Observability for Initial Value Problems with Sparse Initial Data

TL;DR

This paper introduces the concept of s-sparse observability for large ODE systems, addressing the unique determination and reconstruction of sparse initial data from limited observations using matrices with the restricted isometry property.

Contribution

It formalizes s-sparse observability for ODEs and provides conditions for unique recovery and reconstruction of sparse initial states from partial observations.

Findings

Conditions for unique sparse initial data recovery

Reconstruction algorithms for sparse initial states

Application of restricted isometry property in ODE observability

Abstract

In this work we introduce the concept of $s$-sparse observability for large systems of ordinary differential equations. Let $\dot x=f(t,x)$ be such a system. At time $T>0$, suppose we make a set of observations $b=Ax(T)$ of the solution of the system with initial data $x(0)=x^0$, where $A$ is a matrix satisfying the restricted isometry property. The aim of this paper is to give answers to the following questions: Given the observations $b$, is $x^0$ uniquely determined knowing that $x^0$ is sufficiently sparse? Is there any way to reconstruct such a sparse initial data $x^0$?