A Carleman estimate and an energy method for a first-order symmetric hyperbolic system

TL;DR

This paper develops a Carleman estimate for first-order symmetric hyperbolic systems and uses it to establish an observability estimate linking initial data to boundary measurements.

Contribution

It introduces a new Carleman estimate for symmetric hyperbolic systems and applies it to derive boundary observability results.

Findings

Established a Carleman estimate under positivity conditions.

Proved an $L^2$-observability estimate for initial data.

Linked boundary measurements to initial conditions.

Abstract

For a symmetric hyperbolic system of the first order, we prove a Carleman estimate under some positivity condition concerning the coefficient matrices. Next, applying the Carleman estimate, we prove an observability $L^2$-estimate for initial values by boundary data.