Dynamical Sampling for the Recovery of Spatially Constant Source Terms in Dynamical Systems

TL;DR

This paper develops a theoretical framework for recovering stationary sources in dynamical systems using space-time samples, with potential applications in environmental monitoring and sensor placement strategies.

Contribution

It establishes necessary and sufficient conditions for device placement to accurately recover unknown sources in dynamical systems, advancing the field of dynamical sampling.

Findings

Derived conditions for sensor placement to ensure source recovery

Provided a theoretical basis for source identification in dynamical systems

Potential application in environmental pollutant monitoring

Abstract

In this paper, we investigate the problem of source recovery in a dynamical system utilizing space-time samples. This is a specific issue within the broader field of dynamical sampling, which involves collecting samples from solutions to a differential equation across both space and time with the aim of recovering critical data, such as initial values, the sources, the driving operator, or other relevant details. Our focus in this study is the recovery of unknown, stationary sources across both space and time, leveraging space-time samples. This research may have significant applications; for instance, it could provide a model for strategically placing devices to measure the quantity of pollutants emanating from factory smokestacks and dispersing across a specific area. Space-time samples could be collected using measuring devices placed at various spatial locations and activated at different times. We present necessary and sufficient conditions for the positioning of these measuring devices to successfully resolve this dynamical sampling problem. This paper provides both a theoretical foundation for the recovery of sources in dynamical systems and potential practical applications.