Observability Estimate and State Observation Problems for Stochastic Hyperbolic Equations

TL;DR

This paper develops observability inequalities for stochastic hyperbolic equations using Carleman estimates, enabling analysis of state observation problems and establishing unique continuation properties for these equations.

Contribution

It introduces new boundary and internal observability inequalities for stochastic hyperbolic equations with nonsmooth lower order terms, based on global Carleman estimates.

Findings

Derived boundary and internal observability inequalities.

Established a unique continuation property.

Provided tools for state observation in stochastic hyperbolic equations.

Abstract

In this paper, we derive a boundary and an internal observability inequality for stochastic hyperbolic equations with nonsmooth lower order terms. The required inequalities are obtained by global Carleman estimate for stochastic hyperbolic equations. By these inequalities, we study a state observation problem for stochastic hyperbolic equations. As a consequence, we also establish a unique continuation property for stochastic hyperbolic equations.