Partial Observability and its Consistency for PDEs

TL;DR

This paper introduces a quantitative measure of partial observability for PDEs, proves its consistency under approximation schemes, and demonstrates its application through examples like Burgers' equation and shallow water equations.

Contribution

It defines a new measure of partial observability for PDEs and proves its consistency, along with an empirical approximation method using Gramian matrices.

Findings

The unobservability index can be approximated using empirical Gramian.

The measure is consistent under well-posed approximation schemes.

Applications to Burgers' and shallow water equations illustrate the concept.

Abstract

In this paper, a quantitative measure of partial observability is defined for PDEs. The quantity is proved to be consistent if the PDE is approximated using well-posed approximation schemes. A first order approximation of an unobservability index using an empirical Gramian is introduced. Several examples are presented to illustrate the concept of partial observability, including Burgers' equation and a one-dimensional nonlinear shallow water equation.