and inverse problem applications.

Contribution

It introduces a novel global Carleman estimate on Lorentzian manifolds supported outside null cones, advancing control and inverse problem techniques in curved spacetime settings.

Findings

Proves boundary controllability for wave equations on Lorentzian manifolds.

Develops a new Carleman estimate supported outside null cones.

Provides bounds for observability constants and unique continuation results.

Abstract

We prove boundary controllability results for wave equations (with lower-order terms) on Lorentzian manifolds with time-dependent geometry satisfying suitable curvature bounds. The main ingredient is a novel global Carleman estimate on Lorentzian manifolds that is supported in the exterior of a null (or characteristic) cone, which leads to both an observability inequality and bounds for the corresponding constant. The Carleman estimate also yields a unique continuation result on the null cone exterior, which has applications toward inverse problems for linear waves on Lorentzian backgrounds.