Controllability and mixing for acoustic wave motions

TL;DR

This paper investigates the controllability of boundary-driven acoustic waves and demonstrates strong mixing in their stochastic counterparts, establishing an observability inequality as the key link.

Contribution

It proves exact controllability for boundary-driven acoustic waves and shows strong mixing for their stochastic versions, introducing a new observability inequality as the connecting tool.

Findings

Exact controllability of acoustic wave system with boundary control

Strong mixing property for stochastic acoustic wave system with white noise perturbation

Establishment of an observability inequality linking controllability and stochastic mixing

Abstract

This paper concerns the dynamical behaviors of acoustic wave motion driven by a force acting through the boundary. If the boundary force is a suitable control, we show that the dynamical system associated to the acoustic wave motion is exactly controllable. Furthermore, when it is random perturbation of white noise type, we prove that the corresponding stochastic system is strong mixing. The bridge between these two problems is the observability inequality, which will be established in this work.