Quantitative unique continuation for wave operators with a jump discontinuity across an interface and applications to approximate control

TL;DR

This paper establishes quantitative unique continuation results for wave operators with discontinuous coefficients across interfaces, leading to new insights into the controllability cost of wave equations in complex geometries.

Contribution

It introduces a local Carleman estimate for wave operators with discontinuous coefficients and combines it with recent techniques to derive global stability and controllability results.

Findings

Proved a local Carleman estimate for wave operators with jump discontinuities.

Derived a global stability inequality for wave equations with discontinuous coefficients.

Estimated the cost of approximate controllability in complex geometries.

Abstract

In this article we prove quantitative unique continuation results for wave operators of the form $\partial$ 2 t -- div(c(x)$\nabla$$\bullet$) where the scalar coefficient c is discontinuous across an interface of codimension one in a bounded domain or on a compact Riemannian manifold. We do not make any assumptions on the geometry of the interface or on the sign of the jumps of the coefficient c. The key ingredient is a local Carleman estimate for a wave operator with discontinuous coefficients. We then combine this estimate with the recent techniques of Laurent-L{\'e}autaud [LL19] to propagate local unique continuation estimates and obtain a global stability inequality. As a consequence, we deduce the cost of the approximate controllability for waves propagating in this geometry.